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Effect of Molecular Weight, Polydispersity, and Monomer of Linear Homopolymer Melts on the Intrinsic Mechanical Nonlinearity 3Q0(ω) in MAOS Miriam A. Cziep, Mahdi Abbasi, Matthias Heck, Lukas Arens, and Manfred Wilhelm* Department of Chemistry, Karlsruhe Institute of Technology, Karlsruhe, Germany S Supporting Information *

ABSTRACT: Linear mono- and polydisperse homopolymer melts have been investigated with Fourier transformation rheology (FT rheology) to quantify their nonlinear behavior under oscillatory shear via mechanical higher harmonics, i.e., I3/1(ω,γ0). Master curves of the zero-strain nonlinearity, 3 Q0(ω) ≡ limγ0→0 I3/1/γ02, have been created for these linear homopolymer melts, applying the time−temperature superposition (TTS) principle. The quantity 3Q0(ω) is examined for its dependence on molecular weight, molecular weight distribution, and monomer. The investigated nonlinear master curves of 3Q0(ω) for polymer melts with a polydispersity index (PDI) of about 1.07 or smaller display the expected scaling exponent of 3Q0(ω) ∝ ω2 at low frequencies until a maximum, 3 Q0,max, is reached. This maximum 3Q0,max was found to be in the magnitude of the longest relaxation time τ0 and the value of 3 Q0,max to be weakly dependent on molecular weight, or the number of entanglements Z, with 3Q0,max ∝ Z0.35. Within the measured experimental window, the initial slope at low frequencies of nonlinear master curves is very sensitive toward the molecular weight distribution, as quantified through the PDI. The slope of 3Q0(ω) decreases until approximately zero as a limiting plateau value is reached at a polydispersity of around PDI ≈ 2. The experimental findings are also compared to 3Q0(ω) predictions from pom-pom and molecular stress function (MSF) constitutive models. Analytical solutions of 3Q(ω,γ0) for diminishing small strain amplitudes (γ0 → 0) are presented for each model, and an asymptotic solution for 3Q0(ω) is derived for low and high frequencies for monodisperse samples. This simplified equation is a function of Deborah number (De = ωτ0) in the general form of 3Q0(De) = aDe2/(1 + bDe2+k). This equation was fitted to our experimental data of monodisperse homopolymer melts. It is shown that under these conditions the parameters a and b are only functions of the number of entanglements Z and are independent of the investigated monomers. Consequently, a and b can be linked to general polymer properties. With this article, the mechanical nonlinear response of linear polymer melts with regard to molecular weight and distribution as well as monomer type is quantified. The here presented results should be of great interest toward constitutive model development and toward computational polymer physics, e.g., molecular dynamic simulations, as our results seem to quantify general features in nonlinear rheology of linear homopolymer melts.

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of periodic stress σ(t;ω,γ0) at steady state against strain γ(t) or strain rate γ̇(t) are called Lissajous−Bowditch figures.9 They offer the possibility to easily identify nonlinear responses of materials to a large deformation and provide one basis for several other analyzing techniques of LAOS data. In order to give the storage and loss modulus a physical meaning in the nonlinear regime, Cho et al.10 proposed a geometrical interpretation of LAOS data by decomposing the stress into an elastic and a viscous compound. Following this decomposition approach, Ewoldt et al.5,11 expressed the nonlinear stress response by Chebyshev polynoms, where the coefficients were used to quantify nonlinear behavior via Lissajous− Bowditch plots and Pipkin diagrams.12 Rogers et al.13,14 used

INTRODUCTION There is a widespread interest in the linear and nonlinear mechanical behavior of complex fluids.1−5 One established method to achieve mechanical nonlinearity for viscoelastic materials is large amplitude oscillatory shear (LAOS). Oscillatory shear tests lead to a broad spectrum of mechanical response by varying excitation frequency (ω1/2π), strain amplitude (γ0), and temperature independently.6 There are several methods to analyze the results from LAOS tests.7 Hyun et al.8 measured G′(γ0) and G″(γ0) as a function of strain amplitude. They identified four different types of LAOS behavior, depending on the interactions between the microstructures of complex fluids: strain thinning (type I, G′ and G″ decreasing), strain hardening (type II, G′ and G″ increasing), weak strain overshoot (type III, G′ decreasing, G″ increasing followed by decreasing), and strong strain overshoot (type IV, G′ and G″ increasing followed by decreasing). Parametric plots © XXXX American Chemical Society

Received: December 15, 2015 Revised: April 11, 2016

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DOI: 10.1021/acs.macromol.5b02706 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

the pom-pom constitutive model for the prediction of nonlinear behavior of industrial polymer melts. Wagner et al. confirmed the scaling I3/1 ∝ γ02 with the use of the molecular stress function (MSF) model.27,34 They could also show analytically a linear dependence of 3Q0(ω) toward the difference between the orientational effect (parameter α) calculated from the Doi−Edwards (DE) model and the stretching effect (parameter β) obtained from the MSF model via I3/1 ∝ (α − β)γ02. The MSF model of Wagner et al.27,34 combines orientation with isotropic chain stretch. Considering the independent alignment assumption (i.e., DEIAA), a fixed value of 5/21 was used for α, and β was fitted to 0.12, while the prediction was β = 1/10 (linear MSF model) or β = 1/5 (quadratic MSF model), which did not fit. In contrast to this, Abbasi et al.28 used the MSF model with an evolution equation, which presented a scaling of I3/1 ∝ (α − β/10)γ02, where β was numerically redefined and related to the branch topology (β = 1 for linear polymer and β > 1 for branched structures). Using this definition, they were able to predict the LAOS behavior as well as start-up shear and extensional deformations of different autoclave and tubular LDPE samples, using a single set of nonlinear parameters. The behavior of 3Q0(ω) in experiments and predictions for the simplest model systems, i.e., linear homopolymer melts, as a function of monomer, molecular weight, and polydispersity has not been studied systematically yet. With this publication, we evaluate the effect of these parameters toward the intrinsic nonlinearity 3Q0(ω) on specifically synthesized model systems. Linear homopolymer melts were utilized, and the dependence of 3Q0(ω) on the weight-average molecular weight Mw, monomer, and polydispersity (PDI = Mw/Mn) in the MAOS regime was systematically investigated. We examined a broad set of melts of different linear homopolymers to identify possible differences in the intrinsic nonlinearity due to monomer types: polystyrene, cis-1,4-polyisoprene, poly(pmethylstyrene), poly(methyl methacrylate), poly(2-vinylpyridine), poly(ethylene oxide), and polyethylene. Most linear homopolymer systems were specifically synthesized, characterized by SEC and 1H NMR, and examined for their linear and nonlinear mechanical behavior in the melt. Furthermore, the experimental results are compared with predictions of the single mode pom-pom and MSF constitutive models. Asymptotic solutions of the single mode version of the pom-pom and MSF models for 3Q0(ω) for low and high frequencies were derived, and with respect to these simplified equations, a semianalytical equation is presented for 3Q0(ω), where its parameters are related to molecular weight and molecular weight distribution.

