Linear and Extensional Rheology of Model Branched Polystyrenes

Publication Date (Web): July 18, 2017. Copyright © 2017 American Chemical Society ... A Personal Perspective on the Past and the Future. Macromolecul...
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Linear and Extensional Rheology of Model Branched Polystyrenes: From Loosely Grafted Combs to Bottlebrushes Mahdi Abbasi, Lorenz Faust, Kamran Riazi, and Manfred Wilhelm* Institute of Chemical Technology and Polymer Chemistry, Karlsruhe Institute of Technology (KIT), Engesserstraße 18, 76131 Karlsruhe, Germany S Supporting Information *

ABSTRACT: Monodisperse comb polystyrenes (comb-PS) with loosely to densely grafted architectures up to loosely grafted bottlebrush structures were synthesized via anionic polymerization. This comb-PS series, named PS290-Nbr-44, had the same entangled backbone, Mw,bb = 290 kg/mol, corresponding to a number of entanglements along the backbone Zbb ≅ 20, and similar branch length, Mw,br ≅ 44 kg/ mol or Zbr ≅ 3, but varied in the number of branches per molecule, Nbr, from 3 to 190 branches. Consequently, the average number of entanglements between two consecutive branch points along the backbone (branch point spacing), Zs, ranged from well entangled, Zs ≅ 5, to values that were far less than one entanglement, Zs ≅ 0.1. Linear viscoelastic data including the zero-shear rate viscosity, η0, diluted modulus, G0N,s, and a new diluted modulus extracted from the van Gurp−Palmen plot, |G*| at δ = 60°, were analyzed as a function of the Mw of the combs. Scaling of η0 versus Mw revealed three different regions for increasing Nbr or decreasing Zs: (1) loosely grafted combs with Zbr < Zs and η0 ∼ exp(Mw), (2) densely grafted combs with 1 < Zs < Zbr and η0 ∼ Mw−3.4 followed by η0 ∼ Mw−1 for 0.2 < Zs < 1, and (3) loosely grafted bottlebrushes with Zs < 0.2 and η0 ∼ Mw5. The relative maximum in η0 corresponded to a comb-PS with Zs ≅ Zbr, and the relative minimum resulted from a comb-PS with Zs ≅ 0.2, which displayed almost the same η0 as the linear PS290. Strain hardening factors, SHF ≡ ηE,max/ηDE,max, measured in extensional experiments increased with increasing Nbr and reached SHF > 200 for Hencky strains below εH = 4, which is tremendously high and has to the best of our knowledge not been observed yet. Such a high strain hardening is of great fundamental and technical importance in extensional processes, e.g., foaming, film blowing, or fiber spinning.

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Kempf et al.6 synthesized model polystyrene-based combs, 195 < Mw,bb < 275 kg/mol and 15 < Mw,br < 47 kg/mol, with a different number of branches, 2 < Nbr < 29. They found that the strain hardening increased linearly with Nbr and that a maximum SHF ≈ 20 was reached for Nbr = 29. However, they did not investigate combs with a higher number of branches per molecule. Rolón-Garrido and Wagner20 analyzed the extensional viscosity data of grafted brushlike polystyrenes synthesized by the macromonomer technique.21 They concluded that strain hardening strongly increased with Nbr regardless of the branch length. However, the dispersity index of their backbones, 80 < Mn,bb < 160 kg/mol, was rather high at Đ = 1.5 ± 0.1, and the number of branches per backbone was low and varied between 0 and 6.7. Liu et al.22 synthesized well-controlled ultrahigh molecular weight (Mw = 4900 kg/mol, Đ = 1.5) long chain branched (Mw,br = 140 kg/mol and Nbr = 16) comb polystyrenes with equal spacing between two next-neighbor branch points. As a

INTRODUCTION Extensional and shear rheological properties of branched polymer melts are of special interest to academia and industry due to both the strain hardening and the dynamic dilution effects of the branches.1 Linear and nonlinear rheology of wellcharacterized model branched polymers with architectures ranging from star2,3 and H-shaped4 polymers to comb5−9 and pom-pom4,10 structures and even model Cayley-tree topologies11−13 have been investigated. Roovers14 synthesized well-characterized comb polystyrenes (comb-PS), which were then studied by rheology in the linear5,15 and nonlinear16−19 deformation regimes in shear and elongation. Lentzakis et al.8,9 analyzed the uniaxial extensional data of these samples within the framework of the pom-pom model. These comb-PS systems all had a similar number of branches per molecule, 26 ≤ Nbr ≤ 30, where the backbone weight-average molecular weight, Mw,bb, was either 275 or 860 kg/mol and the molecular weight of the branches, Mw,br, was varied from 6.5 kg/mol (unentangled) to 47 kg/mol (entangled). They concluded that increasing both the molecular weight of the backbone segments between two consecutive branch points (branch point spacing), Ms = Mw,bb/(Nbr + 1), and Mbr resulted in more pronounced strain hardening as well as a faster onset of strain hardening. © XXXX American Chemical Society

