Synthesis and Linear and Nonlinear Melt Rheology of Well-Defined

Article pubs.acs.org/Macromolecules

Synthesis and Linear and Nonlinear Melt Rheology of Well-Defined Comb Architectures of PS and PpMS with a Low and Controlled Degree of Long-Chain Branching Michael Kempf, Deepak Ahirwal, Miriam Cziep, and Manfred Wilhelm* Department of Chemistry and Biosciences, Institute of Chemical Technology and Polymerchemistry, Karlsruhe Institute of Technology (KIT), Engesserstr. 18, 76131 Karlsruhe, Germany S Supporting Information *

ABSTRACT: Well-defined, monodisperse homopolymer comb architectures with varied number and length of the branches under linear and nonlinear deformation were synthesized and examined to determine the effect of branching on different rheological properties. The correlation of the rheological properties with the comb topology is of special interest for the determination of the degree of branching. Therefore, well-defined polystyrene-based comb polymers with systematically varied number and molecular weight of the branches, narrow polydispersities, and a controlled, but low, number of branches (typically 0.1−1 mol % branches per backbone) were synthesized and compared with data from polystyrene combs of the Roovers series that have a higher number of branches (>1 mol % branches per backbone). To investigate the rheological properties in detail, various linear and nonlinear techniques were applied. Within the linear regime, the reduced van Gurp−Palmen plot (δ vs | G*|/G0N) was used to identify critical points that illustrated the influence of the branch molecular weight and number of branches on the resulting rheological properties. In the nonlinear regime large amplitude oscillatory shear (LAOS) measurements were performed to obtain the nonlinear parameter Q0(ω) via a quadratic scaling law from FT-rheology. An intrinsic nonlinear master curve based on the Q0(ω) parameter reflected the relaxation hierarchy and was shown to be a sensitive method to extract information on the different relaxation time scales. The nonlinear shear measurements were complemented by uniaxial extensional measurements to quantify the strain hardening effect and how the strain hardening was affected by branch relaxation. The results obtained from the uniaxial extensional measurements could be correlated to relaxation times obtained from the intrinsic nonlinear master curve Q0(ω). Pom-pom constitutive model predictions were performed for the comparison with experimental data for extensional rheology with focus on the strain hardening behavior and for LAOS with focus on the nonlinear parameter Q0(ω) as a function of increasing number and molecular weight of the branches in the pom-pom molecule. A comparison of the applied rheological methods(1) small amplitude oscillatory shear (SAOS) in the linear regime, (2) LAOS in combination with FT-rheology, and (3) extensional rheology in the nonlinear regimeillustrated the detection limits as well as the advantages and disadvantages of each technique toward the investigation of rare, but entangled branched comb polymer topologies.

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INTRODUCTION The rheological properties and processability of branched polymers are highly affected by the degree of long chain branching (LCB). Therefore, there is a strong motivation to better understand the influence of branching on the resulting rheological properties, especially for commercial branched polymers.1 Although rheology is a very sensitive method for the characterization of the effects of long-chain branching, an accurate rheological quantification of the branch length and the overall number of branches is still a challeging task because even a low degree of long-chain branching has a significant effect on the viscoelasticity of the material. Several approaches for the quantification of branching in commercial polymers have been conducted.2−5 However, due to the inherent complexity of these systems, these approaches combined several different analytical techniques. For example, WoodAdams and Dealy2 were able to estimate the degree of long © XXXX American Chemical Society

chain branching (LCB) in polyethylenes (PE) using linear viscoelastic (LVE) data in combination with molecular weight distribution (MWD) determined by high temperature size exclusion chromatography (SEC). Similar approaches were also investigated by Shaw and Tuminello4 and Janzen and Colby5 focusing on a combination of SEC and rheology. The number of short chain branches6 and their length (up to 16 carbons)7 can be determined for polyethylene model systems via 13C melt-state NMR down to 1 branching in 10 000 CH2 under magic-angle spinning. Evaluating branching in commercial polymer systems is especially difficult because the detailed polymer topology is usually unknown and the analysis is additionally complicated due to high polydispersities and the Received: September 28, 2012 Revised: April 28, 2013

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dx.doi.org/10.1021/ma302033g | Macromolecules XXXX, XXX, XXX−XXX

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of the comb and suggested that extensional hardening occurs at rates equal to or greater than its inverse. Step strain experiments were used in various studies to examine the nonlinear response of short and long chain branched comb polymer systems.23,36,37 Kapnistos et al.37 presented stress relaxation master curves covering the entire relaxation of the comb and demonstrated that the hierarchy of relaxation processes observed in the linear regime still existed under strong nonlinear deformations. The two groups of Kapnistos et al.37 and Vega and Milner36 utilized the hierarchical relaxation concept to suggest that if the comb backbone remained well-entangled after dynamic dilution due to the arm relaxation process, the backbone followed the Doi− Edwards38 prediction for a linear chain, i.e., that it acts independent of the polymer architecture. A different approach for quantifying the nonlinear regime in shear is based on medium (MAOS) or large amplitude oscillatory shear (LAOS). When the shear amplitude, γ0, and/or the shear frequency, ω1, are increased in a MAOS/ LAOS experiment, nonlinear effects start to appear in the shear stress response. A nonlinear oscillatory excitation at a frequency of ω1 generates mechanical harmonics occurring at odd multiples of ω1 (3ω1, 5ω1, etc.), where each have different intensities In and phases Φn.39−44 In the literature, the analysis of the stress response was formally performed using Lissajous figures. However, when the relative intensities of the harmonics are small (I3/I1 < 5%), this approach presents some disadvantages for the quantification of the corresponding intensities and phases. In contrast to Lissajous figures, FTrheology40,44 is a very sensitive method, which can detect even very week higher harmonic signals (I3/1 < 10−6).45 To ensure comparability between measurements, the relative intensity of the nth harmonic, given by In/I1 ≡ In/1, which corresponds to the intensity of the nth harmonic (In) normalized to the intensity of the fundamental frequency (I1), is used. The relative intensity of the third harmonic I3/1 is commonly used for the quantification of the nonlinearity under oscillatory shear of polymer melts. The relative phase difference of the nth harmonic, given by Φn = ϕn − nϕ1, where ϕn and ϕ1 are the phases of the nth harmonic and the fundamental frequency, respectively. The phase can be additionally used for the quantification of the nonlinearity.46 Because of its high sensitivity, FT-rheology is a useful method to detect the presence of LCB and to distinguish between branched and linear homopolymer topologies.47−50 Neidhö fer et al.47 were able to differentiate between branched and linear topologies under nonlinear oscillatory shear using the relative intensity of the third harmonic (I3/1) and the phase angle of the third harmonic (Φ3) as a function of frequency at a fixed strain amplitude. It is important to note here that these differences only appeared using nonlinear excitation and that, under linear oscillatory conditions, the concentrated polystyrene solutions of linear and star topologies showed similar G′(ω) and G″(ω) behavior. Schlatter et al.48 investigated I3/1 and Φ3 for linear and sparsely branched polyethylene melts and confirmed that FT-rheology was sensitive to the polymer topology. Vittorias et al.49,51 found optimal conditions (low excitation frequency ω1 ≈ 0.1 Hz, γ0 ≈ 2) for the differentiation between linear and branched polyethylenes and observed that long chain branched polyethylenes displayed less higher harmonics relative to linear polyethylenes with a similar molecular weight. Hyun et al.50,52 studied monodisperse linear and comb polystyrene melts under