a different approach of analyzing Lissajous−Bowditch plots with the sequence of the physical processes (SPP) method. It should be mentioned that the above methods use Fourier transformation (FT) of raw stress time data to reduce noise and obtain maximum sensitivity.5 Fourier transformation in rheology offers the possibility to transform the raw stress time data from oscillatory shear into a frequency spectrum, where the higher harmonics of the excitation frequency can be identified and quantified15 (see also Figure 2). Higher harmonic intensities have been used to analyze complex fluids11,16−19 under large and medium amplitude oscillatory shear (LAOS and MAOS). The intensity of the higher harmonics In changes as a function of strain amplitude γ0 and excitation frequency ω1/2π, where a nonlinear mechanical behavior results in an increase of the odd harmonics of the shear stress at 3ω1/2π, 5ω1/2π, ... (see also eq 6). The third harmonic intensity I3 was chosen as a measure for nonlinearity in several publications, for example by Neidhöfer et al.,20 Fleury et al.,16 Schlatter et al.,21 Vittorias et al.,22 and Hyun et al.,23 who investigated the relative ratio I3/1(γ0,ω)  I3/I1 within a magnitude spectra of the stress versus frequency ω/2π with regard to branched structures and long chain branching (LCB). According to general theories,24,25 the third to first intensity ratio I3/1 is proportional to γ02 at low strain amplitudes. Dynamic strain sweep experiments and also simulations support this assumption, and a scaling exponent of 2, i.e., I3/1 ∝ γ02, can be found for I3/1 at medium strain amplitudes γ0. This transition region between linear behavior (small amplitude oscillatory shear, SAOS), where no higher harmonic contributions are experimentally accessible, and strong nonlinear behavior (LAOS), where I3/1 does not scale as γ02 anymore, is called the MAOS region (Figure 2, scheme 3). This region typically starts at I3/1 ≈ 10−4−10−3 for polymer melts due to instrument limitations to detect smaller nonlinearities.23 The intrinsic nonlinearity 3Q0(ω), as defined by Hyun et al.7,23 (see Theory section), is a function of the excitation frequency (ω1/2π). It allows to quantify the influence of relaxation processes, as for instance reptation, stretching, contour length fluctuation (CLF), and constraint release (CR) mechanisms.26 It is worth mentioning that linear viscoelastic material functions, i.e., G′(ω) and G″(ω), describe the relaxation processes in the linear regime, while 3Q0(ω) also includes the intrinsic nonlinear relaxation processes related to stretch and orientation.27,28 These are strongly affected by molecular weight distribution and topology, even in homopolymers. With the time−temperature superposition (TTS) principle,29 it was possible to generate “nonlinear master curves” for 3Q0(ω), using the shift parameters from respective linear master curves. In this earlier publication, it was shown that these nonlinear master curves are sensitive to molecular topology, e.g. long chain branching (LCB) of polystyrene combs.23,30 Depending on the molecular weight of the backbone (bb) and the molecular weight of side chains, as well as the number of branches (br), the values of characteristic local maxima (Qmax,bb, Qmax,br) and an emerging local minimum (Qmin) in 3Q0(ω) master curves do change.30 Several publications describe simulations of I3/1(γ0), 3 Q(ω,γ0), and especially toward 3Q0(ω), using different constitutive models.26−28,31 Kempf et al.30 compared their experimental measurements of 3Q0(ω) master curves of polystyrene combs with predictions of 3Q0(ω) using the pom-pom model.32,33 Recently, Hyun et al.26 analyzed 3Q0(ω) of branched polymer systems in more detail, using the single mode differential pom-pom model. Hoyle et al.31 re-evaluated

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FT RHEOLOGY Theory. The most simple one-dimensional flow of a viscous fluid under a constant shear rate is described by Newton’s law: σ = ηγ ̇

(1)

For non-Newtonian fluids, the viscosity η is a function of the applied shear rate γ̇. Because of symmetry, the viscosity η being independent of direction of shear, and therefore η = η(γ̇) = η(−γ̇) = η(|γ̇|). For small shear rates, respectively small strain amplitudes, the viscosity can be expanded via an even Taylor series with respect to shear rate: η(γ )̇ = η0 + c1γ 2̇ + c 2γ 4̇ + ... B

(2) DOI: 10.1021/acs.macromol.5b02706 Macromolecules XXXX, XXX, XXX−XXX

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low strain amplitudes γ0, originating from the rheometer. Consequently, when I1 ∝ γ0 and I3 ≡ noise, I3/1(γ0) decreases36 with I3/1 ∝ γ0−1 and reaches a minimum before it increases with the expected scaling of 2, when I3 ∝ γ03 (Figure 1). In our

In oscillatory shear, η0, c1, c2, ... might become complex numbers to account for the relative phase shift. The timedependent strain for oscillatory shear experiments is given in a simplified complex notation: γ(t ) = γ0eiω1t

(3)

The shear rate is given by the time derivative of eq 3: γ(̇ t ) = iω1γ0eiω1t

(4)

Insertion of eq 2 into eq 1 results in the shear stress σ given as a function of shear rate γ̇: σ(γ )̇ = (η0 + c1γ 2̇ + c 2γ 4̇ + ...)γ ̇

(5)

Substitution of the shear rate γ̇ by eq 4 results in a timedependent shear stress: Figure 1. Typical dynamic strain sweep measurement of a linear homopolymer melt. At medium strain amplitudes (MAOS) a proportionality of I3/1 ∝ γ02 is observed (sample PI 87k, T = 20 °C, ω1 = 4 rad/s).

experiments on polymer melts, this scaling was observed at medium strain amplitudes (MAOS region), generally between 0.1 ≤ γ0 ≤ 1.0 (see Figure 1). However, the exact MAOS region also depended on monomer, molecular weight, measurement temperature, and excitation frequency. Finally, I3/1 typically levels off at values around I3/1 ≈ 0.1−0.2 at high strain amplitudes (LAOS).37 The minimum value that can be reached for I3/1(γ0) highly depends on the signal-to-noise ratio (S/N) of the instrument, which limits the application of FT rheology to quantify nonlinearities. For this reason, thorough optimizations have been conducted to maximize S/N up to 107 for I3/1 of emulsions.38 Hyun et al.23 assessed the nonlinear regime for I3/1(γ0) ≥ 5 × 10−3 in their oscillatory shear tests of polymer melts. With newer instrumentation used in our study, it was possible to reach values for I3/1(γ0) of around 1 decade lower, with I3/1(γ0) ≥ 5 × 10−4 (see Figure 1). This value is currently a practical limit for the beginning of the medium amplitude oscillatory shear (MAOS) region, followed by the large amplitude oscillatory shear (LAOS) region, where I3/1(γ0) is not proportional to γ02 anymore. The proportionality I3/1(γ0) ∝ γ02 was observed for all linear homopolymer melt samples in this study, regardless of molecular weight, PDI, or polymer type. The general procedure to obtain nonlinear master curves is illustrated schematically in Figure 2. The raw stress time data of an oscillating shear experiment are recorded and transformed into a frequency spectrum via Fourier transformation (Figure 2, schemes 1 and 2). From the intensities of the first and third harmonics (I1 and I3), I3/1 is calculated and plotted against the strain γ0 (Figure 2, scheme 3). This procedure is repeated for different frequencies and/or different temperatures to cover a maximum experimental range in the 3Q0(ω) frequency space. From each I3/1(γ0) plot, the MAOS region with I3/1 ∝ γ02 is identified, and the parameter 3Q(ω,γ0) = I3/1/γ02 is calculated. In a 3Q vs γ0 plot, a plateau can be identified, whose average value is extrapolated to infinite small strains, γ0 → 0, and eventually yields 3Q0(ω) (Figure 2, scheme 4). Each 3Q0(ω) value is plotted against the frequency, and a nonlinear master curve is obtained via the TTS principle (Figure 2, scheme 5).

