Effect of Finite Extensibility on Nonlinear Extensional Rheology of

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Effect of Finite Extensibility on Nonlinear Extensional Rheology of Polymer Melts Samantha L. Morelly,† Luisa Palmese,† Hiroshi Watanabe,‡ and Nicolas J. Alvarez*,† †

Chemical and Biological Engineering, Drexel University, 3141 Chestnut St., Philadelphia, Pennsylvania 19104, United States Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, Japan 611-0011

Macromolecules Downloaded from pubs.acs.org by UNIV OF LOUISIANA AT LAFAYETTE on 01/17/19. For personal use only.

‡

ABSTRACT: We recently showed that two systems have the same nonlinear flow dynamics if they have (1) the same number of entanglements, Z, (2) the same number of Kuhn segments between entanglements, Ne, and (3) a single monomeric friction reduction between all species. Because differences in polymer chemistry result in different values of (1)−(3), these results show that the likelihood of two melts having the same nonlinear behavior is improbable. This work determines the dependence of the nonlinear extensional flow behavior on Ne for three polymer melts: polystyrene, poly(methyl methacrylate), and poly(tert-butylstyrene). We show that polymer melts with the same Z have almost identical scaled linear viscoelasticity. Extensional rheology at constant strain rate shows strain hardening depends strongly on the value of Ne. More specifically, our data suggest a power-law dependence of steady state stress on the Weissenberg number which increases with increasing Ne.

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INTRODUCTION Predicting the nonlinear extensional flow dynamics of polymer melts and solutions is considered the ultimate test of both molecular and empirical constitutive models.1−9 Recent data have confirmed the universal scaling laws, proposed by the tube model, by demonstrating identical scaled extensional transients and viscosity for polymer melts and concentrated solutions with two distinct chemical repeat units. Universal nonlinear extensional behavior is observed when the polymer chains have the same (i) Z number of entanglements per chain and (ii) Ne Kuhn segments between entanglements (finite extensibility) and (iii) when friction reduction between a solvent/polymer matches the friction reduction between polymer/polymer molecules.10 However, several questions still remain: (i) Do different polymers have the same scaled linear viscoelasticity for the same Z? and (ii) How does steady state extensional stress compare between polymers with same Z and varying Ne? Little steady state nonlinear extensional data exist on amorphous linear polymer melts apart from data on polystyrene (PS).11−17 One exception is the data of Sridhar and co-workers on polyisoprene (PI) and poly(n-butyl) acrylate (PnBA) melts.18 This work compares nonlinear extensional rheology of melts and solutions of PI, PnBA, and PS. Their results are investigated in terms of differences in Z and λmax,18 where λmax is the finite extensibility and ∼Ne1/2. The authors suggest there are three regimes that capture the trends in extensional rheology of polymer melts and solutions. Namely at Wi < 1, normalized steady state stress scales with Wi to the power-law exponent of 1, i.e., the linear viscoelastic regime. Regime 2, 1 < Wi < 100, is depicted by a 0.5 power-law exponent scaling of normalized steady state stress with Wi. A third regime, Wi > 100, shows steady state stress scaling with a power-law exponent n that depends on the value of λmax/Z. Figure 1 shows literature data for normalized steady state stress © XXXX American Chemical Society

Figure 1. Normalized steady state stress as a function of Wi for PS, PI, and PnBA melts from literature data: (a) Sridhar et al.,18 (b) Bach et al.,11 (c) Luap et al.,13 (d) Huang et al.,16 and (e) Bhattacharjee et al.14 The solid line represents a power-law exponent of 0.5, which has been shown to be relevant to PS data.12

as a function of the Rouse Weissenberg number (WiR) for melts of PS, PI, and PnBA as well as a concentrated PS solution. The line in Figure 1 represents a power-law exponent of 0.5. A recent paper by O’Connor et al. shows, via molecular dynamics, the role of Z and Ne on the extensional flow of polymer melts. They determine that in the Newtonian limit there is little to no chain alignment in extensional flow and that chain alignment increases with the Rouse Weissenberg number, WiR. Chain statistics show the degree of chain stretch increases with WiR outside the Newtonian regime, which Received: October 30, 2018 Revised: December 23, 2018