Received: May 17, 2017 Revised: July 7, 2017

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DOI: 10.1021/acs.macromol.7b01034 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules result of the high backbone molecular weight, Zbb = Mw,bb/Me ≈ 190, and long chain branches (LCB), Zbr = Mw,br/Me ≈ 10, a high level of strain hardening (ca. SHF ≈ 100) in uniaxial extension was reached. However, the zero-shear rate viscosity, η0, of the sample at 150 °C was more than 109 Pa·s (could not be reached in the SAOS data), which is too high for industrial melt processing applications, but might be suitable for solution processes, e.g., electrospinning or blending. Inkson et al.23 derived an analytical equation to evaluate η0 for monodisperse comb polymers with different numbers of entanglements along the branches, 2 < Zbr < 10, and backbones, 10 < Zbb < 100, and different numbers of branches per molecule, 6 < Nbr < Zbb, using the reptation theory and considering the contour length fluctuations (CLF or primitive path fluctuations) and the dynamic dilution effect with a dilution exponent α = 4/3. However, the number of branches per backbone was always less than the number of entanglements in the backbone, so that the number of entanglements between the branch points, Zs, was higher than one entanglement, Zs > 1. They concluded that η0 increased exponentially with the branch length but exhibited a quadratic dependence on the backbone length. They also found that an increase in the number of branches, Nbr, reduced η0 due to the dilution effect. A detailed analysis of the simulations in Figure 7c of ref 23 demonstrated that the η0 of a comb polymer, η0,comb, with Zs < 2 lies below the η0 of its linear counterpart. In other words, the scaling of the molecular weight dependence of η0,comb was less than the value of 3.4 obeyed by linear polymers when Zs < 2. Janzen and Colby24 showed that the viscosity of loosely (sparsely) branched polyethylene increases significantly as the distance between branch points becomes much longer than the entanglement molecular weight, Zs > 9, and that the viscosity of densely (highly) branched combs is lower than their linear counterparts. Bottlebrush architectures are comblike polymers with a very high density of branches, usually incorporating up to one grafted side chain onto every backbone repeat unit resulting in a distance between branch points that is much less than one entanglement, e.g., Zs = 0.01. These bottlebrush polymers have a higher entanglement molecular weight, Me ∼ LeD2, compared to linear polymers because of the increasing tube length (Le) and tube diameter (D) of the entangled strands due to their very high branch density.25 For instance, poly(n-butyl acrylate) bottlebrushes with different side chain lengths and similar backbones have an entanglement molecular weight Me ≅ 105− 107 g/mol, which is 5−500 times larger than the corresponding value for the linear topology, Me = 2 × 104 g/mol.26 The linear rheology of bottlebrush polymers synthesized by ring-opening metathesis polymerization (ROMP) with polynorbornene as the backbone and polylactide27 or polyolefin28 as the side chains was previously studied. By increasing the degree of polymerization using a macromonomer technique, the length of the backbone and number of side chains were increased simultaneously. Dalsin et al.28 concluded that bottlebrush polymers with entangled side chains showed a rubbery plateau region due to the formation of entanglements among the crowded branches. However, bottlebrushes with unentangled side chains did not show such a plateau region, despite their high molecular weight Mw = 1500 kg/mol, indicating that there was no side chain or backbone entanglement. These bottlebrushes had different chemical repeating units in the backbone, i.e., polynorbornene, than that of the side chains, i.e.,

polypropylene, which might be very complicated for fundamental molecular theories due to the possibility of microphase separation in the melt state as described by the Flory−Huggins theory. Up to now, there has been no published rheological study on model comb polymers where the same backbone is used for all combs, but there is a systematic change in the number of entangled branches per backbone to cover a wide range of branch point spacing, e.g., 0.1 < Zs < 5. In other words, there is a lack of information and rheological data to quantify the transition from a comb to a bottlebrush configuration even though coarse-grained molecular dynamics simulations and scaling analysis were studied.29 Increasing the number of branches increases the number of points where there is friction between the backbone strand and the neighboring chains. This friction enhances stretching of the backbone sections under deformation and induces strain hardening in extensional deformations. However, increasing the side chains on the backbone above a certain number, which depends on the length of the side chains and backbone, will decrease the linear viscoelastic properties by dilution effects and reduce the modulus. Because of these opposing factors, the question remains as to whether a bottlebrush or comb with a very high number of entangled branches (Nbr ≫ Zbb) would still display increasing strain hardening in extensional flows as increasing the number of branches usually increases the SHF. Alternatively, it is possible that there is a certain limit for the SHF corresponding to an optimum degree of branching for each backbone and side chain length. Anionic synthesis of model comb-PS structures via the Schlenk technique was conducted7 and developed in our group with the aim of producing enough material per batch (10−20 g) to perform both rheological experiments30 and small scale processing methods where elongational rheology is of great importance, e.g., electrospinning,31 fiber spinning, and batch foaming processes.32 In the current study, a series of welldefined comb-PS with loosely to densely grafted structures tending toward bottlebrush architectures were synthesized systematically in order to quantify the effect of the number of branches on the linear viscoelastic, i.e., zero-shear rate viscosity, η0, diluted modulus, G0N,s, and nonlinear viscoelastic properties, specifically extensional viscosity and SHF. To ensure an accessible zero-shear viscosity of the comb-PS over the range of applied temperatures without degradation, i.e., between 130 and 230 °C, the lengths of the entangled backbone and branches were kept at ca. Zbb ≈ 20 and Zbr ≈ 3 entanglements (i.e., Mw,bb ≈ 290 kg/mol and Mw,br ≈ 44 kg/ mol), respectively. Linear viscoelastic properties in SAOS were investigated within the frameworks of the dependence of the zero-shear viscosity on molecular weight, the van Gurp− Palmen (vGP) plot, and the diluted modulus. Extensional behavior was analyzed by evaluating the strain hardening factor, SHF, and compared with predictions from the molecular stress function (MSF) constitutive equation.

