Highly Entangled α-Olefin Molecular Bottlebrushes - ACS Publications

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Highly Entangled α‑Olefin Molecular Bottlebrushes: Melt Structure, Linear Rheology, and Interchain Friction Mechanism Carlos R. Loṕ ez-Barroń ,* Andy H. Tsou, John R. Hagadorn, and Joseph A. Throckmorton ExxonMobil Chemical Company, Baytown, Texas 77520, United States

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S Supporting Information *

ABSTRACT: Linear viscoelastic response and melt microstructure of ultra-high molecular weight poly(α-olefins) (UHMW PO) with bottlebrush architectures, from poly(1-hexene) to poly(1-octadecene) synthesized by metal coordinative insertion polymerization, were measured as a function of side-chain length, Nsc. All these bottlebrush POs are highly entangled, with an average number of entanglements per chain, Z, greater than 50, which allows accurate determination of their rubbery plateau moduli, G0N, and their entanglement molecular weights, Me. Their plateau moduli scale with their side-chain lengths as G0N ∼ Nsc−1.47, in agreement with the scaling theory for the dense bottlebrush limit that predicts G0N ∼ Nsc−3/2. Melt structures of these bottlebrush poly(1-olefin)s and their melt structural changes with temperature were determined by wide-angle X-ray scattering. Concomitant with thermal expansion of these bottlebrush PO melts is a nonmonotonic change in backbone-to-backbone distance (d1) and a monotonic increase in side-chain spacing (d2). Both the melt-flow interchain friction coefficient and the viscosity of these UHMW PO bottlebrushes show a very strong dependence on d2, characterized by two exponential decay regimes, with decay constants having an exponential dependence on Nsc.

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INTRODUCTION Molecular bottlebrushes are branched macromolecules with a high graft density along their backbone. Definitions of comb and bottlebrush copolymers are based on the ratio between coil dimension of their comb arm, or side chain, and the unperturbed backbone spacing in between side chains.1,2 The crossover from comb to bottlebrush architecture occurs once the side chain’s coil size is equal to or greater than the backbone spacer size.2 Therefore, as recently discussed,3 poly(α-olefin)s prepared by polymerization of 1-alkene monomers with alkyl chain length greater than six (i.e., hexene and above) have bottlebrush architecture. Structurally, these are the simplest bottlebrush polymers, with their backbone and side chains consisting of alkanes. Fetters and co-workers reported the chain dimensions and rubbery plateau modulus (G0N) of a series of poly(α-olefin)s, ranging from poly(1-pentene) to poly(1-hexadecene).4,5 They estimated their entanglement molecular weight (Me) using the well-known relation6 Me ≡

ρRT GN0

increasing side-chain length is reflected in the data reported by Fetters et al.4,5 Although plateau moduli of various poly(α-olefin)s, including bottlebrush polyolefins, were determined by Fetters et al.,4,5 neither linear viscoelastic properties nor the molecular weights of these poly(α-olefins) were reported. Therefore, it is unclear whether those polyolefins examined by Fetters et al. were well-entangled or not. A precise and accurate measurement of G0N is not trivial, especially if the plateau modulus is not well-defined in the rheological curves.9 This is typically the case for polymer chains that are not well entangled, i.e., when their molecular weight lower than ∼15Me (i.e., when the average number of entanglements per chain Z ≡ M/Me < 15). In this study, a series of ultra-high molecular weight poly(αolefins) (UHMW PO), ranging from poly(1-hexene) to poly(1-octadecene), were synthesized and their linear viscoelastic properties characterized. The viscoelastic response of molecular bottlebrushes is governed by the ratio between the molecular weight of the arms (or side chains) and the critical entanglement molecular weight of the backbone. In general, their dynamic moduli in the high frequency range (dominated by short-time relaxations) are governed by the length and stiffness of their arms, whereas those in lower frequencies are controlled by the length and stiffness of their backbone. When both the side chains and

(1)

where ρ is the mass density of the polymer, R is the gas constant, and T is the absolute temperature. As discussed by Fetters et al.,7 Me is correlated to the side group “bulkiness”. In general, “skinny” chains (as in polyethylene or polypropylene) have lower values of Me because they entangle more easily than “fat” chains such as poly(4-vinylbiphenyl)8 or bottlebrush polymers. Therefore, the expected increase in Me with © XXXX American Chemical Society