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DOI: 10.1021/acs.macromol.8b02319 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 1. Best Fit BSW Parameters and Estimated Tube Model Parameters sample

Mw [kg/mol]

G0N [kPa]

τc [s]

τm [s]

ρ [kg/L]

Me [kg/mol]

Z

T [°C]

τR [s]

PS-125k PtBS-301k PMMA-86k PS-200k PMMA-100k PS-290k

125 301 86 200 100 290

250 120 720 250 850 250

0.44 3.00 0.17 0.44 0.03 0.44

370 3700 700 1700 320 6800

1.04 0.957 1.18 1.04 1.18 1.04

13.3 38 5.8 13.3 4.9 13.3

9.4 10.1 14.9 14.3 20.4 20.8

130 175 150 130 160 130

38.9 306 37.7 90.0 12.5 190

vacuum die heated from below. The sample temperature was monitored via an in situ thermocouple. The height of the sample was varied between 1 and 2 mm, depending on measurement requirements. Mechanical Spectroscopy. The linear viscoelastic properties of the polymer melts were obtained from small amplitude oscillatory shear, SAOS, and flow measurements. An 8 or 25 mm parallel−plate geometry was used on a DHR-3 rheometer (TA Instruments) with electrically heated plates (EHP). The measurements for PtBS were performed at 175, 180, 190, and 200 °C in air. The measurements for PMMA were performed at 140, 150, 170, and 190 °C under nitrogen. Amplitude sweeps were measured from 0.1% to 10% strain at constant frequency values of 0.1, 1, 10, and 100 rad/s. Frequency sweep measurements were then taken at 1% strain from 0.1 to 100 rad/s. All data for PtBS and PMMA were shifted to 175 and 150 °C using the time−temperature superposition procedure, respectively. Extensional Stress Measurements. The extensional stress under constant extension rate was measured by a VADER 1000 (Rheo Filament ApS, Albertslund, DK). Prior to making a measurement, all samples were molded into cylindrical test specimens with a fixed radius of 3 mm for PtBS and 2.7 mm for PMMA via custom-made vacuum dies. The mold was connected to a vacuum pump to ensure no air bubbles were trapped in the samples. A load of 500 g was applied to the top of the die to ensure compaction of the powder. The height of the final cylindrical sample was controlled by the mass in the die chamber. The PtBS melts were pressed at 200 or 220 °C and annealed for more than 15 min. The PMMA melts were pressed at 190 °C and annealed for more than 15 min. The time was chosen to ensure that all stress applied to the sample had ample time to dissipate via relaxation. All samples were prestretched to a radius Rp at 190 °C prior to the application of extensional flow. The prestretch radius depended on the initial sample size; PtBS had an initial sample radius of 3 mm and was prestretched to 2.5 mm, and PMMA had an initial sample radius of 2.7 mm and was prestreth to 1.5 mm. The prestretching was performed to ensure that the force applied was below the threshold of detachment of the sample from the stainless steel plates. Prior to performing the measurement, the temperature of the VADER 1000 was reduced to 175 °C for PtBS and 150 °C for PMMA 86k and 160 °C for PMMA 100k. The samples were given sufficient time to equilibriate, which was denoted by the return of a zero reading on the force transducer. During the extensional measurements, the force F(t) was measured by a load cell, and the diameter 2R(t) at the mid-filament plane was measured by a laser micrometer. The Hencky strain and the mean value of the stress difference over the mid-filament plane are calculated from