Fourier transformation of the time-dependent stress should consequently consist of peak signals at the odd higher harmonics of the excitation frequency ω1/2π in the frequency spectrum. The intensity of the third harmonic I3 was chosen as a sensitive quantity toward mechanical nonlinearity. To partly compensate experimental variations, I3 is normalized to the stress response I1 at the excitation frequency ω1, which results in an intensive property I3/1(γ0,ω1) ≔ I3/I1. This enhances reproducibility in a way that these experimental errors, as for example inhomogeneities of the sample or incorrect loading and trimming, are mainly compensated, since they equally influence the different harmonic intensities. From eq 6 it is expected, that I1 ∝ γ01 and I3 ∝ γ03, which concludes in a quadratic scaling law of I3/1 ∝ γ02 at low strain amplitudes. Intrinsic Nonlinearity 3Q0(ω). The nonlinear parameter 3 Q(ω,γ0) and the intrinsic nonlinearity 3Q0(ω), at low strain amplitudes, were defined by Hyun et al.23 as follows: lim 3Q (ω , γ0) ≡ 3Q 0(ω)

γ0 → 0

with 3Q (ω) ≡

I3/1 γ0 2

(7)

Using the simplified definition of I1 and I3 from eq 6, an expression for 3Q0(ω) is obtained: c 3 Q 0(ω) ∝ 1 ω 2 η0 (8) From eq 8, it can be seen that the intrinsic nonlinearity 3Q0(ω) is expected to be a function of the zero shear viscosity η0 and therefore a function of the molecular weight and the measurement temperature. With the use of the TTS principle, intrinsic nonlinear master curves can be generated in analogy to linear master curves, using the same WLF parameters.23,30 The TTS principle is expected to be applicable for the intrinsic nonlinearity, since the nonlinear parameter 3Q0(ω) is a backextrapolation of 3Q(ω,γ0) to the linear viscoelastic regime at very small amplitudes (limγ0→0).35 Experimental Aspects. Experimental measurements of polymer melts show that I3 is dominated by a constant noise at C


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Figure 2. Scheme of a five-step procedure from raw data (1) to nonlinear master curve (5). (1) Nonlinear stress time data of an oscillatory shear experiment. (2) After Fourier transformation of the time data, a magnitude frequency spectrum with odd higher harmonics can be obtained. (3) The ratio I3/1(γ0,ω1) of the first and third harmonic is proportional to γ02 in the MAOS region. (4) Extrapolation of 3Q(γ0,ω1) to small amplitudes gives the intrinsic nonlinearity 3Q0(ω1). (5) A nonlinear master curve can be created by plotting several 3Q0(ω1) values of different excitation frequencies, which are shifted to a reference temperature, utilizing the TTS principle.

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INTRINSIC NONLINEARITY IN POM-POM AND MSF CONSTITUTIVE EQUATIONS Pom-Pom Model. The pom-pom model was originally presented to predict polymer melt rheological properties of branched pom-pom topologies with q dangling arms at each side. Hyun et al.26 and Hoyle et al.31 already simulated the single mode pom-pom constitutive equation in the LAOS/ MAOS regime within the framework of FT rheology analysis. Simulation of the single mode pom-pom model in MAOS showed that the stress response depends on the ratio r = τ0/τs of orientation time, τ0 (also longest relaxation time for linear polymers), and the stretch relaxation time, τs, of the backbone, which are proportional to the number of effective entanglements along the backbone. They concluded that for q > 1 the predictions of I3/1(γ0) in LAOS are more sensitive to changes in the ratio of relaxation times, r = τ0/τs, than to changes in the degree of branching, q.31 One method to predict the rheology of a linear molecule using the pom-pom constitutive model is to set the branching degree q = 1 and the stretch relaxation time τs = 0 (r → ∞). Another possible viewpoint is that a linear molecule could also exhibit chain stretch relaxation, and therefore τs > 0. With respect to the latter assumption τs > 0, the ratio of relaxation times, r, in the limit of a linear polymer could be presented as26 τ 4 M r = 0 = 2 w ≈ 0.4Z τs π Me

show a relaxation time spectrum due to different relaxation mechanisms of the polymer chain. Nevertheless, a single mode version was used to gain an analytical solution for 3Q0 in the MAOS region with an explicit formula as a function of molecular weight (respectively Z) and the Deborah number. An asymptotical solution of the pom-pom differential equation at low strain amplitudes (γ0 ≤ 1) results in the intensity of the first and third harmonics of the stress response σ(t) with the Deborah number De = ωτ0 and the number of entanglements Z as follows: γ(t ) = γ0 sin(ωt ) σ (t ) =

∑

In′ sin(nωt ) + In″ cos(nωt )

n = odd

(10)

⎛ De 2 ⎞ I1′ = GN0 ⎜ ⎟γ ⎝ 1 + De 2 ⎠ 0

(11)

⎛ De ⎞ ⎟γ I1″ = GN0 ⎜ ⎝ 1 + De 2 ⎠ 0

(12)

⎛ De 4(1 − 2.5Z −1)(De 2 + 12.5De 2Z −1 − 2 − 2.5Z −1) ⎞ 3 I3′ = GN0 ⎜ ⎟γ0 π (1 + 25De 2Z −2)(1 + 4De 2)(1 + De 2)2 ⎝ ⎠

(13)

(9)

⎛ De 3(1 − 2.5Z −1)(10De 4Z −1 − 5De 2 − 20De 2Z −1 + 1) ⎞ 3 I3″ = GN0 ⎜ ⎟γ0 2π (1 + 25De 2Z −2)(1 + 4De 2)(1 + De 2)2 ⎝ ⎠

where Z = Mw/Me is the number of entanglements of the molecule and proportional to the molecular weight Mw of the linear polymer (eq 9). With respect to this criterion, the pompom model with a single relaxation time is used with Z as the only fitting parameter, to predict the rheological behavior of linear polymer melts in the MAOS regime in the here presented work. Please note that only one relaxation time is used. Single mode models result in an approximation for monodisperse polymers. Even linear monodisperse polymers

where G0N is the rubber plateau modulus and I′1 and I″1 are proportional to the storage and loss modulus, respectively. I′3 and I3″ are the real and imaginary parts of the absolute value of the intensity of third harmonics in the shear stress. It should be mentioned that π in the denominator of eqs 13 and 14 was missed in the publications of Hyun et al.26 and Hoyle et al.31 With respect to the definition of 3Q0(ω) (eq 7) and eqs 11−14, the intrinsic nonlinearity 3Q0(De) is calculated as follows, where

(14)

D


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Macromolecules I1 = I1′2 + I1″2 and I3 = I3′2 + I3″2 (for exact calculation see Supporting Information eqs A1−A22): 3

Q 0(De) =

De 2(1 − 2.5Z −1) 2π(1 + 25De Z ) (1 + 4De 2)0.5 (1 + De 2)0.5 2 −2 0.5

(15)

for Z > 2.5

Figure 3 presents the normalized storage and loss modulus, G′/G0N and G″/G0N, calculated from eqs 11 and 12, along with Figure 4. Single mode pom-pom model predictions of 3Q0(De) for Z = 10 from the exact equation (15) and the simplified asymptotic equation (17). The maximum deviation between the exact and the simplified solution is given at De = 1, with almost 77%.

Molecular Stress Function (MSF) Model. The molecular stress function (MSF) model is based on the Doi−Edwards (DE) theory and includes chain stretching along with the chain orientation during the deformation. This model was extended to predict the nonlinear behavior of linear and long chain branched molecules (e.g. strain hardening in extensional flows) in different nonlinear deformations with a maximum of three nonlinear parameters β, f max, and a2.34 Wagner et al.27 used the MSF model to predict 3Q0(ω) data of entangled linear and model comb polystyrenes, which before had been experimentally provided by Hyun and Wilhelm23 in the framework of FT rheology. They concluded that 3Q0(ω) is proportional to the difference of the orientational effect α and the stretching effect β, whereas f max (maximum tension in a chain segment, governs the steady state extensional viscosity) and a2 (governs the overshoot in transient shear viscosity) had no distinct effect on 3 Q0(ω). The consideration of the DE model for linear polymers with the independent alignment assumption (DE-IAA) resulted in a constant value α = 5/21. The stretching effect β was used as a fitting parameter for polymer combs, whereas β = 0.12 for linear polystyrenes.27 Abbasi et al.28 extended this idea and presented an evolution equation to predict the extensional, transient shear, and MAOS behavior simultaneously with a single set of nonlinear material parameters. In this evolution equation, the parameter β is changed by its numerical value and represents the ratio of number of entanglements in the molecule to the number of entanglements in the backbone and controls the slope of strain hardening in extensional flow. Based on this definition, β is equal to 1 for linear polymers and increases monotonically with increasing LCB content. In the final evolution equation, β = 0.12 of Wagner et al.27 was replaced with β/10 = 0.1 by Abbasi et al.28 to gain a consistency between the predicted β for linear polymer topologies and the fitting of the model on the extensional viscosity and 3Q0 in MAOS deformation. An asymptotical solution of the single mode MSF model for linear polymers in MAOS results, in analogy to the single mode pom-pom model (eqs 10−14), in the intensity of first and third harmonics as a function of Deborah number, De = ωτ0, and of the difference of the orientation and stretch parameters (α − β/10):

Figure 3. Predictions of the pom-pom and MSF model for the intrinsic nonlinearity 3Q0 along with their predictions for normalized loss and storage modulus.

