Molecular Structure, Chain Dimensions, and Linear Rheology of Poly

Article pubs.acs.org/Macromolecules

Cite This: Macromolecules XXXX, XXX, XXX-XXX

Molecular Structure, Chain Dimensions, and Linear Rheology of Poly(4-vinylbiphenyl) Carlos R. López-Barrón,*,† Huaxing Zhou,‡ Jarod M. Younker,† and Jason A. Mann† †

ExxonMobil Chemical Company, Baytown, Texas 77520, United States ExxonMobil Research and Engineering Company, Annandale, New Jersey 08801, United States

‡

S Supporting Information *

ABSTRACT: The linear viscoelasticity and molecular structure of a series of poly(4-vinylbiphenyl) (PVBP) samples with polydispersities ranging from 1.08 to 2.75 are investigated using wide-angle X-ray scattering (WAXS), small-angle neutron scattering (SANS), and rheological measurements. PVBP chain dimensions, such as statistical segment length, persistence length, and packing length, are obtained from SANS measurements. Because of the bulkiness of the biphenyl monomer, PVBP has the lowest chain flexibility and the largest reptation tube diameter among most common thermoplastics, including polystyrene and polyvinylcyclohexane. Nevertheless, PVBP follows the universal power law dependence between elastic modulus and packing length (GN0 ∝ p−3). X-ray diffraction of PVBP consist of two main amorphous peaks: one associated with the interchain (backbone) correlations and a second one arising from phenyl−phenyl correlations. The latter are due to π−π stacking of the phenyl rings, mostly arranged in the T-shaped configuration. Molecular dynamic simulations validate the diffraction peak assignments. The linear rheology of PVBP samples reveals a thermorheologically simple response and a very low value of G0N (0.065 MPa). This is consequence of the low chain flexibility, which also explains the relatively high entanglement molecular weight value for PVBP (58.3 kg/mol).

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which was defined by Witten et al.4 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

INTRODUCTION Fundamental viscoelastic parameters of molten polymers, such as zero-shear viscosity, steady-state compliance, and rubbery plateau modulus, are directly related to molecular characteristics such as persistence length, radius of gyration, and entanglement molecular weight. In turn, polymer rheological properties are closely related to their processability, which dictates their ability to, for instance, fill a mold or form a stable molten film. It follows that structural and rheological characterization of new polymeric molecules is key to fully understand both their flow behavior and dynamic response in the melt state.1 One of the most commonly reported structural parameters is the molecular weight between entanglements, Me, which is defined in terms of the rubbery plateau modulus, G0N, as2 Me ≡

ρRT GN0

p≡

(2) 3,5,6

Fetters and co-workers demonstrated a universal relation between entanglement spacing and packing length of flexible linear Gaussian chains, expressed as

Me = nt 2Navρp3

(3)

where nt is a temperature-independent dimensionless coefficient equal to 21.3 ± 1.6.5,7 This parameter denotes the number of entanglement strands per cubed tube diameter.7 Combining eqs 1 and 3 yields a relation between elastic modulus and packing length

(1)

GN0 =

where ρ is the mass density of the polymer, R is the gas constant, and T is the temperature. As discussed by Fetters et al.,3 Me is directly correlated to the “bulkiness” of side groups in the polymer chain. In general, “skinny” chains (as in polyethylene or polybutadiene) have lower values of Me because they entangle more easily than “fat” chains (as in PS or polydimethylsiloxane). A structure parameter that captures the relative measure of chain thickness is the packing length, p, © XXXX American Chemical Society

M 1 ρNav ⟨h2⟩0

RT −3 p nt 2Nav

(4)

Because of its commercial relevance, polystyrene (PS) is one of the most exhaustively studied polymers, and therefore its Received: July 21, 2017 Revised: October 27, 2017

A

DOI: 10.1021/acs.macromol.7b01564 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules Table 1. Molecular Characteristics and Linear Viscoelastic Parameters of PVBP and PS Samples sample ID

polymerization reaction

Mn (kg/mol)

Mw/Mn

PVBP17 PVBP71 PVBP171 PVBP507 PVBP857 PVBP27 PVBP47 PVBP120 PVBP140 PS

anionic anionic anionic anionic anionic free radical free radical free radical free radical free radical