indicates the existence of confinement. The relative amount of chain stretching is proportional to the value of Ne. While not discussed explicitly, the authors’ data suggest that Ne impacts the power-law exponent relation between the normalized steady state stress and WiR at midrange Weissenberg numbers. This Ne-dependent power-law exponent relation directly contradicts the assertion that for 1 < Wi < 100 the normalized steady state stress has a power-law exponent relation of 0.5 independent of polymer chemistry.19 This work investigates the effect of linear and nonlinear tube model parameters on the linear viscoelasticity and nonlinear rheology of three polymer melts. Poly(methyl methacrylate) (PMMA) and poly(tert-butylstyrene) (PtBS) melts will be compared to PS melts from the literature with the same number of entanglements but different mesoscale flexibility, i.e., number of Kuhn segments between entanglements. We show that polymers with the same number of entanglements have identical scaled linear viscoelastic signatures. On the contrary, in nonlinear dynamics, chains with higher values of Ne show higher degrees of strain hardening before achieving steady state. A steady state stress power-law exponent dependence is observed with increasing Weissenberg number, whose exponent depends on the finite extensibility of the chain. We conclude that the extension-rate thinning behavior of polymer melts is straightforward: this behavior strongly depends on its finite extensibility.

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MATERIALS AND METHODS

PtBS polymer was synthesized using anionic polymerization under vacuum with benzene and sec-butyllithium utilized as the solvent and initiator, respectively. The molecular characteristics were determined from size exclusion chromatography (SEC) combined with low-angle LS, as described previously by Chen et al. and Matsumiya et al.20,21 PtBS (MW = 301000 g/mol, Mw/Mn = 1.02, ρ = 0.957 g/cm3) samples were pressed into cylindrical samples for the measurement of linear and extensional rheology. Samples for linear and extensional rheology were pressed using a custom 6 mm diameter vacuum press die to a height of 2 mm at 200 and 220 °C under vacuum for 12 h under a load of 500 g. The samples were then loaded onto the respective rheometer at room temperature and heated to the experimental temperature. PMMAs (MW = 86000 g/mol (Part No. PSS-mm85k) and 99400 g/mol (Part No. PSS-mm100k)) were purchased (Polymer Standards Service GmbH, Germany) and used as received. PDI was confirmed using an Agilent GPC/SEC via refractive index detector by Polymer Standard Service as 86000/84200 = 1.02 and 99400/92100 = 1.08, respectively. Because of the polymerization technique used by the manufacturer, the tacticity of the PMMAs is different, which is evident when their linear viscoelastic envelopes are compared. The impact of this is discussed more in the Tube Model Parameter Estimation section. The PMMA was dried under vacuum at 50 °C for 1 week to ensure that no adsorbed water would come out during sample preparation. While this is a relatively low temperature given the Tg of PMMA, if the sample was not dry, bubbles would appear during the melt pressing; samples with bubbles were discarded. The samples were melt pressed at 180 °C in either an 8 or 24 mm diameter

ϵ(t ) = − 2 ln(R(t )/R 0)

(1)

and ⟨σzz − σrr ⟩ =

Fe(t ) πR(t )2

(2)

where Fe is the force contribution from uniaxial extensional flow.22,23 The strain rate is defined as ϵ̇ = dϵ/dt. The diameter of the midfilament plane was required to decrease exponentially during stretching. The extensional stress growth function is defined as η̅+ = ⟨σzz − σrr⟩/ϵ̇. B

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Macromolecules Table 2. Nonlinear Tube Model Parameters polymer

⟨R2⟩0/M [Å2]32

mb [g/mol]

b [Å]

N0

M0 [g/mol]

Ne

λmax

a [Å]

PMMA-86k PMMA-100k PS PtBS

0.390 0.390 0.437 0.361

50 50 52 74

15.9 15.9 18.5 21.7

7 7 8 9

700 700 833 1330

8 7 16 29

2.9 2.6 4 5.3

45 42 74 117

Figure 2. (a) Linear viscoelastic response of 125k PS and 301k PtBS melts at 130 and 175 °C, respectively. (b) Normalized linear viscoelastic response of 125k PS and 301k PtBS using parameters in Table 1.