Received: July 5, 2018 Revised: August 22, 2018

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DOI: 10.1021/acs.macromol.8b01431 Macromolecules XXXX, XXX, XXX−XXX

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and diluted with hexane solvent. Scavenger was not used except for the polymerization of 1-nonene, to which a small amount of bis(diisobutylaluminum)oxide was added. At ambient temperature the catalyst solution was then added to the flask, and the mixture was stirred rapidly. After stirring for a designated period of time (typically 0.5 h) the reaction was quenched by the addition of a small amount (10 mg) of Irganox 1076 stabilizer dissolved in toluene. The isolation of polyhexene and polyoctene was performed by evaporation of the volatiles under a stream of nitrogen. The other polymers were isolated and washed on a fritted disk following precipitation in stirring acetone. The samples were all dried under reduced pressure at 50−60 °C overnight. The properties of the UHMW PO bottlebrush polymers are given Table 1. The thermal transitions (glass transition, melting, and

the backbone are entangled, two sequential relaxations arise with two independent rubbery plateaus within the dynamic moduli versus frequency plot.10−12 The absence of entanglements result in relaxation spectra that can be described with a Rouse relaxation spectrum,10 in which the bottlebrushes are idealized as unentangled thick chains. To our knowledge, there are only three previous studies focused on the effects of side-chain length and backbone degree of polymerization on the rheological response of molecular bottlebrushes.10,13 Hu and co-workers reported the linear viscoelastic response of polynorbornene-g-polylactide brush polymers.10 They observed two relaxation processes corresponding to the relaxation of the polylactide side chain and the bottlebrush backbone relaxation only in polymers with side chains having molecular weight from ∼0.5Me to ∼Me. For a bottlebrush polymer with side-chain molecular weight of 0.2Me, side-chain relaxation cannot be found by linear rheology. Dalsin et al. reported the linear rheology of a series of bottlebrush polymers with polynorbornene backbone and either atactic polypropylene (aPP) or poly(ethylene-altpropylene) (PEP) side chains prepared by ring-opening metathesis polymerization.13 For polymers with aPP side chains of Z ∼ 0.5, they did not observe plateau region in the dynamic modulus master curves, indicating no side chain or backbone entanglement. For the polymer with PEP side chains of Z ∼ 3.5, side-chain entanglements were indicated. LópezBarrón et al.14 reported the linear rheology of molecular bottlebrushes with very short atactic polypropylene (aPP) side chains of molecular weight ranging from 119 to 259 g/mol (i.e., with Z < 0.1) and narrow molecular weight distribution (Mw/Mn = 1.02−1.05). The dynamic master curves of these bottlebrushes reveal two sequential relaxation processes corresponding to the segmental relaxation and the relaxation of the bottlebrush backbone. Fast relaxations of their short side chains cannot be distinguished from the glassy relaxation of these bottlebrush polymers. Similar to the polymers reported by López-Barrón et al.,14 the α-olefin molecular bottlebrushes in this study have side chains with molecular weights much lower than their Me (∼1 kg/mol, corresponding to the polyethylene Me), and their rheological responses are similar to those of semiflexible polymers. In this paper, we use X-ray diffraction measurements to quantify the backbone-to-backbone and side-chain distances in the molecular bottlebrush polyolefin melts3,14 as a function of temperature. Structural parameters of these bottlebrush polyolefin melts were then correlated to their dynamic thermal responses, determined by the time and modulus shift factors. Comparing temperature dependences of both the side-chain distance and the rheological properties, direct correlations between chain friction coefficients, melt viscosity, and the sidechain lengths of α-olefin molecular bottlebrushes, or bottlebrush POs, were established for the first time.