presence of additives. Therefore, rheological measurements on commercial samples should be combined with other analytical methods (SEC, NMR, viscosimetry, etc.) to assess the nature of the branching present. Alternatively, several model systems including stars,8−10 H-polymers,11 and combs12−23 with known architectures and relatively low polydispersities were previously synthesized, and their melt rheological properties were analyzed. For the quantification of long chain branching, Trinkle et al.24,25 suggested the reduced van Gurp−Palmen plot26 to correlate the rheological data in the linear regime with the topology of different branched polymers and subsequently developed a topology map to assign different topologies based on this approach. Liu et al.27 used a branch-on-branch constitutive model (BoB model) to calculate the effect of different topological structures on the rheological properties with the aim of identifying characteristic points in the van Gurp−Palmen plot, which can then be used to construct a topology map using theoretically calculated points, in addition to experimental results. Besides the linear, the nonlinear rheological properties are also influenced by long chain branching. For example, long chain branched PE and PS exhibited increased shear thinning behavior when compared to their linear counterparts.28 Long chain branched polymers are also well-known for strain hardening behavior in extensional flow, which plays an important role in processing techniques such as thermoforming,29,30 foaming,31 and, especially, film blowing.30 While the experimental determination of the extensional melt viscosity is more difficult,32 this is nevertheless a highly sensitive technique for distinguishing between different topologies when there is a low degree (Me) branches and well-entangled backbones were examined using linear and nonlinear shear and elongation to determine the effect of branching on the mechanical properties of comb polymers. The objective was to investigate correlations between both the linear and different nonlinear rheological properties and the polymer topology with a special focus on characterizing the number and molecular weight of the branches. In the linear oscillatory regime, the reduced van Gurp−Palmen plot was used to determine the influence of the branch molecular weight and number of branch points. From the nonlinear oscillatory rheological data, values for the nonlinear parameter Q0(ω) were obtained under large amplitude oscillatory shear (LAOS) using the square scaling law in FT-rheology40,50,59,60 for the relative higher harmonics as a function of γ0. An intrinsic nonlinear master curve obtained from these measurements revealed the hierarchy of relaxation processes.50 From nonlinear extensional measurements the strain hardening factor (SHF) was determined for different combs and illustrated how the branch relaxation process affected this factor.

LAOS conditions with the result that the relative intensity of the third harmonic (I3/1) depended quadratically on the strain amplitude I3/1 ∝ γ02 and introduced the instrinsic nonlinear parameter Q0(ω). The Q0(ω) sprectrum reflected the polymer chain relaxation processes, which makes it useful for differentiating between linear and branched topologies. The intrinsic nonlinear master curves experimentally obtained by Hyun et al.50 were recently predicted by Wagner et al. using the molecular stress function (MSF) model,53 where the higher harmonic contribution is directly related to the difference (α − β) of the orientation and the stretch β of the polymer chain. Hyun et al.54 investigated the effect of the polymer topology on the nonlinear parameters Q(ω,γ0) and Q0(ω) with a single mode differential pom-pom model. The linear viscoelastic properties G′(ω) and G″(ω) were compared with the nonlinear viscoelastic property Q0(ω) for a variety of molecular parameters. The intrinsic nonlinearity Q0(ω) displayed two distinct relaxation processes for pom-pom architectures even though G′(ω) and G″(ω) could not distinguish two relaxation processes. It was concluded that the polymer topology had a stronger influence on the nonlinear than on the linear viscoelastic properties. However, one of the major experimental limitations in the rheological testing of model (comb) polymers is the lack of enough material to perform multiple rheological measurements. The most commonly used synthesis route for polystyrenebased homopolymer model combs involves the chloromethylation of the polystyrene backbone to introduce branching points.55 Unfortunately, this approach involves the formation of highly toxic chloromethyl methyl ether and leads to crosslinking side reactions during the functionalization reaction, which makes the synthesis of large quantities (several grams) difficult. However, the synthesis of polyisoprene (PI) and polybutadiene (PBD) model comb homopolymers can be accomplished using the macromonomer method56 or by hydrosilylation.57,58 In the macromonomer method, side chains produced by anionic polymerization are terminated with a polymerizable end group and are then copolymerized with the backbone monomer (isoprene or butadiene) via anionic polymerization. The hydrosilylation approach introduces instead a chlorosilane group at the pendant vinyl groups of the polymer backbone, and then side chains are grafted via nucleophilic attack of living PI/PBD at the chlorosilane group. In both cases, while a low polydispersity of the backbone and the side chains can be achieved through the use of living polymerization, the number of branching points is difficult to

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EXPERIMENTAL SECTION

Materials. Styrene (Acros, 99%) was purified by distillation at reduced pressure after stirring over calcium hydride (CaH2) (Acros, 93%) overnight and then distilled from dibutylmagnesium (Aldrich, 1 M in heptane) into precalibrated ampules and degassed afterward by three successive freezing−evacuation−thawing cycles. The styrene was then stored under argon at −20 °C until needed. Tetrahydrofuran (THF) (Acros, 99+%) was distilled first from CaH2 and then from sodium benzophenone and was finally stored over sodium benzophenone on the vacuum line prior to use. Toluene (Acros, 99.5%) was distilled from CaH2 and stored over living polystyrene on the vacuum line prior to use. Chloroform (CHCl3) was distilled from CaH2 prior to use; sec-butyllithium (sec-BuLi) (Aldrich, 1.4 M in cyclohexane) and n-butyllithium (n-BuLi) (Acros, 2.2 M in cyclohexane) were used as received. The syntheses of the poly(pmethylstyrene) (PpMS) combs61 and the polystyrene (PS) combs PS275k-25-26k and PS275k-29-47k of the Roovers series55 are described elsewhere. The samples used in this study are described C