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values of Z and τc allow for an estimation of the Rouse time, τR, via τR = Z2τc. τR will be used to normalize the extension rates into a Rouse-based Weissenberg number

TUBE MODEL PARAMETER ESTIMATION We analyze the LVE data by fitting a three parameter Baumgaertel−Schausberger−Winter (BSW) relaxation spectrum24−26 to the SAOS data of PtBS and PMMA. Note that this was similar to the procedure used for PS that is welldocumented in Huang et al.16 Typically, this is done using five parameters, but in this case the values of ne = 0.23 and ng = 0.7 were fixed for monodisperse polymer melts.16,27,28 The BSW parameter G0N is exactly the plateau modulus, τc is practically equal to the Rouse time of one entanglement segment, and τm is equal to the disengagement time within a factor of order unity. The best fit values of G0N, τc, and τm are shown in Table 1. Typically, the value of G0N is not expected to change as a function of Mw for a given polymer. Notice in Table 1 that the two PMMA samples have significantly different values. This caused confusion until it was discovered that the plateau modulus of PMMA is strongly dependent on its degree of tacticity.29,30 Unfortunately, the controlled polymerization methods used to synthesize our samples do not control tacticity, and the resulting polymers are an unspecified mixture of isotactic and syndiotactic isomers. Thus, each synthesized sample is expected to have varying values of G0N. The values of G0N are used to calculate the entanglement Mw, Me, via Me = ρRT /GN0

WiR = ϵ̇τR

Additional tube model parameters that are useful for our discussion are the number of Kuhn segments between entanglements, Ne, and the finite extensibility of the chain, λmax. These two parameters are intrinsically linked, Ne ∼ λmax2, and can be estimated from Ne =

2 Me iay = jjj zzz M0 k b {

(6)

where M0 is the molecular weight of a Kuhn segment, a is the tube diameter, and b is the Kuhn length. Following the procedure of Huang et al.,15 b was calculated using the following equation and values from Fetters and coworkers.32 b=

⟨R2⟩0 mb M l cos(θ /2)

(7)

where ⟨R2⟩0/M is the melt chain dimensions from SANS, mb is the average molecular weight per backbone bond, l is the backbone bond length, and θ is the backbone bond angle. For the polymers tested here l and cos(θ/2) are constant at values of 1.5 Å and 0.82, assuming a tetrahedral bond angle of 109.5°, respectively. The number of monomers in a Kuhn segment, N0, is calculated using N0 = b/(2l cos(θ/2)), which leads to the calculation of M0 given by

(3)

where ρ is the density of the polymer, R is the gas constant, and T is the measurement temperature. Me was calculated using the relationship suggested by Ferry.31 The calculated value of Me is tabulated in Table 1 and allows one to define the average number of entanglements per chain. Z = M w / Me

(5)

M 0 = N0 × M w0

(4)

(8)

M0w

where is the molecular weight of a polymer repeat unit. The nonlinear molecular parameters are tabulated in Table 2. Note that from PMMA to PS to PtBS there is an increase in

Notice that Table 1 is organized in terms of increasing Z and shows two polymers with similar Z values for comparison. The C

DOI: 10.1021/acs.macromol.8b02319 Macromolecules XXXX, XXX, XXX−XXX

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Figure 3. (a) Linear viscoelastic envelope of 200k PS and 86k PMMA melts at 130 and 150 °C, respectively. (b) Normalized linear viscoelastic response of 200k PS and 86k PMMA using G0N in Table 1.

Figure 4. (a) Linear viscoelastic envelope of 290k PS and 100k PMMA melts at 130 and 160 °C, respectively. (b) Normalized linear viscoelastic response of 290k PS and 100k PMMA using G0N in Table 1.

the value of Ne and λmax. We understand that the values of M0 for PS and PtBS are smaller than those reported in the literature; however, we have calculated the values for all polymers to ensure consistency in the magnitudes.