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DEFINITIONS OF COMB AND BOTTLEBRUSH TOPOLOGIES Definitions of comb and bottlebrush structures depend on the degree of polymerization of side chains, nbr, as well as the average degree of polymerization between branches, or branch point spacing, ns, in these structures. Using scaling analysis, Sheiko et al.26,29,33 identified a diagram for comb and bottlebrush polymers in terms of nbr and ns in order to B

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Figure 1. Structural characteristics and definitions of loosely and densely grafted comb-PS and bottlebrush-PS according to eqs 5−8. The ranges of Zs, Nbr, and ϕbb values are presented for the investigated combs and bottlebrushes with Mw,bb ≈ 290 kg/mol and Mw,br ≈ 44 kg/mol corresponding to Zbb = 20 and Zbr = 3, respectively.

nm, the following criteria were calculated for the comb and bottlebrush polystyrenes presented here:

distinguish between four different conformational regimes: loosely and densely grafted comb (LC and DC) and loosely and densely grafted bottlebrush (LB and DB) conformations. Equations 1−4 represent the criteria for these definitions: Z br [[mml:lt]] Zs

loosely grafted combs, LC

Z br [[mml:lt]] Zs

densely grafted comb‐PS, DC‐PS

loosely grafted bottlebrush‐PS, LB‐PS (3)

(7)

0.007 [[mml:lt]] Zs [[mml:lt]] 0.021 densely grafted bottlebrush‐PS, DB‐PS

ne−1 [[mml:lt]] Zs [[mml:lt]] ν /(nebl 2) densely grafted bottlebrushes, DB

(6)

0.021 [[mml:lt]] Zs [[mml:lt]] 0.08Z br 0.5

(2)

ν /(nebl 2)[[mml:lt]] Zs [[mml:lt]](Z br /ne)1/2 ν/(bl)3/2 loosely grafted bottlebrushes, LB

(5)

0.08Z br 0.5 [[mml:lt]] Zs [[mml:lt]] Z br

(1)

(Z br /ne)1/2 ν/(bl)3/2 [[mml:lt]] Zs [[mml:lt]] Z br densely grafted combs, DC

loosely grafted comb‐PS, LC‐PS

(8)

However, the DB-PS regime was not experimentally investigated in this study due to the steric hindrance of the reacted entangled side chains. Figure 1 presents the relevant molecular characteristics of this work including the range of Zs, Nbr, and ϕbb for different configurations of comb and bottlebrush PS with a constant Zbb = 20 and Zbr = 3. Figure 1 shows that for these backbone and branch lengths comb architectures with less than 6 branches are LC-PS, while ones with 6 < Nbr < 142 are DC-PS. A bottlebrush conformation with more than 142 branches per molecule has a backbone volume fraction less than 5 vol %.

(4)

where we normalized the nbr and ns to the degree of polymerization of the entanglement molecular weight of a linear polymer, ne. The molecular parameters used in this scaling analysis were monomeric volume, ν, monomeric length, l, and Kuhn length, b (related to the stiffness of the polymer chains). Both backbone and side chains in LC and DC polymers have almost unperturbed random Gaussian conformations; however, in contrast to the LC, the distance between branch points in DC polymers is shorter than the length of the branches, ns < nbr. Further increases in the number of branches toward LB and DB results in the segregation of molecules and reduces the overlap of side chains belonging to neighboring bottlebrushes. The backbone in a LB is stretched due to side chain crowding, while the side chains still maintain their random Gaussian conformations. Further increases in the number of side chains on the backbone results in a DB molecule, where both backbone and side chains adopt stretched conformations. This means that the side chains of a DB molecule theoretically cannot interpenetrate the side chains of the neighboring molecules. According to eqs 1−4 and using styrene as the monomer with ρ = 0.909 g/cm3, M0 = 104.15 g/ mol, Me = 14 500 g/mol (calculated in this paper), ν = 0.19 nm3, Kuhn length b = 1.8 nm, C∞ = 9.5 and l = b/C∞ = 0.19

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EXPERIMENTAL SECTION

Materials. Styrene (Acros, 99%) was purified by first stirring over calcium hydride (CaH2, Acros, 93%), distilled at reduced pressure, and then distilled from dibutylmagnesium (Sigma-Aldrich, 1 M in heptane) into Schlenk-type ampules and degassed afterward by three successive cycles of freezing, evacuation, and thawing. The styrene was then stored under argon at −18 °C until needed. Tetrahydrofuran (THF, Acros, >99%) was stirred over CaH2 and then stored after distillation over sodium in the presence of benzophenone as an indicator under an argon atmosphere until a purple color was observed. The required amount of solvent could be later distilled as needed. Toluene (Acros, 99.5%) and chloroform (CHCl3, Sigma-Aldrich, 99%) were stirred over CaH2 and distilled prior to use. sec-Butyllithium (sec-BuLi, SigmaAldrich, 1.4 M in cyclohexane), n-butyllithium (n-BuLi, Acros, 2.2 M C