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Table 1. Properties of Molecular Bottlebrush Polyolefins sample poly(1-hexene), PH poly(1-octene), PO poly(1-nonene), PN poly(1-decene), PD poly(1-dodecene), PDD poly(1tetradecene), PTD poly(1-octadecene), POD

Mw × 10−6 (g/mol)

Mw/Mn

Tga (°C)

Tmb (°C)

ρ (kg/m3)

6.29 3.03 1.65 3.16 3.96

2.39 2.41 1.76 2.43 2.48

−42.0 −63.5 −67.6 −68.0 −66.5

ND ND ND −0.65 25.9

775 792 777 784 801

4.82

2.12

NDc

43.1

803

5.87

2.50

ND

54.1

762

a Measured as the G″ peak in the DMTA plots. bMeasured as the second (high temperature) peak in the DSC traces. cND: not detected.

crystallization) of the bottlebrush POs were measured by differential scanning calorimetry (DSC) and dynamic mechanical thermal analysis (DMTA). DSC measurements were performed using a DSC2500 (TA Instruments) with a 10 °C/min heating rate. DMTA measurements were performed on an ARES-G2 (TA Instruments) rheometer using serrated 8 mm parallel discs with a 2 °C/min heating rate. A frequency of 1 Hz and strain amplitude of 0.1% were used for the DMTA measurements. All DSC traces along with the DMTA results can be found in the Supporting Information. The glass transition temperature values are plotted as a function of side-chain length (Nsc), in carbon numbers, in Figure 1. Molecular weights and

EXPERIMENTAL METHODS

Synthesis and Characterization. Seven different α-olefins (1hexene, 1-octene, 1-nonene, 1-decene, 1-dodecene, 1-tetradecene, and 1-octadecene) were polymerized under similar conditions in a nitrogen-filled glovebox. The catalyst solution used for the polymerizations was formed by combining hafnium and [N-[2,6-bis(1methylethyl)phenyl]-2-[5,6,7,8-tetrahydro-8-(phenylamino-κN)-1naphthalenyl]-8-quinolinaminato(2-)-κN1,κN8]dimethyl (CAS Registry No. 2170734-21-9) with 1 mol equiv of N,N-dimethylanilinium tetrakis(pentafluorophenyl)borate in toluene. For each polymerization, the purified olefin was loaded into a round-bottomed flask

Figure 1. Glass transition temperature as a function of side-chain length for UHMW PO bottlebrushes. Diamond symbols are data reported previously in the literature.15−19 B

DOI: 10.1021/acs.macromol.8b01431 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules molecular weight distributions of the bottlebrush POs were determined at 145 °C using a gel permeation chromatography instrument (Agilent PL-GPC 220) equipped with a differential refractive index detector. Aldrich reagent grade 1,2,4-trichlorobenzene (TCB) with 300 ppm antioxidant BHT was used as the mobile phase. The dn/dc values used for PH, PO, PN, PD, PDD, PTD, and POD are 0.0785, 0.087, 0.1048, 0.1048, 0.0974, 0.0922, and 0.0815, respectively. Wide-Angle X-ray Scattering. A Linkam TST-350 optical stage with windows removed was used to heat up the bottlebrush POs for X-ray scattering measurements. Wide-angle X-ray scattering (WAXS) measurements were performed on a SAXSLAB Ganesha 300XL+ instrument equipped with a Xenocs GeniX high brilliance microfocus sealed tube X-ray source (energy = 8 keV, Cu K = 1.54 Å), focusing optics, and a Dectris Pilatus 300K detector (487 × 619 pixels, pixel dimension 172). Data processing was performed with the computer program SAXSGUI (JJ X-ray Systems ApS and Rigaku IT, Inc.). Distances were calibrated using silver behenate. The scattering patterns were normalized to the primary beam intensity and corrected for background scattering. Rheometry. Dynamic frequency sweeps (DFS) over a frequency range of 0.01 Hz < ω < 100 Hz, at temperatures ranging from −70 to 210 °C were performed using a strain-controlled rheometer ARES-G2 (TA Instruments) with two sets of parallel plates, one with diameter = 8 mm for measurements at T > Tg + 40 °C and a (custom-made) 3 mm serrated parallel plates for measurements at T ≤ Tg + 40 °C. The use of serrated plates is intended to avoid slip. The 3 mm plates are used in combination with a large sample gap (2.5−3 mm) to minimize instrument compliance effects, following the recommendations made by McKenna et al.20,21 Strains of 1% and 0.1% were used for the two temperature ranges, respectively. The DFS data are shifted vertically and horizontally following the time−temperature superposition (tTs) principle to construct master curves of the dynamic moduli at a given reference temperature, T0 (see Table 2). The time (horizontal) shift factors are fitted to the Williams−Landel−Ferry (WLF) equation:22 log aT = −C1(T − T0)/C2 + (T − T0), where C1 and C2 are empirical constants.