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Figure 4a, which shows LVE of PMMA compared to PS with Z ≈ 20. In both cases, PMMA has a lower value of G0N and τc than PS, which is evident from its lower modulus and slight shift to the right in frequency of the LVE for the reported temperature, respectively. Furthermore, in all cases, the shape of the LVE is nearly identical for the PMMA and PS samples. This is more evident in Figures 3b and 4b, where normalized modulus versus normalized frequency plots for the two corresponding PMMA and PS samples show indistinguishable curves. These results show that chemical repeat unit has little effect on the shape of the LVE. Simply stated, the scaled LVE is only dependent on the number of entanglements. The unscaled LVE is dependent on the value of G0N and τc. This has very important implications for the prediction of LVE for a given polymer Mw. A polymer’s LVE can be accurately predicted from a knowledge of three parameters, G0N, τc, and Me, which are independent of Mw. Strictly speaking, the constraint release time per entanglement (τCR,m) is also necessary to fully specify the LVE behavior of a given polymer species.33 Extensional Rheology. Figure 5 shows the stress growth coefficient, η̅+, for PtBS for varying Rouse Weissenberg (WiR) numbers measured at 175 °C. The solid line represents the LVE prediction for PtBS in extension. Nonlinear strain hardening is represented by deviations from the LVE

RESULTS AND DISCUSSION

Linear Viscoelasticity. Figure 2a shows the measured linear viscoelasticity (LVE) for PtBS-301k compared to PS125k, both having Z ≈ 10. It is evident that the two LVEs have similar shape. PtBS has a lower plateau modulus, G0N, due to its larger tube diameter, a = 11.7 nm, compared to PS, a = 7.4 nm. For the temperatures reported, PS-125k has a high frequency crossover that is slightly shifted to the right compared to PtBS301k. This is evident in the larger value of τc for PtBS reported in Table 1. Figure 2b shows the scaled modulus, G′/G0N and G″/G0N, as a function of scaled frequency, ωτc, for the PtBS301k and PS-125k. The two curves nearly superimpose, signifying a master curve for Z = 10 that is independent of the polymer repeat unit. The same trends can be observed for higher values of Z and different chemical repeat units. Figure 3a shows the LVE of PMMA-86k compared to PS-200k, both having Z ≈ 15. In this case, G0N is higher for PMMA due to a smaller a = 4.5 nm. For PMMA-100k and PS-290k, the same trend is observed in D

DOI: 10.1021/acs.macromol.8b02319 Macromolecules XXXX, XXX, XXX−XXX

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A similar story unfolds with Z ≈ 20. Figure 7a shows the extensional growth coefficient for PMMA 100k for varying Weissenburg numbers at 160 °C. Figure 7b shows the normalized extensional stress growth coefficient as a function of normalized time for PMMA-100k and PS-290k,15 which have identical scaled LVE. It is evident that as in the case above PS-290k experiences higher normalized extensional growth coefficients than PMMA-100k when compared at similar WiR numbers. Figure 8a shows normalized steady state stress as a function of WiR for the polymers examined here. PS follows a power-law trend independent of Z with an exponent of 0.6; it is evident that PMMA and PtBS have unique power-law exponents from PS. Lines are drawn with an approximate power-law exponent through each data set to illustrate the differences. Because the molecular weight and thus Z do not impact the trend seen here, the current tube model theory would suggest that the extensibility (Ne) is responsible for nonlinear differences. One interesting observation is that at Wi = 1 all three polymers have the same normalized stress equal to 3G0N. Note that the data of PtBS are shifted, as indicated in the caption. This suggests that at the onset of the stretching regime, i.e., Wi > 1, steady state stresses begin to exceed 3 times G0N. Incidentally, this is about 2 orders of magnitude below the normalized fracture stress that has been measured for PS.34 These observations suggests that (i) the steady stress at Wi = 1 is predictable irrespective of polymer chemistry and (ii) the critical Wi where fracture occurs is when σss > 100G0N, which is strongly dependent on the power-law exponent. Figure 8b shows the extensional viscosity normalized by 3η0 for all three polymers. While the normalized stress data shows Z independent behavior for each chemistry, this does not hold for normalized viscosity. The same chemistry shows deviations at WiR ≈ 0.5, where higher Mw shows more extension-rate extensional viscosity thinning. This is in direct agreement with MD simulations, which show different power-law exponents with increasing Z.19 This implies that finite extensibility could be used to predict the steady state viscosity of a given polymer chemistry. Furthermore, these results are in agreement with a recent study where we examined, via MD simulations, the effect of λmax on the nonlinear behavior of polymer melts; we showed that decreasing the flexibility of the polymer backbone,