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Figure 2. Reaction scheme for the synthesis of model PS combs. (a) Anionic polymerization of the backbone. (b) Friedel−Crafts acetylation of the backbone. (c) Anionic polymerization of the living side chains. (d) Grafting of living side chains onto the acetylated backbone. in cyclohexane), acetyl chloride (AcCl, Fluka, 99%), and aluminum trichloride (AlCl3, Sigma-Aldrich, 1.0 M in nitrobenzene) were used as received. Anionic Polymerization Method. Backbone and Side Chain Polymerization. Backbone and side chains were synthesized separately using anionic polymerization methods (Figure 2a,c). The reaction vessel was prepared by first taking a Schlenk flask with two stopcocks and distilling in the required amount of dry toluene. Then the flask was directly mounted on the vacuum line. The ampule containing the purified styrene was mounted onto the second ground joint. After adding the styrene (10 vol %) to the toluene (90 vol %), the polymerization was initiated by adding sec-BuLi at room temperature via a syringe. After complete conversion of the monomer for the backbone polymerization, the living anions were terminated using degassed methanol. The backbone polymer was then precipitated in methanol. Residual amounts of solvent were removed by drying the polymer under vacuum at 70 °C. The procedure for the side chain polymerization was similar to that of the backbone polymerization, except that the living anion side chains were kept “alive” so they could later react with the carbonyl groups (see next section) on the PS backbone. However, a small sample of the side chains was removed from the flask using a syringe and then terminated with degassed methanol for the characterization of its molecular weight and dispersity by SEC. Acetylation of the Polystyrene Backbone. Carbonyl groups as branching points were introduced to the PS backbone using the Friedel−Crafts acetylation method (Figure 2b). Linear PS with Mw = 290 kg/mol, the backbone for all comb-PS, was acetylated with different degrees of carbonyl groups ranging from 0.1 to 13.0 mol %. Acetyl chloride was used as the acetylation agent. In a typical functionalization reaction to create for example 30 branching points on the backbone, PS (5 g, Mw = 290 kg/mol) was dissolved in 50 mL of dry chloroform, and then a solution of AcCl (0.08 mL, 1.14 mmol) and AlCl3 (1.14 mL, 1.14 mmol) in 50 mL of chloroform was added. The mixture was stirred at room temperature for 5 h, and afterward the polymer was precipitated in acidified methanol (500 mL, 7 mL of HCl, 6 M). To remove residual reaction products, the polymer was redissolved in THF and precipitated once more. Finally the polymer was dried under vacuum at 70 °C. 1H NMR (CDCl3) was used to quantify the number of acetylated phenyl groups on the backbone. Polystyrene Comb Synthesis (Figure 2d). A two-neck roundbottomed flask containing the partially acetylated backbone was connected to the vacuum line, and the backbone was dissolved in dried THF at a concentration of 60) is consistent with results for LDPE, which have branches statistically distributed along the backbone, and the branch point distance is consequently very similar unlike the loosely grafted structures. According to the extended dynamic dilution theory of Milner−McLeish44 for a comb molecule, given in eq 9.18 of ref 1, the longest arm relaxation time exponentially decreases with increasing volume fraction of branches, ϕbr: τa =

1 3 π τeZ br 2 exp[νZ br(1 − 2ϕbr /3)] 2

Figure 6. Experimental arm relaxation times calculated from the inverse of angular frequency corresponding to relative maxima in δ vs angular frequency in Figure 5c as well as the predictions of eq 15 (solid line) as a function of the number of branches per molecule. Dashed line is guided to the eye to show the deviation of experimental arm relaxation times of combs with Zs < 1 from eq 15 (solid line).

istics of comb and bottlebrush architectures on the equilibrium structure as well as entanglement molecular weight, Me, and plateau modulus, G0N, was studied recently using scaling analysis26,33 and coarse-grained molecular dynamics simulations.29 Figure 5d replots the SAOS data in the framework of the van Gurp−Palmen (vGP) plot in order to discriminate between different LCB features in nearly monodisperse PS combs. As is clearly seen, these PS combs showed the most deviation from a linear PS over the range of 60° < δ < 80°. Trinkle and Friedrich49 used the vGP plot to detect the polydispersity in different linear polymers where the absolute value of the complex modulus |G*| at δ = 60° was defined as the criterion for dispersity index. However, the vGP plot is sensitive to both polydispersity and branched topologies. Bahreini et al.32 examined the same criterion to distinguish the LCB content in blends of industrial linear and branched polypropylenes. It is worth mentioning that no clear diluted modulus (second minimum in δ at low angular frequencies) was observed. According to Figure 5d, δ = 60° is a reasonable criterion for the branching content of these PS combs as there is a single value for |G*| at δ = 60° for each comb-PS. Figure 5d shows this as the intersection (cross points) of the horizontal dashed line at δ = 60° with the experimental data. However, this value could not be detected for PS290-190-44 with a loosely grafted bottlebrush structure, Zs ≅ 0.1. Most of the linear viscoelastic theories present at least two independent rheological characteristics, which are usually the relaxation time and modulus. For instance, for a linear entangled polymer, the plateau modulus is almost constant, G0N, while the longest relaxation time is molecular weight dependent, e.g., τd = kM3.4. The zero-shear viscosity, η0 = G0Nτd, basically depends on these two characteristics, and therefore its scaling with molecular weight for linear polymers should be similar to τd. However, the scaling of the modulus and zeroshear viscosity as a function of molecular weight for a branched polymer strongly depends on the topology (star, comb, and branch on branch) as well as the length of the branches.