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Figure 1. Reaction scheme for the synthesis of (a) model polystyrene (PS) and (b) model poly(p-methylstyrene) (PpMS) combs.61 by the following nomenclature xk-y-zk where x is the weight-averaged molecular weight of the backbone Mw,bb (kg/mol), y is the average number of branches per backbone nbr, and z is the weight-averaged molecular weight of the branches Mw,br (kg/mol) (see Table 1). Synthesis of Polystyrene Combs. Backbone and Side Chain Polymerization. Anionic polymerization of styrene and the grafting procedures were performed using high-vacuum techniques (p ≤ 10−3 mbar). The ampules and the polymerization reactor were equipped with high-vacuum PTFE stopcocks and ground glass joints. To introduce the reagents, the ampules were mounted directly onto the reactor. The dry solvents were introduced directly from the solvent reservoirs on the vacuum line into the reactor. The polystyrene (PS) backbone and the branches were both prepared by room temperature polymerization of styrene with sec-BuLi as the initiator and toluene as the solvent. For the backbone polymerization the residual anions were terminated using degassed methanol after complete conversion of the monomer. The backbone polymer was then precipitated in methanol, redissolved in THF, and precipitated once more. To remove residual amounts of solvents, the polymer was dried under vacuum at 50 °C. The polymerization of the branches was performed in an ampule with high-vacuum PTFE stopcocks and ground glass joints. After completion of the branch polymerization, a small sample was removed from the ampule using a syringe and then terminated with degassed methanol for the branch molecular weight and polydispersity characterization. Functionalization of the Polystyrene Backbone, Acetylation. To achieve a small number of branching points ( 0.75 for the molecular weight range (200−275 kg/mol) of the investigated backbones. Using tan δ as a function of frequency (Figure 4), the effect of backbone dilution was more clearly observed. As shown in Figure 4, the rubbery plateau attributed to backbone relaxation is associated with the minimum in tan δ occurring at low frequencies while the minimum in tan δ at high frequencies is related to branch relaxation. The low frequency tan δ minimum depended on the volume fraction of the branches ϕbr, specifically that the minimum in tan δ occurred at lower frequencies for lower values of ϕbr. For example, this correlation is seen in Figure 4a when comparing the data for PpMS197k-29-15k (ϕbr = 0.69) and PpMS197k-14-15k (ϕbr = 0.52) and also in Figure 4b when comparing PpMS197k-14-42k (ϕbr = 0.75) and PS275k-5-42k (ϕbr = 0.52). However, the relation between tan δ and ϕbr only held up to a certain upper limit of ϕbr because then the Rouse-like behavior of the backbone dominates and tan δ is equal to 1. Rouse-like behavior is displayed in Figure 4b for samples PpMS197k-1442k and PS275k-29-47k, which both had tan δ = 1. Therefore, by using the tan δ as a function of frequency plot, it is not possible to distinguish between their different values for ϕbr. Reduced van Gurp−Palmen Plot. Because of the different relaxation times, comparing polymers with different architectures and, as in this study, different chemistries is a complex multivariable problem. To circumvent this issue, the reduced van Gurp−Palmen plot (red-vGP plot)24−26 was used to aid the data analysis. In the reduced van Gurp−Palmen plot, the tan δ is plotted versus the complex modulus |G*|, which is normalized to the plateau modulus and therefore independent

methyl group adjacent to the carbonyl group, which leads to inactive branching points.63 For the backbones with only two acetyl groups the grafting was quantitative according to SECMALLS measurements. Linear Viscoelastic Data under Small Amplitude Oscillatory Shear. Linear Master Curves. The investigated comb polymers had side chain branches that ranged from slightly to entangled, while the backbone was highly entangled in all cases. The dimensionless number of entanglements is defined as sbr = Mw,br/Me for the number of branch entanglements and as sbb = Mw,bb/Me for the number of backbone entanglements. The values for sbr and sbb as well as the entanglement molecular weight for the different polymers used are given in Table 2. Figure 3 presents the linear viscoelastic data for a series of linear and comb homopolymer melts of PS and PpMS. The

Figure 3. Master curves generated from linear viscoelastic data for PS and PpMS combs.

sample PpMS195k demonstrated the typical behavior for a monodisperse linear polymer with only one rubbery plateau as a result of the entanglement of the polymer backbones. For comb polymers, two rubbery plateaus were instead found where these plateaus correspond to the entanglement of the branches at higher frequencies and, at lower frequencies, the entanglement of the diluted backbone aided by the relaxed branches.16 Figure 3 compares results for comb polymers that have similar branch length, but a varying number of branches. For example, PpMS197k-14-15k and PpMS197k-29-15k had an identical branch molecular weight (15 kg/mol) that was slightly below the entanglement molecular weight (Me ≈ 20.3 kg/ mol61), but the number of branches differed by a factor of 2 (14 and 29 branches). However, the combs PpMS197k-14-42k, PS275k-29-47k, and PS275k-5-42k each had a branch molecular weight of above 42 kg/mol and were entangled, but here the number of branches were 5, 14, and 29 branches, respectively. In contrast to the other combs, PS263k-2-17k and PS263k-2-37k both had a branching degree nbr = 2. The molecular weight of the branches differed from slightly entangled (17 kg/mol) to entangled (37 kg/mol) (Me(PS) = 17.3 kg/mol64). For these two combs, two rubbery plateaus were not determined; unlike results from other combs, they had only the rubbery plateau associated with the entanglement of the backbone. This result was also seen by Hepperle et al.,18 who investigated polystyrene combs of higher polydispersity with 0.5−4 branches per backbone with slightly to wellentangled branches and found one rubbery plateau when the F


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Figure 5. Reduced van Gurp−Palmen plot of PS and PpMS combs. The data of the curves are reduced to two characteristic points, which correspond to the two minima in the plot. The two characteristic points are described by the two values P1 and P2 (exemplarily shown for sample PS275k-29-47k) which correspond to the value of the phase angle at the first and second minimum and the two values G1 and G2 for the reduced complex modulus at the first and second minimum.

G2 (Gred ≈ 0.03) corresponding to backbone relaxation. An indication for the correlation between the first minimum of tan δ and the volume fraction of the branches ϕbr was found in the tan δ as a function of frequency plot (Figure 4). The absolute values of tan δ or the phase angle (P2), in the case of the reduced van Gurp−Palmen plot, gave only indications for the influence of ϕbr for combs with similar backbone and branch molecular weight, but different branching degree. Instead, the comparison of the second corresponding value G2 (the reduced complex modulus of the first minimum) with ϕbr led to a linear relationship between the two values, where the volume fraction of the branches ϕbr was linearly decreasing with increasing G2 (Figure 6). This can be explained in terms of the dynamic

Figure 4. Linear viscoelastic data for tan δ as a function of angular frequency ω for PS and PpMS combs.

of the molecular weight. The information obtained from conventional master curves is retained, but due to the temperature and time invariance of this plot, temperature differences between experiments and changes in relaxation times are compensated.72 In contrast to the linear rheological master curve shown in Figure 3, the different polymers and architectures were directly comparable and differences between the branching degree and branch molecular weight could be elucidated in the reduced van Gurp−Palmen plot (Figure 5). To facilitate the analysis of the van Gurp−Palmen plot, the amount of data is reduced to two characteristic points, which correspond to the two minima in the plot. Those characteristic points are obtained by the intersection of two tangent lines through the inflection points enclosing the minimum; for clarity, only one example for the determination of the characteristic points is shown in Figure 5. The two characteristic points are described by the two values P1 and P2 which correspond to the value of the phase angle at the first and second minimum and the two values G1 and G2 for the reduced complex modulus at the first and second minimum, respectively. As in the tan δ as a function of frequency plot (Figure 4), there were two minima in the phase angle data: one at G1 (Gred ≈ 1) corresponding to the branch relaxation process and a second one at a lower value of the reduced complex modulus at

Figure 6. Linear dependence of the branch volume fraction ϕbr on the reduced complex modulus G2, which corresponds to the value of the first minimum in the reduced van Gurp−Palmen plot and is associated with backbone relaxation. The different trends for combs with wellentangled and slightly entangled branches are illustrated by the dashed lines connecting samples with similar branch molecular weight Mw,br.

dilution concept. As ϕbr increases, the dilution effect and the number of entanglements also increase. Therefore, |G*| and G2 both decrease as G0N, used for normalization, is assumed to be constant for the same monomer type. The reduced van Gurp−Palmen plot was used to compare the combs with only two branches (PS262k-2-17k and PS262k2-37k) with the linear backbone (PS262k) (Figure 7). The comb with slightly entangled branches (PS262k-2-17k) G