Figure 2. (a) WAXS profiles of bottlebrush POs measured at 70 °C and (b) corresponding average distances between backbones (d1) and side chains (d2) as a function of side-chain length. (c) Schematic cartoon of two adjacent bottlebrush molecules.

distance between side chains, d2 = 2π/q*2 (where q*2 is the position of the high-q peak), shows a weak, but measurable, dependence on Nsc for 4 ≤ Nsc ≤ 8, whereas for Nsc > 8 d2 is independent of Nsc. This reflects the tighter packing of the side chains when adjacent backbones are closer together when Nsc < 8. Therefore, going down in olefin carbon number from polyoctene and below would have a strong effect on the frictional drag between adjacent bottlebrush chains, as discussed below. It is critical to characterize the melt structural changes with temperature of these bottlebrush PO molecules to understand their thermorheological behavior. WAXS profiles of three selected bottlebrush POs, namely PH, PDD, and POD, measured at several melt temperatures are shown in Figure 3a. It is clear that increasing temperature results in an increase in the relative intensity between the low-q and the high-q scattering peaks (at q*1 and q*2, respectively). For all three PO bottlebrushes, the intensity of the low-q peak (from backbone scattering) increases, mostly linearly, with temperature. Similar behavior was previously observed in polystyrene24 and poly(4vinylbiphenyl).8 Those two polymers show a pronounced increase in scattering intensity of the interbackbone diffraction peak with temperature. Mitchel and Windle postulated that this is a consequence of the increasing interphenyl ring distances during thermal expansion, which creates areas with electron deficiency (with respect to their surroundings).24 This results in an increase in X-ray contrast between the “void” (electron-deficient) regions and the electron-rich regions near the chain backbones, which in turn manifest as the observed increase in WAXS intensity. We believe that the same mechanism operates in the PO molecular bottlebrushes studied here, namely, electron-deficient regions between side chains are generated by thermal expansion, which is evidenced

Table 2. Rheological Properties sample

C1

C2

T0

G0N (kPa)

Me (kg/mol)

Z

PH PO PN PD PDD PTD POD

6.34 4.91 4.78 4.81 4.51 3.99 3.68

123 136 143 153 161 181 221

25 25 25 25 25 50 70

138 72.5 61.9 51.1 35.9 28.2 20.1

13.9 27.1 31.1 38.0 55.3 74.2 109

451 112 53.1 83.1 71.7 65.0 53.9

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RESULTS AND DISCUSSION Microstructure of the Bottlebrush Melts. It has been established that the X-ray diffraction pattern from “amorphous” polyolefins with bottlebrush architecture consists of two broad peaks.3,14,23 Figure 2a shows the WAXS profiles measured at 70 °C for the UHMW PO bottlebrush series studied here, The low-q scattering peak correspond to the distance between backbones of adjacent chains. The average backbone distance is computed as d1 = 2π/q*1 , where q*1 is the position of the low-q peak. This backbone distance increases with increasing side-chain length, Nsc, as shown in Figure 2b. A linear dependence between d1 and Nsc is observed, which is in agreement with results from an analogous series reported by Turner-Jones,23 although their measurements were performed at different temperatures. The high-q scattering peak in bottlebrush POs arises from van der Waals interactions between side chains of the same or different chains.3,23 The C

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Figure 3. (a) WAXS profiles of PH, PDD, and POD, measured at the indicated temperature. The scattering intensity is normalized to the maximum intensity of the high-q peak (at q*2) measured at 70 °C. (b) Maximum normalized intensity for low-q peaks (at q*1) and high-q peak (at q*2) as a function of temperature.