3

Q0 vs De (eq 15) for various numbers of entanglements (Z = 5, 10, 40, 630, and ∞). It is shown that for low Deborah numbers, De < 1 (De = 1 is the crossover of G′/G0N and G″/G0N), the intrinsic nonlinearity 3Q0 scales with De2, where it behaves like G′/G0N, while at high Deborah numbers, De > 1, 3Q0 scales with De−1, which is the same scaling as G″/G0N. Figure 3 depicts that an increasing molecular weight results in a broader peak for 3 Q0, utilizing the pom-pom model. For polymers with Z ≤ 2.5 the pom-pom model predicts negative intensities (see eq 15), which are clearly without physical meaning. An asymptotical simplification of eq 15 for low and high Deborah numbers De results in 3

Q 0(De) =

3

Q 0(De) =

1 (1 − 2.5Z −1)De 2 2π 1 1 (1 − 2.5Z −1)De−1 2π 10Z −1

for De → 0 for De → ∞

(16a)

(16b)

Combination of eqs 16a and 16b results in a simplified equation with the correct asymptotic limiting function: 3

Q 0(De) =

1 (1 − 2.5Z −1)De 2 2π 1 + 10Z −1De 3

for Z > 2.5

(17)

3

This expression can be used as an approximation of Q0(De) for low and high Deborah numbers. It is worth mentioning that eq 17 cannot predict 3Q0,max around De ≈ 1 properly. Figure 4 shows a comparison of 3 Q0(De) with Z = 10 between the asymptotic equation (17) and the exact solution equation (15) for 3Q0(De) as predicted from the pom-pom model. The maximum deviation, referred to the exact equation, is given at De = 1 with almost 77%. For small and high Deborah numbers the deviation diminishes below 0.1%. A prediction of 3Q0,max(Z) from the pom-pom (and MSF) model is given and compared to experimental values in Figure 7 in the Results section. E

⎛ De 2 ⎞ I1′ = GN0 ⎜ ⎟γ ⎝ 1 + De 2 ⎠ 0

(18)

⎛ De ⎞ ⎟γ I1″ = GN0 ⎜ ⎝ 1 + De 2 ⎠ 0

(19) DOI: 10.1021/acs.macromol.5b02706 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 1. Linear Homopolymer Samples Used in This Study samplea

Mw [g/mol]

Mn [g/mol]

PDI

Z (Mw/Me)b

polymerization method

PS 25k PS 43k PS 83k PS 154k PS 340k PS 107k PS 112k PS 186k PS 602k PI 22k PI 42k PI 87k PpMS 32k PpMS 210k P2VP 83k PMMA 89k PEO 220k PEO 1020k HDPE 155k HDPE 338k

24600 43000 83400 154300 340100 106700 112200 186400 602000 22300 42500 86800 31900 210100 82900 89400 220000 1020000 155000 338400

23700 41800 81100 146100 294500 53600 67500 79900 127400 21600 41100 84300 30500 199700 71500 85400 197000 884000 10300 12500

1.03 1.03 1.03 1.06 1.15 2.00 1.66 2.33 4.70 1.04 1.03 1.03 1.04 1.05 1.16 1.05 1.11 1.15 15.0 27.2

1.5 2.6 5.0 9.2 20.2 6.4 6.7 11.0 35.5 4.5 8.6 17.6 1.4 9.3 3.4 16.7 110 510 187 409

anionic anionic anionic anionic anionic free radical free radical emulsion emulsion anionic anionic anionic anionic anionic anionic anionic anionicc anionicc c c

Sample nomenclature comprises polymer abbreviation (e.g. PS for polystyrene) and rounded weight-average molecular weight Mw (e.g. 25k ≡ 24 900 g/mol). bEntanglement molecular weights of referred polymers in kg/mol: (PS) 16.8, (PI) 4.6, (PpMS) 22.5, (P2VP) 24.3, (PMMA) 5.4, (PEO) 2.0, (HDPE) 0.8. cPEO provided by PSS; HDPE provided by LyondellBasell. a

Using α = 5/21 and β = 1.0 for a linear polymer topology, eq 23 will be simplified to

⎛ De 2 3 4De 2 3De 2 ⎞ 3 I3′ = GN0 (α − β /10)⎜ − + ⎟γ 2 2 4 ⎝ 1 + De 1 + 4De 1 + 9De 2 ⎠ 0

(20)

3

Q 0(De) = 0.207

⎛ De 3 2De De ⎞ 3 ⎟γ I3″ = GN0 (α − β /10)⎜ − + ⎝ 1 + De 2 4 1 + 4De 2 1 + 9De 2 ⎠ 0

Equations 11 and 12 in the pom-pom model and eqs 18 and 19 in the MSF model show that these models predict similar Maxwell behavior for the loss and storage modulus at low enough strain amplitudes. With respect to the definition of the intrinsic nonlinearity 3Q0(De) (eq 7 and eqs 18−21), 3Q0(De) is calculated for the single mode MSF model (for detailed calculation see Supporting Information eqs A23−A35):

3

Q 0(De) = a

(22)

Figure 3 compares eq 22 from the MSF model (with constant parameter α = 5/21 and two different β = 1.0 and 1.75) with pom-pom model predictions (eq 15). The MSF model predictions, as well as the pom-pom model, scale with De2 at low frequencies. After the crossover point of the normalized loss and storage modulus, the MSF predictions level off at a plateau value of 3Q0,max = 0.25(α − β/10), where for a linear polymer 3Q0,max ≈ 0.0345, with α = 5/21 and β = 1 (see Figure 3). This leveling off at high frequencies is similar to the predictions of the pom-pom model (3Q0,max = 1/4π ≈ 0.08) with infinitive number of entanglements (Z → ∞). It should be mentioned that an increasing parameter β in the MSF model decreases the maximum plateau value of 3Q0(De), 3Q0,max, linearly. An asymptotical simplification of eq 22 for low and high De, in analogy to eqs 16a and 16b, results in 3 De 2 (α − β /10) 2 1 + 6De 2

De 2 1 + bDe 2 + k

(25)

where a, b, and k are the material characteristics that might depend on the monomer type, molecular weight, and/or polydispersity. These parameters a, b, and k will be quantified experimentally in the Results and Discussion section. For the pom-pom model, a and b show a dependence on the entanglement number Z, a = 1/(2π)(1 − 2.5Z−1) and b = 10Z−1. In the MSF model, a and b are constants instead, a = 1.5(α − β/10) and b = 6, with α = 5/21 and β = 1. In other words, for a linear polymer, the MSF model predicts 3Q0(De) as a function only of the longest relaxation time τ0. This longest relaxation time τ0 can be predicted by polymer reptation theories (τ0 ∝ M3.4) or by direct measurements of loss and storage modulus data.39

3 De 2 Q 0(De) = (α − β /10) 2 (1 + 9De 2)0.5 (1 + 4De 2)0.5

3

Q 0(De) =

(24)

The deviation of eq 24 compared to the exact eq 22 is below 2%. Therefore, an analogue to Figure 4 is not shown here. A comparison between eqs 17 and 24 shows that a general equation could predict the intrinsic nonlinearity 3Q0(De) at low and high De as

(21)

3

De 2 1 + 6De 2

⎛ G″ ⎞ lim ⎜ ⎟ = ω⎠

ω→ 0⎝

⎛ G′ ⎞ lim ⎜ 2 ⎟ = ω→ 0⎝ G″ ⎠

∑ Giτi

(26)

i

∑ i

Giτi 2 (Giτi)2

(27)

⎛ G″ G′ ⎞ ⎛ G′ ⎞ ⎟ = τ ⎟ = lim ⎜ lim ⎜ 0 2 ω → 0⎝ ω G″ ⎠ ω → 0⎝ ωG″ ⎠

(23) F

(28)


Article

Macromolecules In contrast to the MSF model, 3Q0(De) depends not only on the longest relaxation time τ0 in the pom-pom model but also on the molecular weight Mw.