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Figure 7. Reduced van Gurp−Palmen plot for two combs with two branches each (PS262k-2-17k: Mw,br = 17 kg/mol; PS262k-2-37k: Mw,br = 37 kg/mol) in comparison with the linear backbone. Only the comb PS262k-2-37k with entangled branches shows a deviation from the behavior of the backbone. This is the result of the dynamic dilution of the backbone, leading to a decrease in phase angle at low Gred values.

followed the behavior of the linear backbone. In the case of the comb with entangled branches (PS262k-2-37k) the phase angle value decreases at low reduced complex moduli, which is due to the retraction of the branches and consequent dynamic dilution. The reduced van Gurp−Palmen plot can be useful to distinguish between branched and linear topologies even for very low branching degrees. Nonlinear Data under Large Amplitude Oscillatory Shear. Fourier Transformation (FT)-Rheology and Nonlinear Parameter Q0. Measurements in medium (MAOS) and large amplitude oscillatory shear (LAOS)44 were performed in straincontrolled mode by conducting strain sweep experiments in the nonlinear regime at various frequencies and temperatures. The obtained stress data were converted from the time to the frequency domain using FT-rheology.40,44 The relative intensity of the third harmonic I3/1 = I(3ω1)/I(ω1) (with ω1 as the excitation frequency) was used to quantify the nonlinearity. In the nonlinear regime, I3/1 increases with increasing strain amplitude and exhibits a quadratic scaling relationship as a function of strain amplitude both at small and medium strain amplitude, eq 1.50,59,60 I3/I1 ≡ I3/1 ∝ γ0 2

Figure 8. Nonlinearity parameter I3/1 and Q parameter as a function of the strain amplitude for a poly(p-methylstyrene) (PpMS) comb PpMS197k-14-42k at 190 °C: (a) the strain dependence in the MAOS and LAOS regions, where the quadratic relationship I3/1 ∝ γ02 was fulfilled in the MAOS region (ω1 = 1 rad/s); (b) the quadratic strain amplitude dependence at frequencies varying from 0.5 to 5 rad/s; and (c) the Q parameter (Q = I3/1/γ02) obtained from (b).

Instrinsic Nonlinear MAOS Master Curve. The intrinsic nonlinearity Q0 depends on both frequency and temperature. It was experimentally verified that time−temperature superposition (TTS) using the shift factors obtained from the linear master curve can be applied to generate a new intrinsic nonlinear MAOS master curve for Q0(ω) (see Figure 9). For linear polymers the intrinsic nonlinear MAOS master curve showed only one maximum at Qmax,bb as a result of the backbone relaxation. For branched polymers a second maximum at Qmax,br appeared at higher frequencies, corresponding to the branch relaxation processes. To investigate the influence of the molecular structure, the maxima (Qmax,bb(Q0), Qmax,br(Q0)) and the minimum (Qmin,bb(Q0)) values were analyzed in detail. For example, with increasing molecular weight of the branches Qmax,bb(Q0) decreased (Figure 9a), and when two combs had a similar branch molecular weight and larger number of branches (e.g., PpMS197k-14-42k and PS275k-29-47k) the absolute Qmax,bb(Q0) values were also similar. However, when the number of branches was lower (nbr = 5), but the branch molecular weight was the same, the absolute value for Qmax,bb(Q0) was a factor of 3 higher (see Figure 9b). This means that while the absolute value of Qmax,bb(Q0) was sensitive to the branch molecular weight, it was also a function of the number of branches. The correlation of the number and molecular weight of the branches and

(1)

In this study, I3/1 was investigated as a function of strain amplitude γ0 at frequencies changing from 0.5 to 5 rad/s. In Figure 8a, the dependence of the nonlinearity parameter I3/1 on the strain amplitude is illustrated. In the MAOS region, I3/1 depended quadratically on the strain, while at higher strain amplitudes, it deviated from this behavior and levels off. This quadratic dependency was valid at all frequencies up to 5 rad/s in the MAOS regime (see Figure 8b). As the strain dependence is quadratic, the data can be used to calculate the nonlinear coefficient Q definied as Q = I3/1/γ0 2

(2)

and by extrapolation for low strain amplitudes a nonlinear zero strain value Q0 (the instrinsic nonlinearity) can be achieved

lim Q ≡ Q 0

γ0 → 0

(3) H


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Figure 9. Intrinsic nonlinearity Q0 for PS combs (PS275k-29-47k, PS275k-25-26k, PpMS197k-14-42k, and PS275k-5-42k at Tref = 180 °C for (a) combs with a high number of branches, the two maxima (Qmax,bb(Q0) and Qmax,br(Q0)) displayed are due to backbone and branch relaxation, respectively, where Qmax,bb(Q0) decreased and shifted to lower frequencies as the branch molecular weight increased (b) a comb with a low number of branches (PS275k-5-42k with 5 branches), the ratio between the maxima and the minimum decreased significantly, which is a result of the reduced backbone chain stretch and the reduced dilution of the backbone.

Qmax,bb(Q0) is shown in Figure 10a,b. In each case a linear decrease of Qmax,bb(Q0) can be observed for combs with similar number of branches (Figure 10a) or with similar molecular weight of the branches (Figure 10b). Only combs that vary in the number or molecular weight of the branches deviate from the linear decrease; this difference is indicated by the arrows in Figure 10a,b. Comb polymers with a low amount of branches, such as PS275k-5-42k, seemed almost to behave like linear polymers (Figure 9b). It displayed a slight minimum, which was confirmed by the small differences observed between the maxima and minimum resulting in a scattered curve (Figure 9b). The ratio of Qmax,bb(Q0) and Qmin(Q0) was different for each of the three combs in Figure 9a and can therefore be correlated with the number of branches. For example, when the number of branches was almost tripled from 5 (PS275k-5-42k) to 14 (PpMS197k-14-42k), the value for RQ12 := Qmax,bb(Q0)/ Q min (Q 0 ) was doubled, i.e., RQ 12 (PpMS197k-14-42k)/ RQ12(PS275k-5-42k) = 4.4/1.9 = 2.3. As the number of branches is further increased to 25 (PS275k-25-26k), the RQ12 is increased by a factor of 4, i.e., RQ12(PS275k-25-26k)/ RQ12(PS275k-5-42k) = 4. Figure 10a shows this linear dependence for the value of RQ12 on the number of branches. The Qmin,bb(Q0) value for the PS275k-29-47k comb could not be evaluated accurately due to the data scattering present around the minimum and is therefore not shown. For the combs PS275k-29-47k and PpMS197k-14-42k that have a similar branch molecular weight (Mw,br = 42−47 kg/mol), Qmin(ω) occurred within the same range of frequency (Figure 9a), while for PS275k-25-26k, which had a lower branch

Figure 10. Linear dependency of the Qmax,bb(Q0) value on the molecular weight (a) and number of branches (b) can be found for similar comb structures. The ratio RQ12 between the maximum Qmax,bb(Q0) and the minimum Qmin(Q0) (c) revealed as well a linear correlation with the number of branches for the combs PpMS197k-1442k, PS275k-5-42k, and PS275k-25-26k. The double arrows in (a) and (b) indicate differences in Qmax,bb for similar comb structures, which differ in the number or in the molecular weight of the branches and deviate therefore from the linear decrease in Qmax,bb.