by the increase in both characteristic distances d1 and d2, as discussed below. The decrease in the high-q peak intensity observed in PH (shown in Figure 3b) is the expected behavior for polymer melts that undergo an increase in thermal disorder as the temperature is raised.8 In other words, the side chains are less correlated at higher temperatures. Interestingly, this decreasing high-q scattering intensity with temperature is not found in other two polyolefin bottlebrushes. Instead, their high-q scattering intensities stay constant with increasing temperatures. To quantify the molecular thermal expansion, we use the characteristic backbone-to-backbone and side-chain distances (d1 and d2, respectively), normalized to the corresponding values measured at 70 °C (d1,70 °C, and d2,70 °C). These are plotted as a function of temperature in Figure 4. The sidechain distance increase monotonically, and linearly, with temperature, in accordance with volumetric expansion of these bottlebrush POs. This increase in side-chain distance correlated to the decrease in interchain friction coefficient with temperature, as discussed below. However, a monotonic increase in the backbone-to-backbone distance with temperature can only observed in PO bottlebrushes with Nsc ≤ 7. For the rest of the bottlebrushes, a slight decrease in d1 is observed at low temperatures, followed by a modest increase at high temperatures. This suggests that the thermal expansion of the bottlebrush POs with long side chains may involve a reconfiguration of the backbone contour without effectively

Figure 4. Normalized characteristic backbone-to-backbone and the side-chain distances as a function of temperature.

changing the backbone distance. This proposed reconfiguration mechanism is not well understood at this time. D

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Figure 5. Master curves of the dynamic moduli for the UHMW PO bottlebrushes constructed via the tTs principle, using the reference temperatures listed in Table 2 and the time and modulus shift factors shown in Figure 6.

Linear Rheology. The master curves of dynamic moduli (G′ and G″, or elastic and viscous moduli) versus frequency for all UHMW PO bottlebrushes studied here are shown in Figure 5. Due to the fact that the molecular weight of the side chains is far below the entanglement molecular weight of polyethylene (Msc ≪ Me,polyethylene), these dynamic master curves do not show a plateau modulus associated with the side chains. These master curves were constructed using the tTs principle over the whole relaxation spectrum (from the glassy to the terminal regimes) for the amorphous bottlebrushes (PH, PO, and PN) and in the melt for the rest of the bottlebrushes. The scale multiplicative factors for time and modulus, aT(T) and bT(T), respectively, with reference temperatures, T0, given in Table 2 are plotted as a function of temperature in Figure 6. The successful superposition of the DFS data measured at different temperatures indicates that all PO bottlebrushes evaluated behave as thermorheologically simple materials,25 within the temperature region examined. Considering that the failure of the tTs principle in the rubbery-to-glass transition zone has been reported for several polymers,26−28 we further verify the tTs validity near the glass transition temperature for the amorphous PO bottlebrushes. The frequency-dependent tan δ master curves (shown in the Supporting Information) were obtained with successful implementation of tTs superposition. As shown in Figure 6, the modulus multiplicative factor is weakly dependent on temperature, indicating negligible density changes with temperature, which is the behavior observed in most molten polymers. The exponential decay of the time multiplicative factor, aT, with temperature is well described by the WLF equation as shown in Figure 6, where the solid lines are best fits to the WLF equation using the constants C1 and C2 listed in Table 2, along with the reference temperatures. As shown in Figure 5, the very well-defined elastic modulus plateaus can be found in all UHMW PO bottlebrushes examined in this study, suggesting well-entangled bottlebrush melts. Therefore, their plateau modulus, G0N, can be directly determined from the value of the elastic modulus, G′, at the frequency, ωmin, where the viscous modulus, G″, reaches a

Figure 6. Time and modulus shift factors versus temperature used to construct the dynamic master curves shown in Figure 5. Solid lines are best fits to the WLF equation with constants C1 and C2 given in Table 2.