Figure 5. Stress growth coefficient for 301k PtBS (Z = 10) for different extension rates at 175 °C [0.022 (6.73), 0.075 (23.0), and 0.15 (45.9) s−1] and at 200 °C [0.0022 (0.0153), and 0.15 (1.14) s−1] shifted to 175 °C using TTS. WiR for each strain rate is noted in parentheses after the rate. Inset graph shows stress growth coefficient as a function of Hencky strain for three strain rates.

prediction. PtBS experiences strain hardening at all WiR measured. Unfortunately, extensional results for PS-125k are not available, and the authors did not have access to material for extensional measurements. Steady state values are determined by examining the data as a function of Hencky strain instead of time. The steady state plateau is much more evident in strain than in log time. An example of this is shown as an inset of Figure 5 for three strain rates. Steady state is defined when the stress is relatively constant over at least 1 unit of strain. Figure 6a shows the extensional growth coefficient for PMMA 86k for varying WiR measured at 150 °C. A qualitative comparison with Figure 5 shows that PMMA has less apparent strain hardening than PtBS. Figure 6b shows the normalized extensional stress growth coefficient, η̅+/(G0Nτc), as a function of normalized time, t/τc, for PMMA-86k and PS-200k, which have identical scaled LVE. The PS-200k data are taken from Huang and co-workers. 17 PS-200k achieves a higher normalized extensional growth coefficient than PMMA at similar WiR numbers.

Figure 6. (a) Stress growth coefficient for 88k PMMA (Z = 15) for different extension rates at 150 °C: 0.01 (0.377), 0.03 (1.13), 0.07 (2.64), and 0.2 (7.54) s−1. (b) Normalized stress growth coefficient for 88k PMMA and 200k PS17 using parameters in Table 1. WiR for each strain rate is noted in parentheses after the rate. E

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Figure 7. (a) Stress growth coefficient for 100k PMMA (Z = 20) for different extension rates at 160 °C: 0.01 (0.125), 0.03 (0.375), 0.06 (0.75), 0.2 (2.5), and 0.6 (7.5) s−1. (b) Normalized stress growth coefficient for 100k PMMA and 290k PS15 using parameters from Table 1. WiR for each strain rate is noted in parentheses after the rate.

Figure 8. (a) Normalized steady state stress as a function of WiR for PtBS 301k, PS 200k and 290k, and PMMA 86k and 100k with power-law lines for reference. The number next to the polymer label is the power-law exponent. (b) Normalized steady state extensional viscosity as a function of WiR for PtBS 301k, PS 200k and 290k, and PMMA 86k and 100k. Note that PtBS normalized stress is scaled by a factor of 2 for clarity.

i.e., increasing the stretchability of the chain, increases the transient strain-hardening and normalized steady state stress.19 Figure 9 shows the power-law exponent as a function of λmax for PMMA, PS, and PtBS. There appears an increasing trend of power-law exponent with λmax. The open symbols in Figure 9 show the observed power-law exponent from MD simulations (see steady state stress data in the inset). While both experiments and simulations show an upward trend of power-law exponent with λmax, it is evident that the trends are not identical. This could be due to the fact that the MD simulations represent two polymers with identical monomeric friction but different values of Ne, whereas the experiments are based on different polymers. We have discussed in other reported works that the monomeric friction reduction is an important parameter in controlling the nonlinear response of a material.10,15 In solutions composed of molecular solvents, there is a hypothesized difference between the monomeric friction reduction felt by polymer−polymer interactions versus polymer−solvent interactions. This disparity in monomeric friction reduction leads to highly different strain hardening responses. In the case of this work, the systems are all melts and only experience monomeric friction reduction via polymer−