molecular weight (Mw,br = 26 kg/mol), Qmin(ω) was shifted to higher frequencies. However, as already pointed out, the absolute value of Qmax,bb(Q0) is dependent on the number and molecular weight of the branches. The value decreases with increasing number and molecular weight of the branches as a result of dilution. Consequently, the absolute value of Qmax,bb(Q0) can give qualitative information about the degree of LCB. The Qmin(Q0) value is most probably affected by the polydispersity and is therefore only applicable to monodisperse polymers. Comparison of the Linear and Instrinsic Nonlinear Master Curves. An advantage of the intrinsic nonlinear MAOS master curve is the direct access to relaxation times and processes, which cannot be obtained directly from the linear master curve in most cases. To illustrate this, the linear (Figure 11a) and the intrinsic nonlinear MAOS (Figure 11b) master curves for PpMS197k-14-42k are shown in comparison. The first maximum at low angular frequency (point A) in the nonlinear master curve is not in agreement with the crossover point of I


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Table 3. Comb Relaxation Times, T = 180 °C τd,expa [s]

sample PS62k

0.0093

PS275k-5-42k PpMS197-14-42k PS275k-29-47k

14.3 50.8 93.5

τd,theoryb [s]

τR,bb,expa [s]

τbra [s]

0.45 2.69 2.14

0.02 0.14 0.05

b

0.004 0.0093c 51.9 17.6 32.8

a

Determined by the minima and maxima of the instrinsic nonlinear master curve at 180 °C. bDetermined using η0 = τdG0N (G0N: plateau modulus of the diluted backbone). cDetermined using the Likhtman− McLeish model.74

η0 = τd,theoryGN0

The theoretical reptation time τd,theory for the linear polymer was determined using the Likhtman−McLeish model,74 included in the Reptate75 software package used. Since eq 6 is based on the Doi−Edwards theory, the theoretically calculated values are only valid for the case of linear polymer topologies and do therefore not correspond with the experimental values determined from the intrinsic nonlinear master curves. However, for the linear polymer, the reptation time calculated using the Likhtman−McLeish model is identical with the value obtained from the intrinsic nonlinear master curve. The value determined by eq 6 differs by a factor of about 2, which is still comparable with the experimental value. The huge difference between the experimental and the theoretical values of the comb polymers is a result of the restriction of the theoretical approach to linear polymers. The comparability of the theoretical and experimental values of the linear polymer gives besides the pom-pom prediction another confirmation for the accurate determination of the reptation time using the intrinsic nonlinear master curve. Relaxation times can be thereby obtained for complex architectures, which are otherwise not accessible. Qualitative Analysis of the Intrinsic Nonlinear Parameter Q0 in Combination with Pom-Pom Model Simulations. For the consideration of the effect of the comb architecture on the nonlinear rheology, a theoretical model for a specifically branched molecule (the pom-pom topology) was used to analyze the extra stress present.76 A pom-pom molecule is constructed out of a combination of one long backbone segment and identical multiple branches at each end of the backbone segment.77 This is the simplest molecule that contains multiple branches, allowing for segments of the molecule to become “buried” and, thus, causing a hierarchy of relaxation processes. The two dominant relaxation processes are the backbone stretch relaxation and the backbone reorientation. The backbone stretch relaxation time, τs, is the characteristic time for the path length of the backbone to return within the tube to its equilibrium length, while the orientation relaxation time, τd, is the characteristic time of the backbone to reptate out of a tube of unstretched length via branchpoint diffusion along the backbone tube. The backbone stretch relaxation time, τs, and the orientation relaxation time, τd, were determined from the branch disentanglement time. The original differential form proposed by McLeish and Larson76 was improved via the local branch-point displacement proposed as by Blackwell et al.78 The equations were further modified for the backbone reorientation time caused by very fast nonlinear flows proposed by Lee et al.79 which resulted in the revised pom-pom model equations as given by

Figure 11. Comparison between (a) the linear and (b) the intrinsic nonlinear LAOS master curve for PpMS197k-14-42k (Tref = 180 °C). The maxima and minima present in the intrinsic nonlinear master curve corresponded to relaxation times observed in the linear master curve and, thus, experimentally determined relaxation times. The following points are of interest: (A) reptation time τd, (B) backbone Rouse time τR,bb, and plateau modulus G0N,bb, and (C) branch relaxation time τbr.

G′/G″ in the linear master curve, although it was found to correspond with the reptation time τd, which was in agreement with findings of pom-pom model simulations. This result is important for an accurate determination of the reptation time for polymers with complex architectures or high polydispersity, such as LDPE. The minimum (point B) in the nonlinear master curve corresponded to the Rouse time of the backbone τR,bb, which cannot be accurately determined using the minimum of tan δ, due to the overlay of G′/G″, which is the case for highly branched architectures. The second maximum (point C) corresponded to the branch relaxation time τbr, which is in good agreement with the transition point from branch to backbone relaxation in the linear master curve. In the case of a low number of branches or slightly entangled branches the branch relaxation time τbr cannot be determined using the linear master curve, since a crossover point does not emerge. It is likely that a second minimum, corresponding to the branch relaxation process, might exist in the nonlinear master curve. Because of experimental challenges, this frequency region is so far difficult to access. The different relaxation times obtained from the intrinsic nonlinear master curve are listed in Table 3. The experimental reptation times were obtained using the following equations: N

η0 = τd =

∑ Giτi i=1

(4)

12 η0 π 2 GN0

(5)

(6)

The discrete relaxation spectrum parameter set Gi, τi was obtained from the linear master curves using the IRIS73 software. The theoretical reptation time τd,theory was obtained using J


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stress tensor: σ = G0ϕbβ λ 2(t )S (t )

(7)

orientation tensor: S (t ) = A /tr A

(8)

A ̇ − κij·A − A ·κij = −

1 (A − I ) τd*

(9)

stretch: Dλ(t ) 1 = λ(t )κij: S − [λ(t ) − 1] exp(v*(λ(t ) − 1)) Dt τs (10)

time scales: τs =

5 qs bbϕb β − 1τa 2

τd =

75 qs bb 2ϕb 2(β − 1)τa 2π 2

(11)

reversing flow correction: ⎧1 for 1 ≤ λ ≤ q ⎪ ⎪ τd 1 =⎨ τd* ⎪ 1 λ̇ ⎪ τ + λ − κ : S for λ < 1 ⎩ d

(12)

Figure 12. Pom-pom model simulations for the nonlinear parameter Q0: (a) effect of changing the molecular weight of the branches sbr with sbb = 14, q = 3 and (b) effect of the number of branches q with sbb = 14 and sbr = 3.