minimum: G0N = G′(ω)G″→min.9,29 These plateau modulus values can be found in Table 2. To corroborate the accuracy of these values, a second method to calculate G0N, based on the analysis of the van Gurp−Palmen curves30 (plots of the loss angle, δ = arctan(G″/G′), versus the complex modulus, G* = G′2 + G″2) shown in Figure 7a, is used. This method consists on simply reading the value of G0N at the location of the minimum in δ in the rubbery region.31 These two methods yield nearly identical G0N values, as shown in Figure 7b. These E

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Figure 8. Rubbery plateau modulus and entanglement molecular weight as a function of side chain length. Solid lines are fit to the power law relations given by eqs 3 and 4. Nsc is plotted in linear (left panel) and logarithmic (right panel) scales. Figure 7. (a) van Gurp−Palmen plots of the UHMW PO bottlebrushes constructed using data in Figure 5. (b) Comparison of elastic plateau modulus values the two methods described in the text. The solid straight line is the identity line.

aimed to perform SANS measurements of these polymers is warranted. As indicated in Figure 8, the dependence of the G0N on Nsc follows a power law decay function (solid line in Figure 8) of the bottlebrush side-chain length:

plateau modulus values were then used to compute the entanglement molecular weight, Me, by29 Me =

ρRT GN0

GN0 = 1.05Nsc−1.47 (2)

(3)

and, as a consequence of eq 2, the entanglement molecular weight follows a power law growth dependence on Nsc:

where R is the universal gas constant, T is the absolute temperature, and ρ is the polymer melt density.29 G0N and Me values for all the PO bottlebrushes are listed in Table 2 and plotted as a function of Nsc in Figure 8. Both parameters show a smooth dependence on N sc for the UHMW PO bottlebrushes studied here, in contrast to the scattered values reported by Fetters et al.4,7 (also plotted in Figure 8). Note that molecular weights of the polyolefins used in the study by Fetters et al. were not reported. Therefore, it is not possible to elucidate whether the discrepancy between the two data sets arises from different entanglements levels. However, it should be noted that both sets of data display the same trend. An important consequence of having accurate values of G0N is that they can be used to assess the universal relation between this parameter and the packing length (p), namely, G0N ∼ p−3,32 which has been verified for an extensive number of linear polymers.7,8 The packing length is a structure parameter that captures the relative chain thickness which defined by Witten et al.33 as the ratio between the volume occupied by a chain (M/ρNav, where Nav is Avogadro’s number) to its mean-square end-to-end distance, ⟨h2⟩0. The latter parameter requires measurements of chain conformations via small-angle neutron scattering (SANS) of hydrogen−deuterium isotope blends. Unfortunately, deuterated versions of the α-olefin bottlebrushes are not available at the moment. However, due to the importance of the packing length scenario, further research

Me = 1.85Nsc1.47

(4)

Fitting of eqs 3 and 4 to the plateau modulus and entanglement molecular weight data, respectively, by the same, but negative, exponent implies a good approximation to density dependent on reciprocal temperature is a reciprocal function, i.e., ρ ∝ T−1. Note that the coefficients in eqs 3 and 4 have units of MPa and kg/mol, respectively. The predicted relation between entanglement plateau modulus and degree of polymerization of side chains derived by Sheiko and coworkers1,34 is shown in eq 5 and is the same as our experimental finding of eq 3. GN0 ∼ Nsc−3/2

(5)

This relation is derived from eq 2 using two scaling relationships where the first one is the relationship between bottlebrush persistence length lp and size of the side chains, ⟨Rsc2⟩1/2, namely, lp ∝ ⟨Rsc2⟩1/2 ∝ Nsc1/2, and the second one is the relationship between Me and the dimensions of the entanglement strand, given by the Kavassalis−Noolandi conjecture35 as Me ∝ D6/lp3, where D is the diameter of the entanglement strand (D = 2⟨Rsc2⟩1/2). Fetter et al. proposed that the plateau modulus of a polyolefin could be estimated using the following empirical equations:5 F

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(6)

GN0 = 41.84mb−1.58 (mb = 35−56 g/mol)

(7)

the side-chain interactions rather than interactions between backbones. Therefore, it is of great interest to assess the correlation between the distance between side chains and the friction coefficient. It is know that temperature dependences of relaxation times, τ, in polymer melts and solutions are controlled by the ratio between the friction coefficient, ζ, and the absolute temperature:36

where mb is the average molecular weight per backbone bond of the polymer. The coefficients in eqs 6 and 7 have units of MPa. Fetters and co-workers clearly stated that eqs 6 and 7 are strictly empirical, and they had “no justification for these expressions, except as fits to the data”.5 Equation 7 is plotted, as a dashed line, along with the data for our UHMW bottlebrush POs in Figure 9. This equation does provide an approximate fit to our data but does not accurately describe the data across the whole mb range.