■

synthesized with free radical polymerization or with emulsion polymerization. For free radical polymerization azobis(isobutyronitrile) (AIBN; Aldrich, 98%) was used as initiator. Styrene was given into a small flask, and after adding the initiator, the mixture was stirred under argon at 60 °C for 24 h. The reaction was continued for another 2 days at room temperature, with stirring as long as possible, before THF was used to dissolve the product. The polymer was precipitated in fresh methanol and dried under reduced pressure at 70 °C. For emulsion polymerization, distilled water, a 10 wt % aqueous solution of sodium dodecyl sulfate (Aldrich, 99%) as emulsifier, and styrene were given into a flask with a sealed precision glass (KPG) stirrer. After stirring for 30 min under argon, the emulsion was heated to 80 °C, and the reaction was started by adding solid potassium peroxodisulfate (K2S2O8; Aldrich, 99%) as initiator. Heating and stirring were continued for 6 h. The reaction was stopped by adding a 4% aqueous solution of 1,4-dihydroxybenzene (hydroquinone; Acros, 99,5%) and cooling with an ice bath to room temperature. To remove the emulsifier, concentrated hydrochloric acid (Acros, 37% in water) was added. The sediment was then washed with distilled water until it was neutral (6 ≤ pH ≤ 7) and washed with methanol until styrene smell could no longer be observed. Molecular Characterizations. The number- and weight-average molecular weight (Mn and Mw) and polydispersity index (PDI = Mw/ Mn) of the samples were determined with size exclusion chromatography (SEC). The SEC system consisted of an Agilent 1100 pump, an Agilent 1200 differential refractive index (DRI), and UV detector with two PSS SDV Lux 8 mm × 300 mm columns (103 and 105 Å pore size). The SEC analysis was done in THF at 25 °C with a flow rate of 1 mL min−1. 1H NMR spectroscopy was performed in deuterated chloroform (CDCl3) at 25 °C with a Bruker Avance III Microbay 400 MHz spectrometer and 512 scans. The amount of cis/trans-1,4 and -3,4 microstructure in the polyisoprene samples was determined by the peak intensity ratio of the CH− signal from the 1,4 microstructure between 4.95 and 5.10 ppm and the CH3 signal from the 3,4 microstructure between 4.55 and 4.71 ppm.41 Differential scanning calorimetry (DSC), using a DSC30 from Mettler Toledo, with a heat rate of 10 °C/min over two separated heat runs was used to obtain glass transition temperatures Tg of each sample. Rheological Measurements. All rheological characterizations were conducted with an ARES G2 strain-controlled rotational rheometer from TA Instruments. Parallel plate geometries (13 and 25 mm diameter) were used to measure the SAOS behavior. Nonlinear oscillatory (MAOS) tests were partly conducted with cone−plate geometries (13 and 25 mm diameter, both with 0.1 rad cone angle) to obtain homogeneous shear fields, partly with a self-made 10 mm partitioned plate geometry together with a 13 mm, 0.1 rad cone geometry.42 The partitioned plate setup was chosen to obtain high shear rates without edge fracture affecting the measurement. The linear viscoelastic region of each sample was determined with strain sweep experiments under oscillatory shear. Linear master curves of G′(ω) and G″(ω) were obtained with frequency sweep tests at small amplitude oscillatory shear (SAOS) and the use of the time− temperature superposition (TTS) principle. The horizontal shift factors were calculated with the Williams−Landel−Ferry (WLF) equation: log aT = [−C1(T − Tref) ]/[C2 + T − Tref] (see Table 2). Nonlinear master curves of 3Q0(ω) were obtained for all samples, using the same shift factors as for the linear master curves, with the procedure, which is pictured in Figure 2, based on the underlying theory described above. Time data from an oscillatory shear test were Fourier transformed, and the magnitude of first and third harmonic intensities, I1 and I3, were used to obtain the parameter 3Q at a given excitation frequency ω1/2π. The ARES G2 rheometer, used in this work, is provided with the TA Instruments TRIOS software, which is capable of directly calculating I3/1 from raw stress time data. The nonlinear master curves were then created from 3Q0 values with the TTS principle, utilizing WLF parameters from corresponding linear master curves.

EXPERIMENTAL SECTION

Materials. All polymer samples have been specifically synthesized with anionic polymerization,40 free radical polymerization, or emulsion polymerization, with the exception of high-density polyethylene (HDPE), which was donated and characterized (Mw, Mn, PDI) by LyondellBasell, and poly(ethylene oxide) (PEO), which was kindly donated and characterized by Polymer Standards Service GmbH (PSS). Table 1 lists the investigated samples. Monomer and Solvent Purification. Styrene (Acros, 99%), pmethylstyrene (Acros, 98%), methyl methacrylate (Acros, 99%), and 2-vinylpyridine (Aldrich, 97%) were first degassed by three freezing− evacuation−thawing cycles and then stirred over calcium hydride (CaH2; Acros, 93%) overnight. Afterward, it was distilled at reduced pressure and stirred over di-n-butylmagnesium (Aldrich, 1 M in heptane) for 20 min. It was then distilled at reduced pressure into precalibrated ampules and stored under argon at −18 °C until needed. Isoprene (Acros, 98% stabilized) was stirred over n-butyllithium (nBuLi; Aldrich, 2.5 M in hexanes) for no longer than 20 min in an ice bath. Then it was distilled at room temperature, first into a liquid nitrogen cooled flask at reduced pressure and then into precalibrated ampules, and stored under argon at −18 °C until needed. Tetrahydrofuran (THF; Carl Roth, 99.5%) was first distilled from CaH2 and then from sodium/benzophenone (Acros, 99%). It was stored over sodium benzophenone on the vacuum line prior to use. Toluene (Carl Roth, 99.5%) was distilled from CaH2 and stored over sodium benzophenone in a flask on the vacuum line. Synthesis of Linear Homopolymers. Monodisperse Polymers by Anionic Polymerization. A high-vacuum technique (≤10−3 mbar) was used to synthesize linear polystyrene (PS), poly(p-methylstyrene) (PpMS), polyisoprene (PI), poly(methyl methacrylate) (PMMA), and poly(2-vinylpyridine) (P2VP) samples with PDI < 1.2. The monomer ampules were directly installed on the vacuum line with ground glass joints. After removal of the argon used for storage, the monomer was carefully freezed with liquid nitrogen. Then, toluene was distilled over the vacuum line and freezed into the ampule. Once the compounds melted completely, they were thoroughly mixed, and sec-butyllithium (s-BuLi; Acros, 1.3 M in cyclohexane/hexane) was added as initiator with a syringe in an argon counterstream at room temperature. A deep orange (polystyrene) or a slightly yellow (polyisoprene) color indicated the living anions of the polymers. After stirring at room temperature for several minutes (10 min for PS and PpMS, 30 min for PMMA, and 60 min for PI), the reaction was terminated with degassed methanol (Carl Roth, 99%), at which any color disappeared immediately. The polymers were then precipitated in methanol (PMMA in water), redissolved in THF, and precipitated again in methanol. For the synthesis of P2VP an initiator solution of s-BuLi in toluene was prepared with an excess of 10 mol % 1,1-diphenylethylene (DPE; Acros, 98%), which resulted a deep red liquid. Pyridine (Carl Roth, ≥99.5%, p.a.) and THF (10:1 pyridine/THF) were distilled into a baked out and argon flushed reactor, where the 2-vinylpyridine ampule and also the initiator ampule were connected with ground glass joints. The reactor was cooled with an ice bath, and under stirring, first the initiator solution was dropped in thoroughly and then the 2vinylpyridine monomer was added fast. A red color indicated the living polymer. After stirring for 24 h at room temperature, the reaction was stopped by adding degassed methanol until all color was vanished. P2VP was precipitated in petroleum ether (Acros, boiling range 40−60 °C). The final products were filtered and dried under reduced pressure at 70 °C and PMMA at 150 °C for at least three days to remove residual amounts of solvents. Polyisoprene samples were further stabilized with 0.5 wt % 2,6-di-tert-butyl-4-methylphenol (BHT; Acros, 99%). Polydisperse Polymers by Free Radical and Emulsion Polymerization. Polystyrene samples with a polydispersity higher than 1.2 were G