where sbb (Mw,bb/Me) and sbr (Mw,br/Me) are the dimensionless molecular weights of the backbone and arms, respectively, κij is the deformation rate tensor, with i indicating the flow direction and j the velocity gradient direction, τa is the arm relaxation time, ϕb is the backbone fraction in the pom-pom molecule, the dilution parameter β = 2, and v* = 2/(q − 1); the overdot used in the equations indicates a material derivative.71 The number of branching points q corresponds to a total number of branches of 2q. The prediction of the pom-pom constitutive model in LAOS was examined using the nonlinear parameter Q0 for different model parameters. In Figure 12a, the qualitative behavior of the Q0 parameter with changing molecular weight of the pom-pom branches is shown. The maximum of Q0(ω) was observed at the reptation frequency 1/τd. The maximum value of Q0(ω) decreased as well as the molecular weight of the branches (sbr) increased and was shifted to higher frequencies. In Figure 12b, the behavior of Q0 with increasing number of branches q is shown, where q = 1 refers to the linear topology, which displayed no decrease in Q0(ω) after reaching the maximum. A decrease of the maximum value of Q0(ω) as well as a decrease after reaching the maximum can be observed for the branched architectures with q ≥ 2 in contrast to the linear one. The decrease of Q0(ω) with increasing number of arms as well as the decrease after reaching the maximum of Q0(ω) corresponds with the experimental findings. In the case of the pom-pom molecule the predicted shift of the maximum of Q0(ω) to higher frequencies for increasing branch molecular weight sbr is a result of dilution and the therefore decreasing effective molecular weight of the backbone. The maximum of Q0(ω) for combs is instead shifted to lower frequencies in the experiment, which is a result of the hierarchical relaxation process and the resulting retardation of the backbone relaxation process. From the above analysis of the simulated intrinsic nonlinear master curves, it was concluded that this was a very useful tool for experimentally evaluating the reptation time (τd). Furthermore,

an increase in the number of branches q or in the molecular weight of the branches sbr resulted in a decrease in the separation between the two time scales and thus in a decrease in the Q0 maximum as stretching of the polymer backbone chain became dominant. Extensional Rheology. Uniaxial extensional experiments were performed both on polystyrene and PpMS-based comb structures with different number and molecular weight Mw,br of the branches. Of particular interest was the resulting strain hardening behavior and its relationsship to the molecular structure of the combs. Quantitative data on the effect of strain hardening were obtained via the strain hardening factor (SHF), which is defined as the ratio of the steady-state extensional viscosity ηE(t,ε), ̇ obtained from the experimental data, over the Doi−Edwards (DE) prediction for η+E(DE), which reflects the behavior of a linear polymer chain: ηE+(t , ε)̇

SHF =

ηE+(DE)

(13)

The following equation was used for the Doi−Edwards (DE) transient uniaxial viscosity: t

σ=

∫−∞ dt′m(t − t′)Q [E(t , t′)]

(14)

with the memory function m(t−t′) given by ∞

m(t − t ′) =

∑ i=1

Gi exp[−(t − t ′)/τi] τi

(15)

For the nonlinear strain measure Q[E(t,t′)] the Currie approximation was used: K


Macromolecules ⎤ ⎡ ⎛ 5 ⎞ 5 ⎥C Q≈⎜ ⎟B − ⎢ ⎝ J − 1⎠ ⎣ (J − 1)(I2 + 13/4)1/2 ⎦

Article

(16)

The transient extensional viscosity at the curve maximum η+E,max was used to compare the different samples, since a steadystate extensional viscosity cannot be obtained by the extensional technique used. The results for the transient extensional viscosity as a function of the extensional time are illustrated in Figures 13−15; an overview of the strain hardening factors for

Figure 15. Extensional rheology measurements for combs with two branches, but varying molecular weight of the branches. The filled spheres represent the experimental data, the unfilled spheres represent the Doi−Edwards (DE) prediction, and the dashed line indicates the linear viscoelastic (LVE) prediction (T = 180 °C).

Figure 13. Extensional rheology measurements for combs that were identical except for the number of branches; Mw,bb and Mw,br were almost identical. The filled spheres represent the experimental data, the unfilled spheres represent the Doi−Edwards (DE) prediction, and the dashed line indicates the linear viscoelastic (LVE) prediction (T = 180 °C).

Figure 16. Behavior of the strain hardening factor depending on Hencky strain rate ε̇H for combs with slightly entangled branches. The SHF increases with ε̇H and the number of branches (T = 180 °C).

Figure 14. Extensional measurements for combs that have a different branch molecular weight, but an identical backbone molecular weight (Mw,bb = 197 kg/mol) and number of branches (nbr = 14). The molecular weight of the branches ranged from slightly entangled (Mw,br = 15 kg/mol) to entangled (Mw,br = 42 kg/mol). The filled spheres represent experimental data, the unfilled spheres represent the Doi− Edwards (DE) prediction, and the dashed line indicates the linear viscoelastic (LVE) prediction (T = 180 °C).

Figure 17. Strain hardening factor depending on Hencky strain rate ε̇H for combs with entangled branches. An increase of the SHF with the branching degree can be observed for ε̇H < 0.3 s−1. While for ε̇H > 0.3 s−1, the corresponding Rouse time of the backbone τR,bb is exceeded and branch relaxation occurs. A decrease or constant value of the SHF is observed in this case (T = 180 °C).

the different combs is given in Figures 16 and 17. The accuracy of the extensional measurements was tested by comparing the experimental data with its corresponding linear viscoelastic prediction, which is shown as a dashed line in Figures 13−15. Both the LVE and DE predictions were obtained from dynamic-mechanical data using self-written MATLAB programs. The discrete relaxation spectrum parameter set Gi, τi was obtained from the linear master curves using the IRIS software.

Influence of the Number of Branches. The influence of the average number of grafted side chains on the strain hardening behavior was investigated using a series of combs that were almost identical except that the number of branches increased from 5 to 14 and, finally to 29 (Figure 13). In Figure 13, it is L


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for detectable strain hardening. The lowest number of branches, introduced via synthesis within our investigation, was on average two branches per backbone (Figure 15). The branch molecular weight was chosen, as in the other samples, from slightly entangled (17 kg/mol) to entangled (37 kg/mol). In contrast to the other combs, only the highest extension rate of 3 s−1 showed even a slight strain hardening effect (SHF ≈ 1.4). The reason for this low limit is probably due to the underlying molecular topology as a two arm comb can be a mixture of asymmetric stars, H-polymers, and combs. However, star polymer topologies cannot contribute to strain hardening due to their lack of backbone chain stretch.80,81 It is also possible that the number of branches was too low to reach a sufficient number of entanglements and efficient backbone stretch. Strain Hardening Behavior of Combs with Slightly Entangled Branches (Mbr ≈ Me). In Figure 16 the strain hardening factors for combs with slightly entangled branches PpMS197k-14-15k and PpMS197k-29-15k and for the combs with low number of branches PS262k-2-17k and PS262k-2-37k at different Hencky strain rates ε̇H are shown. Those combs show a linear increase of the SHF with increasing Hencky strain rate ε̇H and increasing number of branches. Strain Hardening Behavior of Combs with Entangled Branches (Mbr > Me). The combs with entangled branches PpMS197k-14-42k, PS275k-29-47k, and PS275k-5-42k show also an increase in the SHF with increasing number of branches, but only for ε̇H < 0.3 s−1 (see Figure 17). As soon as the Hencky strain rate ε̇H of about 0.3 s−1 is exceeded, a decrease or a plateau for the SHF can be found. In contrast to the combs with slightly entangled branches the decrease in SHF for entangled branches is more pronounced the higher the number of branches. This might be a result of the reduced chain stretch, due to the occurrence of branch relaxation for extension rates ε̇H ≥ 0.3 s−1, which are similar to or higher than the inverse backbone Rouse time τR,bb−1. For combs with slightly entangled branches the inverse backbone Rouse time τR,bb−1 is not exceeded for the applied Hencky strain rates, and therefore a decrease in SHF cannot be observed. On the other hand, this might explain the occurrence of a pronounced maximum of strain hardening with subsequent downturn of the transient extensional viscosity η+E for high ε̇H, where the extension time is closer to the backbone Rouse time τR,bb and the branch relaxation takes place. For extension times lower than the backbone Rouse time τR,bb a plateau at steady state or even no maximum is reached instead. This phenomenon was already observed for pom-pom molecules measured by Nielsen et al.82 Similar maxima were reported for LDPE melts by Wagner et al.83 and Rasmussen.84 The origin of the occurrence of a maximum in η+E is still under discussion. Wagner et al.85 explained the observed downturn after reaching the maximum in η+E for a pom-pom molecule by the hypothesis that after retraction of the branches into the tube of the backbone the pom-pom behaves like a linear polymer, due to the loss of efficient entangled branches. The implementation of the branch point withdrawal in the MSF model the predictions corresponded well with the experimental data. However, recent findings of Burghelea et al.86 gave indications, that the maximum in η+E is related to inhomogeneities of the deformation. Wagner et al.85 also noted that a high level of strain hardening can be achieved with a high number of branches, even if the branches are too short to be entangled. This