τ∼

ζ T

Additionally,the tTs principle assumes that all relaxation times associated with every relaxation mode in a polymer melt have the same temperature dependence, which makes possible to superimpose linear viscoelastic data at different temperatures using the time and modulus multiplicative shift factors (aT and bT, respectively). Accordingly, the time shift factor can be derived from eq 11 as aT =

ζT τ = 0 τ0 ζ0T

(12)

where τ0 and ζ0 are the relaxation time and the friction coefficient at the reference temperature T0, respectively. Considering that the temperature dependence of the modulus, at any relaxation time, is given by the product of the mass density and the absolute temperature (i.e., G(τ) ∼ ρT), the modulus shift factor is determined as

Figure 9. Rubbery plateau modulus and entanglement molecular weight as a function of mb. The dashed line is a plot of eq 7, and solid lines are plots of eqs 9 and 10. Inset: log−log plot of G0N and Me as a function of 1/7mb − 2.

bT =

ρT ρ0 T0

(13)

where ρ0 is the density at the reference temperature. The viscosity is related to the product of the density and the friction coefficient by

A more accurate relation between G0N and mb can be derived by combining the theoretical scaling relationship between G0N and Nsc (eq 5) and the exact relation between Nsc and mb (mb = 14 + 7Nsc). This leads to the following equation

η ∼ ρζ

i1 y GN0 ∼ jjj mb − 2zzz (8) k7 { which reveals that a simple power law function, such as eq 7, cannot describe the relationship between G0N and mb for molecular bottlebrushes. If, instead of using the theoretical relation given by eq 5, we use the fitted relations between G0N, Me, and Nsc, given by eqs 3 and 4, we obtain −3/2

i1 y GN0 = 1.05jjj mb − 2zzz k7 { and

(11)

(14)

Now, the temperature dependence of the viscosity can be computed from the time and modulus shift factors, using eqs 12−14, as η ρζ = = aT bT η0 ρ0 ζ0

(15)

As shown in Figure 6, the time shift factor show a strong dependence with temperature. Because the side-chain distance is weakly dependent on temperature (see Figure 4), a strong dependence of d2 on ζ could be expected and is shown in Figure 10. A parametric plot of the normalized friction coefficient, ζ/ζ70 °C (where ζ70 °C is the friction coefficient at 70 °C) against the normalized side-chain distance, d2/d2,70 °C (given in Figure 4), is illustrated in Figure 10 for all UHMW PO bottlebrushes evaluated. The normalized friction coefficient and viscosity are calculated using the temperaturedependent time and modulus shift factors in Figure 6 and eqs 12 and 15, respectively. Remarkably, both friction coefficient and viscosity are lowered 2−3 orders of magnitude upon a modest increase of only ∼6% in d2. This suggests a very strong effect of the side-chain interactions on the interchain friction and on the dynamic properties of bottlebrush PO melts. The origin of such strong effects could be a combination of shortrange van der Waals and steric interactions. The latter could potentially be associated with side-chain interdigitation as recently discussed.14

−1.47

(9)

i1 y Me = 1.85jjj mb − 2zzz (10) k7 { where the coefficients in eqs 9 and 10 have units of MPa and kg/mol, respectively. As shown in Figure 9, eqs 9 and 10 deliver exceptional fits to our UHMW PO bottlebrush data. Correlation between Molecular Friction Coefficient and Melt Structure. The intrachain friction coefficient, ζ, describes the average force per chain required to pull it through its local surroundings at a unit velocity. Therefore, chain flexibility and intermolecular forces between surrounding chains determine this friction coefficient value. Because of the large distance separating backbones, one would expect that the molecular friction between bottlebrush chains is dominated by 1.47