Figure 9. Maximum stretch, λmax, as a function of the normalized steady state stress power-law exponent. Filled squares represent PS, PMMA, and PtBS from Figure 8, and open squares represent the literature modeling data from Figure 1b from O’Connor et al. The inset shows the MD simulation data from O’Connor for two polymers with Z = 9 and varying λmax.

polymer interactions. Thus, we expect and observe simple power-law exponent responses. We do not argue that the F

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monomeric friction reduction is the same for these particular polymers. In fact, monomeric friction reduction via polymer− polymer interactions have also been proposed to be chemistry dependent. Matsumiya et al. studied nearly unentangled PtBS and PS under uniaxial flow conditions to investigate the impact of monomeric friction differences from chemistry in the absence of entanglements. They demonstrate more strain hardening for PtBS than PS, which the authors argue is due to a difference in monomeric friction reduction between the two chemical repeat units.35 The different exponents (Figure 8a) could partly reflect the magnitude of monomeric friction reduction, which is expected to depend on λmax and, more importantly, on chemistry.15,16,36 We now test the idea that the differences between polymers can be scaled via λmax. For fully stretched molecules, we expect σ̅ = σ/G0N ∝ λmax2, as discussed previously in Huang et al.16 For PMMA86k 2 ) is PS 200k and PMMA 86k, the ratio of (λPS200k max /λmax PS200k PMMA86k ≈2. At a WiR ≈ 1, the ratio of σ̅ / σ̅ is ≈0.88, and for WiR ≈ 3 the ratio of stresses is ∼1.5. For PS 290k and PMMA100k 2 PMMA 100k, the ratio of (λPS290k ) is ≈2.3. At a max /λmax WiR ≈ 1, the ratio of σ̅ PS290k/ σ̅ PMMA100k is ≈1.22, and for WiR ≈ 3, σ̅ PS290k/ σ̅ PMMA100k is ≈1.75. It is evident that the ratio of σ̅ is increasing with increasing Wi for all three Polymer melts, and therefore there appears to be no consistent scaling with λmax. These results are consistent with concentrated polymer solutions where no scaling of stress with λmax2 was observed.15 Our results are also consistent with the results from MD simulations where the ratio of λmax2 for M2 to M1 ≈ 1.8. However, even deep into the fully stretched chain regime (at WiR ≈ 50), the ratio of σ̅ M2/ σ̅ M1 is only 1.05.19 We must conclude that the scaling of stress is in fact more complicated than a scaling with λmax2 and most likely follows the scaling arguments put forth in O’Connor et al.19

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Ole Hassager and Qian Huang for helpful discussions.

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CONCLUSIONS We demonstrate that the normalized linear viscoelasticity of monodisperse linear polymer melts is solely dependent on the number of entanglements, Z. While this has been shown for polymer melts and polymer solutions, we show clearly that a linear polymer’s unscaled LVE can be readily predicted via the three parameters G0N, τc, and Z regardless of whether it is in the melt or solution state. Our results also show that polymer melts with the same Z do not have the same nonlinear extensional rheology. In fact, our data suggest that the strain hardening of a given polymer is dictated by the value of λmax, i.e., the finite extensibility of the chain. In other words, the more flexible the polymer chain (larger number of Kuhn segments between entanglements), the more normalized stress is required to achieve steady state and the weaker the extension rate thinning behavior. Furthermore, our results are in excellent agreement with MD simulations of polymer melts in uniaxial extensional flow. This unique data set should help in testing, understanding, and refining polymer constitutive models.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; Ph +1 215 571 4120. ORCID

Hiroshi Watanabe: 0000-0003-0826-2454 Nicolas J. Alvarez: 0000-0002-0976-6542 G

DOI: 10.1021/acs.macromol.8b02319 Macromolecules XXXX, XXX, XXX−XXX

Article

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DOI: 10.1021/acs.macromol.8b02319 Macromolecules XXXX, XXX, XXX−XXX