clearly seen that the absolute values of the maximum extensional viscosity using the highest Hencky strain rate (ε̇H = 3 s−1) were nearly identical for the two polystyrene combs PS275k-5-42k and PS275k-29-47k, ηE (PS275k-5-42k, ε̇H = 3 s−1) = 648 kPa·s and ηE(PS275k-5-42k, ε̇H = 3 s−1) = 656 kPa·s, even though the number of branches differed by a factor of 6. However, the comb with 14 branches (PpMS197k-14-42k) had a slightly higher value for ηE, ηE(PpMS197k-14-42k, ε̇H = 3 s−1) = 725 kPa·s, but was still comparable to the other two combs. A decrease in the Hencky strain rate ε̇H caused the maximum + extensional viscosity ηE,max to increase for combs with comparable values. In general, when comparing the three combs in Figure 14, the comb with the least number of branches PS275k-5-42k (5 branches) also had the lowest degree of strain hardening (SHF ≈ 4). However, when the number of branches increased by a factor of about 3 (PpMS197k-14-42k, 14 branches), the strain hardening factor increased 3-fold (SHF ≈ 12), but further increase in the number of branches (PS275k-29-47k, 29 branches) did not result in a further large increase in the SHF. The SHF was comparable for PpMS197k-14-42k and PS275k-29-47k at high values for the Hencky strain rate (ε̇H = 3 s−1) as mentioned before. In contrast, the ratio between the amount of strain hardening for PS275k-29-47k and PpMS197k-14-42k did increase as the Hencky strain rate decreased. In general, for this series of combs with slightly entangled branches (Mw,br ≈ 15 kg/mol), the absolute transient extensional viscosity increased as the Hencky strain rate was increased. The highest SHF was recorded for PpMS197k-29-15k (29 branches) at the highest Hencky strain rate used (SHF = 19.6 at ε̇H = 3 s−1). From the above results, it was concluded that an increase in the number of branches did not have a large influence on the absolute transient extensional viscosity but did affect the amount of strain hardening, which increased significantly up to a critical number of branches (∼14) and then leveled off. Influence of the Branch Molecular Weight. The influence of the branch molecular weight on the transitional extensional viscosity and the SHF is shown in Figure 14 by the comparison of two combs that have a similar backbone molecular weight and the same number of branches, but the molecular weights of the branches were designed to be slightly below the entanglement molecular weight (Mw,br = 15 kg/mol for PpMS197k-14-15k) or entangled (Mw,br = 42 kg/mol for PpMS197k-14-42k). The absolute maximum transient extensional viscosity of PpMS197k-14-15k (lower branch molecular weight) was lower, ηE(PpMS197k-14-15k, ε̇H = 3 s−1) = 371 kPa·s, than that of PpMS197k-14-42k, ηE(PpMS197k-14-42k, ε̇H = 3 s−1) = 725 kPa·s. The SHF showed also a lower value for PpMS197k-14-15k (SHF ≈ 8.8) than for PpMS197k-14-42k (SHF ≈ 12); the SHF overall increased with increasing extension rate. As the branch molecular weight for PpMS197k-14-15k was below the critical molecular weight and slightly below the entanglement molecular weight, the reduced entanglement most likely caused the observed decrease in both strain hardening and viscosity when compared with results for PpMS197k-14-42k (Mw,br > Mc). Lower Limit for the Number of Branches to Determine Strain Hardening. Based on the results shown in Figure 13, there was a upper limit beyond which increasing the number of branches did not increase strain hardening. This was in the range between 14 and 29 branches. Therefore, it is likely that there exists a lower limit to the number of branches required M


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corresponds to the findings for the comb polymers with slightly entangled branches PpMS197k-14-15k and PpMS197k-29-15k, where the strain hardening increases with increasing number of branches (PpMS197k-14-15k: SHF = 8.8 at ε̇H = 3 s−1; PpMS197k-29-15k: SHF = 19.6 at ε̇H = 3 s−1). It can be concluded that the introduction of a high number (nbr ≥ 1%) of slightly entangled (Mbr ≈ Me) branches can result in a higher strain hardening behavior than in the case of a low number of long-chain branching. In the case of the combs with slightly entangled branches PpMS197k-14-15k and PpMS197k-29-15k, the extension rate is below the reptation rate τd−1 at low ε̇H < 0.3 s−1. Though it should be assumed that for ε̇H < τd−1 the extension rate is too low to significantly affect the chain configuration, and the uniaxial extensional viscosity should consequently correspond to the linear viscoelastic behavior; this is not the case and strain hardening occurs. From previous results, it can be concluded that by an increase of the number of branches the amount of strain hardening increases until an upper limit for the number of branches is reached. This is the case for the combs with entangled branches, where the limit of SHF is reached for a number of branches in the range from 14 to 29. For combs with slightly entangled branches a limit in SHF could not be observed for the range from 14 to 29 branches. By lowering the number of branches a lower limit for strain hardening can be reached, where strain hardening cannot be experimentally observed anymore. The branches need to be at least slightly entangled (≈Me) to contribute to the strain hardening and the SHF increases with the molecular weight of the branches for high extension rates. The strain hardening behavior is caused by the restricted stretching of the backbone between the branch points connecting the branches.87 It can be assumed that for only two branches the stretching of the backbone is not effective to result in strain hardening behavior. The molecular weight of the intersection between two branches plays a role. In the case where the intersection molecular weight is too high, it cannot be effectively stretched. For the two arm comb a slight emerging of strain hardening could already be observed at high extension rates, while already with five branches a pronounced strain hardening occurs. The number of branches when the transition from linear to strain hardening behavior takes place should therefore be in the range between two and five branches, in case the branches are substantially entangled. Qualitative Analysis of Nonlinear Elongation Rheology in Combination with Pom-Pom Model Simulations. The pompom constitutive model predictions for extensional rheology were examined with particular focus on the strain hardening factor (SHF) behavior as a function of increasing number and molecular weight of the branches in the pom-pom molecule. With an increasing number of branches SHF increased for a constant Hencky strain rate ε̇H (see Figure 18a). The SHF increased as well for increasing Hencky strain rates for constant numbers of branches q. The increase in the SHF with ε̇H at constant q was explained by the first term in the stretching equation λκ·S or λε̇(S11 − S22), which shows that, with increasing ε̇H, a large chain stretch can be sustained at steady state. When the number of branches was increased, the separation between the two time scales was reduced and the stretching time τs increased, causing the pom-pom molecules to stretch earlier. In Figure 18b, the pom-pom branch molecular weight was instead changed, resulting in a decrease in the SHF