(15)

where ν = 3/2 and the term 2ϕbr/3 in the exponential function takes into account the dilution mechanism of branches on the effective branch length, Zbr. The presence of a relative maximum in δ between the angular frequency 1 to 10 rad/s in Figure 5c (e.g., this peak is shown by an arrow for PS290190-44) could be related to the dominant relaxation time of branches, τa, in the molecular theories.8,45 By increasing the number of branches, this peak shifts to higher angular frequencies (shorter arm relaxation times) due to the higher dilution effect of the branches. However, a further increase in the number of branches toward the bottlebrush-like structures, i.e., Nbr ≥ 60 and Zs < 0.3, shifted the relative maximum in δ to lower frequencies. Figure 6 shows the inverse of frequency related to these maxima as τa along with the predictions of eq 15 as a function of Nbr (bottom axis) and Zs (top axis). The decrease in the experimental arm relaxation times of comb-PS with Nbr < 30 (Zs > 0.6) is in agreement with the predictions from eq 15. In contrast to this, for combs with Nbr > 30, τa increases with increasing Nbr. Equation 15 only includes the exponential effect of side chain length and dynamic dilution on the arm relaxation time, while the effect of side chain crowding, especially when Zs < 1, on the arm relaxation time was not taken into account (see Figure 6). This behavior indicates that there is an additional intermolecular interaction between the neighboring side chains, which results in a longer arm relaxation time. This type of interaction originates from the densely branched topologies where the distance between the branches is much less than one entanglement. To the best of our knowledge, such interactions have not been taken into account in the predictions from the developed tube models based on the reptation theory, e.g., pom-pom,35 hierarchical,46 branch-on-branch (BoB),47 and time marching algorithm (TMA)48 models. However, the effect of molecular characterH

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proportional to Mw−2 and, at a large enough number of branches, scales with Nbr−2, which is in agreement with G0N,s ∼ ϕbb1+α with a dilution exponent α = 1, where ϕbb = Mbb/Mw. From this figure, it is observed that |G*| at δ = 60° scales with Mw−1 and Nbr−1 (at high enough Mw) for this series of PS combs and therefore almost scales with ϕbb1. Scaling of the parameter |G*|δ=60° might also be useful for industrial branched polymers, e.g., LCB-polypropylene, where the second minimum cannot be observed due to its high polydispersity.32 Figure 7 reveals that comb PS with Nbr between 3 and 10 branches per backbone (or 2 < Zs < 5) have zero-shear viscosity η0 values that are higher than their linear-PS counterparts. This is in agreement with star molecules where η0 ∼ exp(Mbr); however, in the current series of combs, the length of the side chains was kept constant. It seems that η0 of loosely grafted comb structures exponentially increases with molecular weight of comb, η0 ∼ exp(Mw), above the linear reference. A relative maximum in η0 can be predicted in Figure 7, which is related to a comb structure with Nbr ≈ 6 or Zbr ≈ 3. This is in agreement with the simulation results of Inkson et al.23 for model comb polymers with Zbb = 20 and a minimum of 6 branches per molecule. Scaling analysis in Figure 1 and eq 5 shows that Zs = Zbr is a transition point from LC to DC conformation. With respect to the backbone length, Zbb ≈ 20, and the number of entanglements between branch point spacing, Zs, it can be concluded from Figure 7 that combs with Zs > 2 have a similar or higher zero-shear viscosity than their linear analogues, η0,comb ≥ η0,lin, which is in agreement with our analysis of Figure 7c from ref 23. Increasing Nbr to more than 6 branches, but still keeping the entangled branch point spacing, 1 < Zs < 3, resulted in a continuous decrease according to η0 ∼ Mw−3.4 where combs with Nbr > 9 or Zs < 2 have lower values for η0 than their linear counterparts. This drastic decrease is related to an effective CLF mechanism of branch tips and their intensive dilution effects on the backbone relaxation time and modulus. Therefore, CLF, dynamic dilution, Rouse relaxation of backbone segments, and reptation of the diluted backbone are the dominant relaxation mechanisms for DC-PS configuration. Considering an diluted entangled linear polymer solution,51,52 the plateau modulus and longest relaxation time scale with the volume fraction of polymer, ϕ2 and ϕ1.4, respectively. Consequently, the zero-shear viscosity of a polymer solution is proportional to ϕ3.4, i.e., η0 = GNτ ≡ G0Nτdϕ3.4, where G0N and τd correspond to the polymer melt (ϕ = 1). This scaling is in agreement with the results for a DC configuration with 1 < Zs < 3, if the relaxed side chains are considered as a hypothetical solvent. Our analysis on the simulations of η0 as a function of the total molecular weight in Figure 7c of ref 23 for combs with Zbb ≥ 20 and 2 < Zbr < 10 revealed that η0 ∼ M−3.4±0.2, which is valid when the number of branches is between 6 < Nbr < Zbb. This behavior could be anticipated from the results shown in Figure 7 for combs with 10 and 14 branches per molecules. An increase in Nbr to above 20, where Zs < 1, still decreased η0 of the PS combs, but with a lower power law dependency around −1, η0 ∼ Mw−1 ∼ ϕbb1, meaning that PS290-60-44 with Zs ≅ 0.3 had a η0 that was similar to its backbone, PS290. This type of behavior is similar to that of a diluted unentangled polymer solution. In other words, a high amount of side chains (0.75 < ϕbr < 0.9) acts as a solvent and dilutes the unentangled backbone, which is already relaxed by Rouse motions. However, this gradual decreasing in zero-shear viscosity, η0 ∼ Mw−1, might not have this exact

Figure 7 shows experimental data of the diluted modulus, the absolute value of the complex modulus |G*| at δ = 60°,

G0N,s,

Figure 7. Zero-shear viscosity, η0, diluted modulus, G0N,s, and complex modulus |G*| at δ = 60° as a function of the weight-average molecular weight (bottom x-axis) and number of branches per molecule (top xaxis). Filled symbols are the samples from this work (see Table 1) with Mw,bb = 290 kg/mol, while open symbols are taken from Kempf et al.6 with Mw,bb = 275 kg/mol, which are PS275-5-42 and PS275-29-47. Dashed lines were drawn to guide the eye, and scaling exponents are explained in the main text.