Article

Macromolecules

and a PDI ≤ 1.2. Values from the literature44 were used for samples that did not allow to determine Me with the mentioned method (i.e., HDPE and PEO). The average number of entanglements for each sample, Z = Mw/Me, was calculated, and results are presented in Table 1. MAOS Experiments. Nonlinear mechanical data were obtained for all samples with Fourier transformation of measured stress time data in medium amplitude oscillatory shear (MAOS). The obtained 3Q0(ω) values yielded nonlinear master curves for each polymer sample, when the WLF parameter from corresponding linear master curves were used (for parameters see Table 2). A typical measuring routine started at a certain temperature with a fresh sample and a first strain sweep test at a low frequency (usually 1 rad/s). The test ended with a high frequency at the same temperature and with the same sample, at which a scaling of 2 (I3/1 ∝ γ02) was no longer observed. Reasons for the nonquadratic scaling could be sample failure, for example due to edge fracture. The experiment could also not be continued, when the torque was too high for the instrument. Longer waiting times (up to 20 min) between two measurements improved our data with regard to the observation of the expected scaling of I3/1 ∝ γ02. Samples had to be fully relaxed before the next measurement, or else a scaling exponent smaller than 2 was found (typically I3/1 ∝ γ01.4−1.7). Reproducibility of the nonlinear master curves was given in our experiments, with a typical standard deviation for 3Q0(ω) values of estimated 4% for low frequencies and up to 15% for higher frequencies. Appearing scattering of 3Q0(ω) values at higher frequencies after the observed local maximum, 3 Q0,max, occurred because of experimental difficulties. These intricacies originated from low measurement temperatures close to the glass transition temperature Tg of the polymer melts or high measurement frequencies, which led to possible wall slip and other sample failures, torque overload, or low S/N of the higher harmonics, i.e., I3. Nonlinear Master Curves of Chemically Different, Linear Monodisperse Polymers. Nonlinear master curves of linear homopolymers with a low PDI ≤ 1.08 were investigated with respect to molecular weight and monomer. Investigated linear homopolymers in this study were polystyrene (PS), 1,4-cispolyisoprene (PI), poly-p-methylstyrene (PpMS), poly(2-vinylpyridine) (P2VP), poly(methyl methacrylate) (PMMA), poly(ethylene oxide) (PEO), and high-density polyethylene (HDPE). For detailed sample properties see Table 1. In the measured frequency range, the nonlinear master curves of samples with a low PDI showed a scaling at limω→03Q0(ω) ∝ ωn with typically n = 2 ± 0.4 (n = 2 expected from eq 25) and scaling exponents k between 0.06 and 0.49 for higher frequencies, limω→∞3Q0(ω) ∝ ω−k, after reaching the maximum 3 Q0,max (Figure 6; see also eq 25). The large variety of different scaling exponents for 3Q0(ω) at higher frequencies is based on the experimental problems described above, so that only few measurement points could be acquired in this region. However, high frequency 3Q0(ω) data could be obtained for a couple of polymer melt samples (PS 154k, PS 83k, and PpMS 210k). These measurements resulted a scaling exponent k between 0.30 and 0.37. A characteristic feature in the experimental data of all nonlinear master curves of monodisperse, linear homopolymer melts is the maximum 3 Q0,max . Set expectations from constitutive modeling are either a dependence on the entanglement number Z (pom-pom) or a constant value (MSF). The experiment reveals a correlation to Z; however,

Table 2. Linear Rheological Data of the Used Polymers sample

Tref [°C]

−C1

C2 [K]

G0N [kPa]

Tg [°C]

τ0a [s]

PS 25k PS 43k PS 83k PS 154k PS 340k PS 112k PS 107k PS 186k PS 602k PI 22k PI 42k PI 87k PpMS 31k PpMS 210k P2VP 83k PMMA 89k PEO 220k PEO 1020k HDPE 155k HDPE 338k

160 160 160 160 160 160 160 160 160 −10 −10 −10 160 160 160 160 80 80 140 140

7.455 5.396 6.087 6.153 6.199 6.149 6.044 6.146 5.853 6.245 6.651 6.370 5.374 6.770 6.828 10.782 2.469 1.206 2.246 2.835

131.4 98.8 104.6 104.7 104.8 105.5 104.4 103.6 103.6 104.3 106.6 104.7 84.2 96.8 121.4 140.4 203.2 91.5 212.4 270.9

287.7 217.1 178.8 153.0 169.2 166.0 140.0 135.2 167.6 339.1 335.6 357.7 156.6 129.9 135.0 605.0 na na na na

103.6 105.4 105.5 107.8 106.7 105.7 106.0 106.3 106.7 −65.0 −61.5 −58.2 90.6 113.5 86.5 125.6 na na 133.9b 130.3b

0.0013 0.0085 0.13 1.3 33 na na na na 0.0048 0.066 0.84 0.058 33 2.7 10 0.25 na na na

Relaxation times are listed for Tg + 60 °C with exception of PEO 220k, which is given for T = 80 °C. bNo Tg was measured for HDPE; instead the melt temperature Tm is listed. a

■

RESULTS AND DISCUSSION SAOS Experiments. Linear master curves of G′(ω) and G″(ω) have been acquired via the TTS principle29 for all samples in this work (see also Table 1). The resulting WLF parameters C1 and C2 (Table 2) were used in the creation of nonlinear master curves of the intrinsic nonlinearity 3Q0(ω) for each respective sample. The storage modulus G′ of monodisperse samples showed a plateau modulus G0N in the rubber plateau region, while for samples with a PDI ≥ 1.2 only a shoulder could be examined (Figure 5). For all monodisperse

Figure 5. Storage shear modulus G′ of linear polystyrene melts with different molecular weight and PDI (given in parentheses) as a function of angular frequency ω, nominated to a reference temperature of Tg + 60 °C.

polymers, the expected proportionalities of G′ ∝ ω2 and G″ ∝ ω1 were reached at low frequencies. The plateau value G0N was determined by the minimum of tan δ = G″/G′ in the rubber plateau region.43 From this plateau value G0N, the entanglement molecular weight Me was calculated with Me = (4/5)ρRT/ G0N.44,45 For samples synthesized from the same monomer, Me was taken from the sample with the highest molecular weight H


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Macromolecules

Figure 8. Comparison of the longest relaxation time τ0 with the relaxation time τQ = 1/ωmax (see Figure 6). The experimentally determined scaling of 0.9 and the expected scaling of 1 (τQ = τ0) are displayed. Entanglement range from Z = 1.4 to 17.6 (also see Table 1).

Figure 6. Nonlinear master curves of low PDI polystyrene (PS) and cis-1,4-polyisoprene (PI) samples. For low frequencies approximately 3 Q0(ω) ∝ ω2 was found for all monodisperse polymer melts (dotted lines).

neither of the models could predict the found scaling of approximately 3Q0,max ∝ Z0.35 (Figure 7; for detailed calculation of 3Q0,max(Z) for each model see the Supporting Information). Figure 9. Nonlinear master curves of monodisperse (PDI ≤ 1.07) linear melts and related fits via eq 25.

The obtained picture can be compared with the predictions of the constitutive model theories as seen in Figure 3. Equation 25 was then fitted to all nonlinear master curves of low PDI polymers, with k = 0.35 as mean value for the decreasing slope after 3Q0,max (Figure 9). This yielded the parameters a and b, which were plotted against the number of entanglements Z (Figure 10). A weak scaling law, related to the entanglement number Z for a and b, was found for all investigated polymers, a = AZ−0.5 and b = BZ−1.