G

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brushes with longer side chains, suggesting weaker interactions per unit mass. As Nsc increases, less fractions of the side chains may interdigitate with adjacent bottlebrush molecules, resulting in less side-chain interactions and, consequently, less friction. However, this molecular interpretation requires further confirmation from additional experimental or simulation work. On the basis of the data presented in Figure 10, we postulate that the friction coefficient is universally determined by the side-chain distance for bottlebrushes with short side chains. A very interesting question that warrants further research is whether such relation between side-chain distances and monomeric friction holds for polyolefins with even shorter side chains (i.e., polypropylene to poly(1pentene)), where the backbone-to-backbone interactions become much stronger and, for α-olefin bottlebrushes with very long side chains (Nsc ≥ 17), were backbone-to-backbone interactions are strongly hindered.

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CONCLUSIONS We report the first rheological study of UHMW polyolefin bottlebrushes with side-chain lengths ranging from 4 to 16 carbons and average number of entanglements per chain, Z, ranging from 53 to 451. Because of their extensive entanglements, these bottlebrush polymers exhibit a welldefined plateau modulus which made accurate determination of G0N and Me possible. The theoretical scaling between plateau modulus and side-chain length predicted for molecular bottlebrushes bottlebrushes (G0N ∼ Nsc−3/2) was confirmed by our experimental data:G0N ∼ Nsc−1.47. Based on these scaling relations, a new equation between plateau modulus and average molecular weight per backbone, G0N ∼ (mb/7 − 2)−1.47, was obtained which describe our results more accurately than the empirical relation reported by Fetters et al.5 Using temperature dependences of the rheological properties and the structural parameters (measured by WAXS) allowed us to establish relationships between friction coefficient, viscosity and side-chains distance, d 2. Dependences of friction coefficient and viscosity on d2 for all bottlebrush POs examined exhibit two exponential decay regimes with decay constants that, in turn, follow exponential dependence on Nsc. To our knowledge, this is the first experimental evidence of the strong dependence of interchain friction forces and dynamic properties on structural parameter of molecular bottlebrushes, which can be summarize as ζ, η = f(d2(T),Nsc).

Figure 10. Normalized friction coefficient versus normalized characteristic length d2.

The dependence of both friction coefficient and viscosity on the side-chain distance for all the PO bottlebrushes can be segmented into two regimes. Each regime describes an exponential dependence on d2/d2,70 °C log log

log log

ζ ζ70 °C η η70 °C ζ ζ70 °C η η70 °C

| o o o o o d 2, 70 °C o o o } o (d 2/d 2, 70 °C < D) d2 o o o o ∼ −K1 o o d 2, 70 °C o ~ | d2 o o o ∼ −k 2 o o d 2, 70 °C o o o } (d 2/d 2, 70 °C ≥ D) o d2 o o o o ∼ −K 2 o o d 2, 70 °C o ~ ∼ − k1

d2

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(16)

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b01431. Plots of dynamic moduli and heat flow (DSC) as a function of temperature and plot of dynamic master curves of tan δ (PDF)

k1 = 8.94 + 158e−0.16Nsc

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k 2 = 9.67 + 104e−0.23Nsc K1 = 6.81 + 165e−0.14Nsc K 2 = 10.6 + 110e−0.21Nsc

ASSOCIATED CONTENT

S Supporting Information *

where D is the intersection point between the two regimes. The values of the decay constants, ki and Ki (i = 1, 2), are, in turn, a function of the side chain length, Nsc, as shown in the insets of Figure 10. These are fitted with the exponential decay functions

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (C.R.L.-B.).

(17)

ORCID

By combining eqs 16 and 17, the overall dependences of the friction coefficient and viscosity of PO bottlebrushes on the side-chain distance and the side-chain length can be described. Interestingly, weaker dependences are observed for bottle-

Carlos R. López-Barrón: 0000-0002-9620-0298 Notes

The authors declare no competing financial interest. H

DOI: 10.1021/acs.macromol.8b01431 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

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DOI: 10.1021/acs.macromol.8b01431 Macromolecules XXXX, XXX, XXX−XXX