and the zero-shear viscosity, η0, of all the PS combs as a function of the total molecular weight. Two additional PS combs from Kempf et al.6 with Mw,bb = 275 kg/mol, and Mw,br = 42 and 47 kg/mol were added to Figure 7, which had 5 and 29 branches per backbone and are called PS275-5-42 and PS27529-47, respectively. The η0 value was defined by extrapolation of |η*| to very low angular frequencies. However, for PS290190-44, this region was not fully reached; therefore, the relaxation spectra, gi and λi, obtained by fitting the Maxwell model to G′ and G″ (reported in Table S1) were used to calculate an estimate value for η0 = ∑giλi. The fact that this is only a lower estimate is indicated by an up-arrow on its data point in Figure 7. The solid line represents η0 = 3.31 × 10−14Mw−3.4 for linear-PS at the reference temperature of 180 °C,50 while dashed lines were drawn to guide the eye. The diluted modulus, G0N,s, is related to the second plateau of G′ at low frequency (see Figure 5a), which corresponds to the second minimum of the phase angle, δ, in Figure 5c. The top xaxis of Figure 7 shows the corresponding number of branches per molecules and is consistent with the bottom x-axis, Nbr = (Mw − Mw,bb)/Mw,br. Although increasing the number of branches on the backbone confines the backbone motions and enhances the reptation time of the backbone resulting in a higher viscosity, the entire molecule has more end-points than its linear counterpart making the CLF relaxation mechanism more effective. Increasing the number of end-points in a molecule leads to a lower viscosity as well as lower modulus relative to the linear analogues. Relaxation of side chains by the CLF mechanism dilutes the backbone media and reduces the plateau modulus of the backbone, G0N,s ∼ ϕbb1+α, according to dilution theories with α = 1 or 4/3. In the series of combs considered here with almost similar backbone (Mw,bb = 275 or 290 kg/mol) and side chain lengths (Mw,br = 42−47 kg/mol), the diluted modulus, G0N,s, is I

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Figure 8. Tensile stress growth coefficient data of linear and comb-PS with the same backbone length, 290 kg/mol, and side chains 44 kg/mol, but different numbers of branches per molecule, PS290-Nbr-44, along with predictions from the MSF (solid lines) and DE model (dashed lines) at different extensional rates and T = 180 °C.

scaling relation or there might even be a transition region with a variable slope present. Further increasing Nbr to 120 and 190 drastically increases the scaling in this region to above 5 as shown by the solid line in Figure 7 which scales as η0 ∼ Mw5 ∼ Nbr5 ∼ ϕbb−5. However, this last result should be treated with caution because the very long relaxation time of PS290-190-44 makes it difficult to obtain an accurate measurement of the zero-shear viscosity, and this scaling is based on just two data points. An extrapolation of the last two asymptotic lines results in a crossover at a relative minimum η0 corresponding to Zs ≅ 0.2 or Nbr ≅ 100. This strong dependency of η0 on number of

branching is related to the intramolecular interactions between neighboring entangled side chains due to the tight spacing between branch points, which is much less than one entanglement, e.g., Zs ≅ 0.1 for PS290-190-42. Although this architecture effectively precludes the formation of backbone entanglements in the polymer melts (diluted backbone), it causes an intrinsic stretching of branch point spacing and side chains,26 which results in longer arm relaxation times τa than those predicted by eq 5 for LC or DC configurations with the same side chain length (see Figure 6). Therefore, these two opposing effects of side-chain crowding (dilution and enhanced J

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the Rouse time of PS290-10-44 is almost 0.2 s, while its relevant arm relaxation time in Figure 6 is τa ≈ 0.5 s, which means that at extensional rates ε̇ > 2 s−1 the arms have been already stretched. In the present PS combs, the length of the entangled side chains and the backbone were kept constant. Hence, by changing the number of branches, the distance between the branch points, Zs, varied from well-entangled, 3 < Zs, for LC-PS to 0.2 < Zs < 3 for DC-PS and finally to mostly unentangled, Zs < 0.2, for LB structures. It is obvious from Figure 8b that the presence of loosely branched structures, Nbr = 3, induced clear strain hardening behavior in extension. Increasing the Nbr to 10, 14, 30, and 60 in Figures 8c, 8d, 8e, and 8f, respectively, reduced the onset extensional rate of strain hardening from ε̇ = 0.01 to 0.003 s−1 corresponding to stretching times τs ≈ 100 to τs ≈ 333 s, whereas the maximum degree of strain hardening (the ratio of extensional viscosity to LVE line) increased significantly with the number of branches. This means that Nbr or Zs has a larger effect on the maximum stretching of the backbone than the stretch relaxation time corresponding to the onset of strain hardening. A further increase in Nbr toward loosely grafted bottlebrush topologies, i.e., PS290-120-44, as shown in Figures 8g, resulted in a very large degree of strain hardening and an almost unobtainable steady state extensional viscosity at the maximum Hencky strain of 4. However, Figure 8h shows that almost all extensional experiments for PS290-190-44 broke at a Hencky strain below 3 and did not display a plateau region in the stress growth coefficient data. The MSF constitutive equation was used to quantify the strain hardening in the extensional data with its two nonlinear parameters β and f max. Equations 11−14 were fitted to the stress growth coefficient data using the relaxation spectra and adjusting the two fitting parameters. Values for β and f max2 are reported in Figure 8 and plotted in Figure 9 as a function of Nbr and as a function of the branch point spacing, Zs = Zbb/(1 + Nbr).The parameters β and f max govern the slope and maximum stress growth coefficient, respectively, and Figure 9 shows that both of them increase monotonically with Nbr. According to the molecular definition of β for branched polymers, this parameter