Figure 7. Experimentally determined nonlinear master curve maxima 3 Q0,max of monodisperse, linear homopolymer melts presented in this study (see Figure 9). 3Q0,max scales with the entanglement number (∝ Z0.35). Predictions from the simplified pom-pom (eq 17) and MSF models (eq 23) as well as expectations from the finalized empiric equation (eq 30) are shown for comparison.

The nonlinear relaxation time τQ is calculated via τQωmax = 1 at the maximum 3Q0,max and is within experimental limits, similar to the longest linear relaxation time τ0 (Figure 8), which is obtained from respective linear master curves (eq 28). This similarity between nonlinear and linear relaxation times shows that they can be associated with similar molecular dynamics and processes. For further analysis with respect to the pom-pom and MSF constitutive models, as well as the proposed general equation (25), all nonlinear master curves are plotted against the Deborah number, De = ωτ0. This results in a shift on top of each other of those 3Q0(De,Z) master curves, where the low frequency slopes, with a scaling of 2, coincide (Figure 9).

Figure 10. Parameters a and b from eq 25 for polymer samples with PDI ≤ 1.16, along with predictions from the pom-pom constitutive model (pointed line) and MSF model (point-dashed lines) as a function of Z. For exact parameter expressions see also Table 3. Dashed lines indicate the power law behavior of each parameter from the experiment as a ∝ Z−0.5 and b ∝ Z−1. I


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Table 3. Parameters for Eq 25, As Predicted by the Pom-Pom and MSF Constitutive Model, as Well as Results from Fits on Experimental Data As Presented in Eq 30 a

model

−1

1/(2π)(1 − 2.5Z ) 1.5(α − β/10) 0.32Z−0.5

pom-pom MSF experiment a

b −1

10Z 6 33.8Z−1

k 1 0 0.35

3

Q0,max

1/(4π)a 0.25(α − β/10) ∝Z0.35

For Z → ∞; see also Figure 7.

semiempiric equation (30) in comparison. For low frequencies, all three methods are able to depict the expected scaling of 2, but for high frequencies, only the semiempiric expression is able to describe the full behavior of the experimental data. It should be mentioned that the pom-pom model cannot predict 3Q0(De) for samples below a certain threshold of Mw ≤ 2.5Z (for example PS25k), since the entanglement number is too low or nonexistent (see also Theory section). In conclusion, it was experimentally shown that a and b from eq 25 do not depend on the monomer for monodisperse, linear homopolymers, but rather only on the entanglement number Z. Therefore, eq 30 seems to be a unifying quantification of nonlinear shear under MAOS conditions for these kinds of polymer melts. The investigated constitutive models are able to approximately predict these parameters. Nevertheless, for small polymers below a molecular weight with Z ≤ 3 major discrepancies occur, especially for the pom-pom model (Figure 10). It was also shown that k is experimentally around 0.35 for monodisperse samples, where its predicted value is either k = 0 (MSF model) or k = 1 (pom-pom model). We want to note that the sample PEO 1020k, which has an extremely high number of entanglements with Z = 510, is excluded from the here presented study. The reason is that the measured 3Q0(ω) master curve of this sample could not be predicted by the found semiempiric equation (30), using the parameter values of A = 0.32 and B = 33.8. Fetters and coworkers46,47 stated that the transition from a dominating Rouse behavior and contour length fluctuation (CLF) to pure reptation of a polymer chain takes place at a reptation molecular weight Mr, which is a function of the polymer specific packing length p. From all investigated polymer samples in our study, only the sample PEO 1020k exceeds the threshold of Mr. Therefore, we assume a change in molecular dynamics due to the extremely high molecular weight48 (Mw = 1020 kg/mol ≫ Mr ≈ 680 kg/mol), which needs further investigation. Effect of Polydispersity. Polydisperse, linear polymer melts were investigated with the MSF constitutive model. Fix values were chosen for the nonlinear parameters, α = 5/21 and β = 1.0. The summation of a relaxation spectrum from each investigated sample was used to calculate the real and imaginary parts of the higher harmonic intensities I1 and I3, i.e., I1′ , I3′ , I1″, and I″3 , for every sample individually. Each spectrum was calculated, using relaxation spectra of Gi and Dei = ωτi, instead of G0N and De = ωτ0 in eqs 18−21. The relaxation spectra (Gi and τi, see Table 4) were experimentally obtained from the storage and loss moduli of respective linear master curves. Finally, a linear summation of the obtained spectra were used to calculate the 3Q0 parameter. Figure 12 depicts the simulation along with experimental 3Q0(ω) data. Because of the finite number of relaxation spectra, which were used (see Table 4), a wavy form is obtained for the simulation of 3Q0(ω) instead of a smooth curve. Direct comparison shows the accordance of the multi mode MSF model prediction with the experimental data within our experimental window. However, as mentioned

In a next step, an iteration (eq 29) of eq 25, with the new scaling laws for a and b, was fitted on the experimental data with respective entanglement numbers for Z and resulted average values for A and B. Q 0(De , Z) = AZ −0.5

3

De 2 1 + BZ −1De 2 + 0.35

(29)

The parameter values of A and B can be used in a final, empiric equation for 3Q0(De,Z), where the molecular weight, i.e., the entanglement number Z, is a fitting parameter, and the Deborah number De, specifically the angular frequency ω, is used as variable for a unifying equation (eq 30). Q 0(De , Z) = 0.32Z −0.5

3

3

Q 0,max(Z) ≈ 0.01Z 0.35

De 2 1 + 33.8Z −1De 2 + 0.35

(30) (31)

Equation 30 approximately describes all monodisperse, linear homopolymer melts, independent of chemical composition within the investigated samples. Detailed calculation of the maximum 3Q0,max is given in the Supporting Information (eqs A36−A44). The different parameters a, b, and k from eq 25 are presented in Table 3 for the pom-pom and MSF model and are compared to the semiempiric equation (30). Figure 11 shows the predictions from the pom-pom and MSF model with experimental data and the here presented

Figure 11. Predictions of the simplified single mode pom-pom (dashed lines, eq 17) and simplified single mode MSF (dotted lines, eq 24) model, measured data (symbols) and predictions from the here presented semiempiric equation (full lines, eq 30). J


Article

Macromolecules Table 4. Relaxation Spectra of Polydisperse Samples at T(PS) = 160 °C and T(HDPE) = 140 °C PS 602k

PS 340k τi [s]

Gi [Pa] 1.13 3.81 9.90 3.77 3.39 3.83 4.39 3.42 1.35 3.12

× × × × × × × × × ×

× × × × × × ×

10 105 104 104 104 104 104 104 104 103 PS 112k

9.45 3.94 4.31 5.28 5.22 4.26 3.47 3.02 2.22 1.56

7

10 105 105 104 104 104 103

× × × × × × × × × ×

−6

10 10−4 10−3 10−2 10−1 100 101 102 103 104

1.89 5.85 1.69 5.59 4.08 4.50 4.90 4.83 1.04

× × × × × × × × ×

7

× × × × × × × × ×

10 105 105 104 104 104 104 104 104

6.65 2.11 1.94 1.75 1.46 9.46 5.43 3.06 1.05

τi [s]

Gi [Pa]

PS 107k τi [s]

Gi [Pa] 1.34 5.13 1.68 9.08 5.48 1.89 2.08

7

PS 186k τi [s]

Gi [Pa]

8.08 2.57 2.58 2.27 1.52 8.44 4.46

× × × × × × ×

Gi [Pa] −6

10 10−4 10−3 10−2 10−1 10−1 100

1.21 4.73 1.53 7.95 3.97 1.09 9.93

× × × × × × ×

7

10 105 105 104 104 104 102

τi [s]

Gi [Pa] −6

10 10−4 10−3 10−2 10−1 10−1 100 101 102

8.87 5.09 1.53 6.57 4.97 3.67 2.47 1.75 7.87 1.27

× × × × × × × × × ×

1.11 × 2.26 × 2.13 × 2.00 × 1.68 × 1.23 × 8.40 × 6.02 × 4.47 × 3.24 × HDPE 338k

6

10 105 105 104 104 104 104 104 103 103

HDPE 155k

8.53 2.88 3.05 2.69 1.81 1.03 5.57

× × × × × × ×

−6

10 10−4 10−3 10−2 10−1 100 100

2.34 5.35 1.76 3.39

× × × ×

τi [s] 6

10 105 105 104

4.27 3.87 2.81 2.54

× × × ×

τi [s]

Gi [Pa] −3

10 10−2 10−1 100

2.39 9.43 4.42 1.30

× × × ×

10−5 10−4 10−3 10−2 10−1 100 100 101 102 103

5

10 104 104 104

1.26 1.10 1.01 1.19

× × × ×

10−2 10−1 100 101

Figure 13. Nonlinear master curves of linear polymer melts with different polydispersities. With increasing PDI, the slope at low frequencies decreases.