Figure 18. Pom-pom model simulations for SHF as a function of (a) variation of the number of branches q with sbb = 14, sbr = 3 and (b) variation of the molecular weight of the branches sbr with sbb = 14, q = 3.

as the branch molecular weight increased. This effect is caused by a decrease in the backbone fraction of the molecule as the branch molecular weight increases, resulting in a decrease in τs, which causes the SHF to decrease as τs is directly proportional to the backbone fraction ϕb.

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CONCLUSION The goal of the present study was to correlate the molecular architecture of polystyrene (PS) and poly(p-methylstyrene) (PpMS) comb homopolymer melts with their linear and nonlinear rheological behavior in shear and elongation. The proposed synthesis methods yielded well-defined comb structures with low polydispersities (PDI < 1.1) and a low amount of branches (0.1−1 mol %) with molecular weights in the range between 15 and 42 kg/mol. In the linear viscoelastic regime, the reduced van Gurp−Palmen plot was applied to the data obtained in small amplitude oscillatory shear (SAOS) experiments. A linear correlation of the volume fraction of the branches with the value of the reduced modulus, corresponding to the minimum of the backbone relaxation, was found. Even for low branching degrees (two branches per backbone) a deviation from the behavior of linear polymers could be determined. Using the nonlinear parameter Q0 obtained via large ampliude oscillatory shear (LAOS) measurements in combination with Fourier transformation (FT)-rheology, an intrinsic nonlinear master curve was created and correlated with the relaxation processes deduced from the linear master curve. From the intrinsic nonlinear master curve, different relaxation N


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Table 4. Advantages, Disadvantages, and Detection Limits for the Applied Rheological Methods Regarding the Investigation of Comb Polymer Topologies method SAOS, redvGP

advantage low amount of sample necessary

disadvantage

detection limit

medium time exposure (a few hours)

data already available from SAOS MAOS, FT

low amount of sample necessary

elongation

determination of nonlinear simulation parameters Q0(ω) investigation of strain hardening

time consuming (several hours)

higher amount of sample necessary in contrast to shear measurements

easy to perform (for the case of high melt viscosity polymers) low time exposure (several minutes)

differentiation between linear and branched topology for >2 branches (if entangled) semiquantitative determination of the branch volume fraction ϕbr differentiation between linear and branched topology for >2 branches (if entangled) qualitative determination of the branching degree high sensitivity toward branching, detection of >2 branches (at least slightly entangled) qualitative determination of the branching degree using the strain hardening factor for comparison

backbone molecular weights in the range from 200 to 300 kg/ mol). The molecular weight of the branches needs to be at least slightly entangled (Me < Mw,br < Mc) to contribute to the strain hardening. The SHF increases with the molecular weight of the branches. The comb polymers with slightly entangled branches (Mw,br ≈ 15 kg/mol) showed a linear increase of the strain hardening behavior with the extension rate, while no decrease was observed for the extension rates applied. For combs with entangled branches (Mw,br > 40 kg/mol) the SHF levels off with extension rates exceeding the inverse Rouse time of the backbone, due to branch relaxation. To reach substantial strain hardening (SHF > 5) slightly entangled (Me < Mw,br < Mc) branches between 5 and 10 branches per backbone (Mbb ≈ 15− 20 × Me) are necessary. The advantages, disadvantages, and detection limits of the herein presented rheological methods (master curves obtained in small amplitude oscillatory shear (SAOS), reduced van Gurp−Palmen plot (red-vGP), extensional rheology, and large amplitude oscillatory shear (LAOS) in combination with Fourier transformation (FT)-rheology for the investigation of comb polymer topologies are listed in Table 4.

times (reptation time, Rouse time of the backbone, and the branch relaxation time) were directly extracted from the corresponding maxima and minimum. It was found that the reptation time extracted from the nonlinear master curve did not correspond to the crossover point of G′ and G″ in the linear master curve in the case for branched polymers. The correspondence of the reptation time with the maximum Qmax,bb(ω) was confirmed via pom-pom model simulations for branched polymers. It can be concluded that the reptation time can be extracted from the nonlinear master curve in contrast to the values obtained from the linear measurement data. Therefore, reptation times and relaxation times (Rouse time, branch relaxation time) can be obtained using the nonlinear master curve, even if those times are not accessible from the linear data. The experimental accessibility of relaxation times clears the way for a better physical understanding of the underlying relaxation processes and can also be used to improve linear and nonlinear rheological modeling. Using the maximum Qmax,bb(Q0) in the nonlinear master curve, even branched polymers with a small number of branches (here in the case of two branches) could be distinguished from the linear polymer topology. A linear dependency of the Qmax,bb(Q0) value with the number of branches was found for comb polymers with similar molecular weight of the branches and respectively for the molecular weight of the branches for combs with similar number of branches. Comparing the different rheological measurement techniques, it can be concluded that this technique is highly sensitive to determine even low degrees of branching and qualitative correlations can be established. From uniaxial extensional measurements it was found that the number as well as the molecular weight of the branches have an influence on the strain hardening behavior of branched polymers. The strain hardening factor (SHF) increases linearly with increasing number of branches. After a certain number of branches a maximum is reached, the number of branches has then only a low influence on the strain hardening. This transition appeared for combs with the number of branches in the range from 14 to 29, with molecular weights above the critical molecular weight (Mw,br > Mc, with Mc ≈ 35 kg/mol). The lowest amount of strain hardening was determined for a comb with only two branches per backbone, where slight strain hardening can only be observed for the highest extension rate (ε̇H = 3 s−1) applied. The number of branches, when the transition from slight to extensive strain hardening takes place, is in the range between 2 and 5 branches (for branch molecular weights in the range from 15 to 40 kg/mol) per backbone (for

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ASSOCIATED CONTENT

S Supporting Information *

Figure 19; Tables 5−8. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.W.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the German Research Foundation (DFG) (grant WI 1911/6-1) is gratefully acknowledged. M.K. thanks Prof. Dimitris Vlassopoulos (University of Crete) for the kind hospitality and support during his research stay at FORTH as well as for providing samples of the Roovers series, Prof. Kyu Hyun (University of Pusan) for fruitful discussions, and Dr. Jennifer Kübel for proofreading the manuscript and fruitful discussions.

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REFERENCES

(1) Gahleitner, M. Prog. Polym. Sci. 2001, 26, 895−944.

O


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