arm relaxation time) result in a minimum zero-shear viscosity (roughly equal to its backbone value) for a comb structure with Zs ≅ 0.2. This means that tube-based models should consider the effect of this stretching or the non-Gaussian conformation of side chains and branch point spacing in the relaxation mechanisms of a bottlebrush polymer. It can be concluded from Figure 7 that diluted modulus G0N,s, virtually scales with ϕbb2 for all of the comb-PS; however, the scaling of the longest relaxation time and consequently zeroshear viscosity depended strongly on the conformation of the molecule. This means that the relaxation time of the whole molecule depends not only on the dilution effect of side chains but also on the topology and conformation of the branches. While the diluted modulus G0N,s and |G*|δ=60° are not sensitive to the conformation of the side chains, both gradually decreased as the number of branches for comb and bottlebrush architectures increased. Hence, according to η0 = GNτd, the longest relaxation time of the comb-PS, τd, gradually scales with exp(ϕbb−1), ϕbb1.4, ϕbb−1, and ϕbb−7 from LC to LB structures. With respect to the relative minimum and maximum in η0, we can distinguish between three different configurations for combs topologies with entangled side chains and backbone: (1) LC with Zbr < Zs where η0 increases exponentially with Nbr, (2) DC with 0.2 < Zs < Zbr where η0 decreases according to a power law dependency in a similar manner to a dilute entangled solution and followed by an unentangled solution, and (3) LB structures with Zs < 0.2 where η0 again increases with increasing Nbr. However, DB with Zs < 0.1 were not investigated here due to the synthesis limitations in Friedel−Crafts acetylation step. Extensional Rheology. Uniaxial extensional experiments were carried out with an extensional viscosity fixture (EVF) at different extensional rates and a maximum Hencky strain of εH = 4. However, some of the samples did not break after this deformation. Lentzakis et al.9 have conducted extensional experiments on similar PS combs synthesized by Roovers et al.14,15 They used a Sentmanat extensional rheometer (SER), which is similar to an EVF, and verified the uniaxial deformation by video microscopy. They also confirmed the reliability of the SER measurements up to a Hencky strain of εH = 4 using a filament stretching rheometer (FSR) developed by Hassager et al. at the Technical University of Denmark, DTU.53 Figure 8 depicts the experimental data of the tensile stress growth coefficient versus time for different extensional rates at 180 °C for linear and the PS combs described in Table 1. With respect to the estimated formula for the Rouse relaxation time, τR,lin = τeZ2, PS290 at 180 °C has a Rouse time of 0.08 s, leading to an onset rate of strain hardening τR,lin−1 ≡ 12.5 s−1. However, this rate is outside the range of the experimental window (see Figure 8a). The experimental data show that all of the PS combs underwent strain hardening over the range of imposed extensional rates. According to Figure 6, the minimum value of estimated arm relaxation time is τa ≈ 0.25 s for PS290-30-44, which means that all PS combs should show strain hardening when ε̇ > 4 s−1. The PS combs synthesized with PS290 as the backbone have longer Rouse times than the backbone, τR,comb = τeZbb(Zbb + qZbr), due to the increased monomeric friction of the side chains. However, Lentzakis et al.9 concluded that the stretching time, τs, of comb-PS with entangled side chains is even longer than τR,comb because the Rouse time only includes the effect of monomeric friction, while friction due to the entanglements in the side chains is not taken into account in τR,comb. For instance,

Figure 9. Nonlinear parameters β and f max2 of the MSF model as a function of the number of branches per molecule, Nbr, as well as the distance between the branch points for PS combs with the same backbone, Mw,bb = 290 kg/mol, and similar branch molecular weight, Mw,br ≅ 44 kg/mol. K

DOI: 10.1021/acs.macromol.7b01034 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules is equal to the ratio of the total molecular weight of branched polymer to the molecular weight of the backbone:37 βdefinition =

Z N M ≡ 1 + br br ≡ ϕbb−1 Mbb Z bb

(16)

However, this definition overestimates the experimental extensional data, e.g., for PS290-120-44, eq 16 predicts βdefinition = 19, whereas β = 3 resulted a good fit to the stress growth coefficient data. In this paper, we do not want to focus on the molecular definition of the β and f max parameters for comb polymers as the MSF model was only used to quantify the extensional data, especially the strain hardening. However, according to Figure 9 and the definition of β in eq 16, a logarithmic function correlates empirically to the fitted values of β as a function of Nbr or ϕbb: 1.7 ⎛ Z brNbr ⎞ β = 1 + log⎜1 + ⎟ ≡ 1 + log ϕbb−1.7 Z bb ⎠ ⎝

Figure 10. Strain hardening factor, SHF(ε̇) ≡ ηE,max(ε̇)/ηDE,max(ε̇), as a function of the extensional rate for different PS combs each with the same backbone and similar side chain lengths. Symbols depict the maximum experimental stress growth coefficient data, and solid lines are the predictions of the MSF model for each specific sample.