Figure 12. Experimental 3Q0(ω) data of polydisperse linear polymer melts along with respective molecular stress function (MSF) model predictions (PDI given in parentheses). The low frequency region, where 3Q0(ω) ∝ ω2, is hardly accessible for some polymer measurements because of instrumental and experimental limitations. This is especially valid for high polydispersities and high molecular weights.

linear terminal regime to very low frequencies, where very high measurement temperatures are needed. Therefore, it was not possible to obtain reliable data for the longest relaxation time τ0 via the zero shear viscosity η0 with eq 28 for all samples, and τco was used instead. A scaling of approximately 3Q0(De) ∝ De2 for low frequencies, before the maximum 3Q0,max, was observed for all linear monodisperse polymers in this work (see Figure 6 or 9). With increasing PDI, the slope of 3Q0(De) for low frequencies decreases, until it levels off at a constant value, which equals 3 Q0,max, within the measured frequency range and accessible intensity range of I3/1(γ0) for samples with a PDI ≥ 2. Our experiments show a dependence of the low frequency scaling 3 Q0(De) ∝ Den on the PDI. The following empiric equation can describe this observation with a minimum set of parameters at De = 0.02, where the scaling of n = 2 for monodisperse samples (PDI = 1) is still given (Figure 14). 2 n= for De = 0.02 (32) PDI2.2

before, the MSF constitutive model cannot predict the decrease of nonlinear master curves, with 3Q0(ω) ∝ ω−0.35, for high frequencies after the maximum 3Q0,max, even for monodisperse samples. For smaller frequencies (3Q0(ω) with ω < ωmax) the MSF prediction shows the expected scaling exponent of 2, 3 Q0(ω) ∝ ω2. For increasing ω, a decreasing slope in 3Q0(ω) is observed until 3Q0(ω) = constant at 3Q0,max and for ω > ωmax. The MSF constitutive model therefore predicts the same scaling behavior for all linear polymer melts, independent of dispersity. The experimental window however is typically limited to 10−3 ≤ De ≤ 103, and the low frequency ranges, that are needed to obtain the expected scaling of 2 (see also Figure 13), could not be approached with our instrumental setup. Within the experimental window, the prediction overlaps approximately with our experimental data. In Figure 13 several 3Q0(De) master curves for polymer samples with different polydispersities are presented. In this figure, the relaxation time τco, from the crossover of G′ and G″, was used to calculate De. The high polydispersity shifts the

It is shown by the MSF model for polydisperse polymer melts (Figure 12) that 3Q0(De) ∝ De2 for extremely low frequencies. K


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Macromolecules 3

Q0,max, and a scaling of k = 0.35 for the decreasing slope after the maximum for low PDI polymer melts. Polymer melts with a PDI higher than 1.1 experimentally revealed a scaling exponent 3Q0 ∝ Den at low frequencies that could be experimentally described by n = 2/PDI2.2 for a fixed Deborah number of 0.02. Model prediction from the MSF constitutive model for high PDI polymers (PDI > 1.1) show that at very low frequencies the scaling of 3Q0 ∝ ω2 should be seen again. This prediction, however, is currently not accessible experimentally due to the potentially very high measurements temperatures and the broad relaxation time distributions that are needed for such an experiment in addition to an instrument sensitivity that is not available at the moment. The nonlinear mechanical behavior of viscoelastic polymer melts and solutions is a field of ongoing research. Recent developments in instrumentation and access to more powerful computation offer the possibility for better insight into the macroscopic and microscopic level of polymeric materials. The detailed investigation and quantification of the intrinsic nonlinearity 3Q0(ω) can lead to new developments in constitutive modeling and computer simulations, especially molecular dynamic simulations of polymer melts. These simulations need to be correlated and tested via experimental nonlinear mechanical response of model systems under welldefined nonlinear conditions.

Figure 14. Scaling exponents n from 3Q0(ω) ∝ ωn for De = 0.02 as a function of PDI (see related Figure 13).

However, to be able to detect this behavior, a sensitive measurement at a high temperature above the Tg in combination with a high activation energy and low thermal degradation is desirable but might not always be possible experimentally.

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CONCLUSION The goal of this publication was to experimentally quantify the intrinsic nonlinearity 3Q0(ω) with medium oscillatory shear (MAOS) measurements in combination with model calculations, i.e., predictions from the pom-pom and molecular stress function (MSF) constitutive models, for linear homopolymer melt model systems, varying molecular weight, chemical composition, and polydispersity. Linear homopolymer melt model systems of polystyrene (PS), cis-1,4-polyisoprene (PI), poly(p-methylstyrene) (PpMS), poly(methyl methacrylate) (PMMA), and poly(2vinylpyridine) (P2VP) were synthesized with different molecular weights (from Z = 1.5 to 38 entanglements) and different polydispersities (PDI = 1.03 to 4.7). High-density polyethylene (HDPE) with higher polydispersities (PDI = 15 and 27.2) and monodisperse poly(ethylene oxide) (PEO) were also included but not synthesized. For synthesis, anionic, free radical, and emulsion polymerizations have been utilized. Oscillatory shear experiments resulted in linear (SAOS) and nonlinear (MAOS) master curves of 3Q0(ω) of all samples. These master curves have been created via the TTS principle, whereas for the nonlinear master curves, obtained WLF parameters from the respective linear master curves could be used. For all presented samples, a scaling of I3/1 ∝ γ02 was observed in MAOS, and therefore 3Q0(ω) values for the nonlinear master curves could be obtained. It was observed that all nonlinear master curves of low PDI samples (PDI < 1.1) show a scaling of 2 (3Q0 ∝ ω2) for low frequencies, until a maximum 3Q0,max is reached. This result was also predicted by the pom-pom and MSF constitutive models. For high frequencies, we experimentally found that 3Q0(ω) scales with 3 Q0(ω) ∝ ω−k, with k = 0.35, which is in between the values predicted by the two constitutive models that forecast either a behavior with a scaling of −1 (pom-pom) or a constant plateau value (MSF). A general, combined, and simplified equation (25) from the pom-pom and MSF model was able to describe the nonlinear master curves of linear homopolymer melts with low PDI, independent of the investigated monomer, by fitting this equation to our experimental data of 3Q0(De). The proposed semiempirical eq 30 includes two molecular weight dependent parameters a = 0.32Z−0.5 and b = 33.75Z−1, a quadratic scaling of 3Q0(ω) for frequencies below the maximum

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b02706. Step-by-step model calculations of the simplified equations for the intrinsic nonlinearity 3Q0(ω) in the single mode pom-pom and single mode molecular stress function (MSF) constitutive models; calculations of 3 Q0,max for the simplified single mode pom-pom model, the single mode MSF model, and the semiempiric equation (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Phone +49 (0)721 608-43150; Fax +49 (0)721 608-994004; e-mail [email protected] (M.W.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Dr. I. Vittorias (LyondellBasell) and Dr. T. Hofe (PSS) for donating polymer samples. Additionally, we thank L. Faust and D. Marschner for helping in the laboratory and also our colleagues for fruitful discussions about LAOS methods, in particular D. Merger and L. Schwab. Dr. M. Abbasi thanks the Alexander von Humboldt Foundation for financial support.

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REFERENCES

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