(17)

It is worth mentioning the Taylor expansion of eq 17 at ϕbb ≈ 1 (linear polymer) is the same as eq 16. From the data in Figure 9, it was obvious that f max2 plotted on a logarithmic scale is also a monotonically increasing function of Nbr in a similar manner to the parameter β plotted on a linear scale. Therefore, a power law function was proposed for f max2 as a function of Nbr or ϕbb: fmax

2

1.7 ⎛ Z brNbr ⎞ = 4⎜1 + ⎟ ≡ 4ϕbb−1.7 Z bb ⎠ ⎝

loosely grafted bottlebrush were synthesized using anionic polymerization. The number of side chains per molecule varied between Nbr ≅ 3 and 190, corresponding to the number of entanglements between branch points, Zs ≅ 5 and 0.1, respectively. Analysis of the linear viscoelastic and extensional properties within the frameworks of the zero-shear viscosity, η0, diluted modulus, G0N,s, vGP plot and the strain hardening factor resulted in the following conclusions: 1. Scaling of η0 versus total weight-average molecular weight, Mw (or volume fraction of backbone, ϕbb = Mw,bb/Mw, where Mw,bb is constant at 290 kg/mol), resulted in a relative minimum and maximum, which corresponded to three different configurations of side chains: loosely grafted combs (Zbr < Zs) where η0 ∼ exp(Mw) ∼ exp(ϕbb−1), densely grafted combs with an entangled, diluted backbone (1 < Zs < Zbr) where η0 ∼ Mw−3.4 ∼ ϕbb3.4 followed by densely grafted combs with unentangled, diluted backbones (0.2 < Zs < 1) where η0 ∼ Mw−1 ∼ Nbr−1 ∼ ϕbb, and, finally, the loosely grafted bottlebrush conformation (Zs < 0.2) where η0 ∼ Mw5 ∼ Nbr5 ∼ ϕbb−5. 2. Increasing the number of branches exponentially decreased the arm relaxation time, τa, to a minimum value, which corresponded to Zs ≈ 0.6, while further increases in the number of branches enhanced the arm relaxation time monotonically. 3. The diluted modulus scaled (almost) as G0N,s ∼ Mw−2 ∼ ϕbb2 for the comb and bottlebrush-PS investigated here. 4. A new modulus that takes into account the dilution effect of branches, |G*|δ=60°, was extracted from the van Gurp− Palmen plot at constant δ = 60°. This modulus is similar to the diluted modulus G0N,s, but has a different scaling behavior, | G*|δ=60° ∼ Mw−1 ∼ ϕbb. 5. For the first time, a strain hardening factor, SHF, above 200 at Hencky strains below 4 for comb-PS was achieved by increasing the number of branches to 120−190 branches per molecule, corresponding to the branch point distances Zs ≅ 0.16−0.1. These scaling relationships for zero-shear viscosity and diluted modulus give an insight into the dominant relaxation mechanisms of comb architectures with different branch densities from loosely grafted combs to bottlebrushes.

(18)

Equations 17 and 18 plotted in Figure 9 show that parameters f max and β might be dependent parameters, which are functions of the backbone volume fraction. The ratio of the maximum (steady state) tensile stress growth coefficient to the one predicted by the DE model is defined as the strain hardening factor, SHF(ε̇) ≡ ηE,max(ε̇)/ ηDE,max(ε̇). This parameter was calculated for the experimental data as well as for the MSF model in Figure 8. However, as a steady state extensional viscosity might not be achieved experimentally in all samples at Hencky strains below εH = 4 in the EVF fixture or samples might break before achieving the plateau value, the maximum achieved stress growth coefficient was used to calculate the SHF as shown in Figure 10. It should be mentioned that in the experimental data (symbols) and the model predictions (solid lines) for the SHF the DE model was based on eqs 11−14 where no stretching was considered, i.e., f = 1. Figure 10 shows that by increasing the Nbr to 120, the SHF increases to above SHF = 200, which to the best of our knowledge has never been achieved before at Hencky strains below 4. However, a further increase in the number of branches resulted in very stiff samples that break at Hencky strains below 3 (see PS290-190-44 in Figure 8h). This means that bottlebrush structures might not have good processability compared to comb topologies in the melt state under extensional deformations. However, solution processing of such bottlebrushes, e.g. electrospinning, might be a possible alternative.31

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CONCLUSION A series of monodisperse comb polystyrenes with the same entangled backbone, Mw,bb = 290 kg/mol, and similar branch lengths, Mw,br ≅ 44 kg/mol, but different numbers of branches per molecule from loosely to densely grafted combs and a L

DOI: 10.1021/acs.macromol.7b01034 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

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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.7b01034. 1 H NMR spectra for all backbones and PS-combs, TTS shift factors at reference temperature Tref = 180 °C, relaxation spectra of Maxwell model (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.). ORCID

Mahdi Abbasi: 0000-0001-6099-421X Manfred Wilhelm: 0000-0003-2105-6946 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Mahdi Abbasi gratefully acknowledges the Alexander von Humboldt Foundation for financial support. Kamran Riazi acknowledges financial support from the German Federation of Industrial Research Associations (AiF, KF2473305AG3). The authors thank Prof. Dimitris Vlassopoulos for helpful advice on rheological characterizations and Dr. Nico Dingenouts, Dr. Jennifer Kübel, Dr. Michael Krämer, and Dr. Wolfgang Radke for fruitful discussions on GPC-MALLS, and Dr. Pavleta Tzvetkova for 1H NMR measurements with a 600 MHz spectrometer.

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REFERENCES

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DOI: 10.1021/acs.macromol.7b01034 Macromolecules XXXX, XXX, XXX−XXX