17.5 71.2 171 507 857 27.2 47.3 120 140 69.7

1.07 1.08 1.12 1.71 1.67 2.05 1.90 2.10 2.75 2.3

viscoelastic and molecular characteristics are well documented.1 However, rheological and structural characterization of closely related aromatic polymers, such as poly(4-vinylbiphenyl) (PVBP) or poly(2-vinylnaphthalene), has not been reported to date. PVBP has a significantly higher glass transition temperature than polystyrene, which makes this polymer commercially attractive for high-temperature applications (e.g., in microwave-safe expanded polystyrene foams). However, the structure, dynamics, and viscoelasticity of this polymer are not known to date. One would expect that due to the “bulkier” nature of the bicyclic monomer 4-vinylbiphenyl, compared to styrene, the persistence length of PVBP should be greater than that of PS. Consequently, the elastic modulus of PVBP should be lower and its entanglement molecular weight greater than those of PS. Furthermore, one would expect different dynamics on PVBP, compared to other amorphous polymers, due to the numerous π−π interactions. This paper reports the linear viscoelastic response and structural characterization of a series of PVBP samples with varying molecular weight and polydispersity values. A combination of small-angle neutron scattering (SANS) and wide-angle X-ray scattering (WAXS) measurements and molecular dynamic simulations are used to determine structural parameters, such as Rg and backbone-to-backbone distance, which are correlated with rheological properties to compute Me and p. To our knowledge, this is the first systematic study of a linear polymer having strong π−π interactions. Structural analysis of a PS sample is carried out in parallel, for comparison.

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η0 (Pa·s)

C1

C2

Tg (°C)

× × × × × × × × ×

6.97 8.21 8.53 8.89 8.85 9.01 8.99 8.41 8.70

97.8 103 99.2 104 106 114 116 105 115

150.3 154.4 156.0 155.6 154.5 136.8 148.6 154.2 151.8

3.10 2.38 2.33 6.03 3.78 7.12 1.55 1.41 5.19

104 105 106 107 108 104 105 106 106

τs (s) 9.92 9.52 1.16 1.22 1.46 5.98 4.98 1.54 5.58

× × × × × × × × ×

10−6 10−6 10−5 10−5 10−5 10−6 10−6 10−5 10−6

to initiate the polymerization. The reaction was kept at room temperature for 2 h and then quenched by degassed methanol. Polymer was collected and purified by precipitating in methanol. Final polymer was collected by filtration and dried in a vacuum oven. The molecular weight (MW) of the polymers was determined by gel permeation chromatography (GPC) analysis with tetrahydrofuran as eluent. MW was determined using narrow polydispersity polystyrene standards and the universal calibration method.8 Mn and polydispersity (PDI = Mw/Mn) values of all the polymers used in this investigation are listed in Table 1. A small amount (∼0.2 wt %) of butylated hydroxytoluene (BHT) was added to each sample to delay the onset of thermal degradation at high temperatures during the rheological measurements. All the PVBP samples studied here are amorphous (as assessed by X-ray diffraction measurements, discussed below) and therefore transparent. Their glass transition temperatures (Tg), measured by differential scanning calorimetry (DSC), are listed in Table 1. The Tg values were also determined by dynamic mechanical spectroscopy, which produced values nearly identical to those obtained by DSC, as shown in Figure S1 of the Supporting Information. H−D Exchange Reaction. Sample PVBP171 was partially deuterated following a hydrogen−deuterium (H−D) exchange process reported by Willenberg9 for deuteration of atactic polystyrene. In short, a 10 wt % solution of PVBP171 in hexadeuterobenzene (99.6%, Sigma-Aldrich) is prepared by stirring at room temperature. 50 μL per gram of polymer of a 1 M solution of ethylaluminum dichloride in hexane is added to the solution. After 4 h of stirring at room temperature, the solution was poured into methanol to precipitate the polymer, which is collected by filtration, washed with additional methanol, and dried in a vacuum oven at 40 °C. The deuterated sample (dPVBP171) was characterized by GPC and 1H NMR. NMR spectra of both PVBP samples (before and after isotope exchange) are shown in Figure S2, and the corresponding GPC traces are shown in Figure S3. From the aromatic/aliphatic integration area ratio, it was deduced that the deuterium exchange level (DL) is 95%. SEC measurements of the sample before and after deuteration indicate a small decrease in MW without broadening the molecular weight distribution. A 90/10 PVBP171/dPVBP171 isotope blend was prepared by dissolving in THF and precipitation in cold methanol. The precipitate was washed three times with methanol, after which 0.1 wt % of BHT was added to avoid thermal degradation. The sample was dried under vacuum at 130 °C and compression molded at 190 °C into discs with 1 mm thickness for SANS measurements. Small-Angle Neutron Scattering (SANS). SANS measurements were performed on the PVBP171/dPVBP171 isotope blend. Measurements were performed at the National Institute of Standards and Technology (NIST) Center for Neutron Research, using the NG7 30m SANS instrument with incident beam wavelength λ = 6 Å and wavelength spread Δλ/λ = 11%. Three configurations (sample-todetector distance) were used: 1, 4, and 13.5 m to cover a wide range in momentum transfer vector q (= |q| = (4π/λ) sin θ) of 0.003−0.6 Å−1. Raw data were corrected for sample transmission, background radiation, sample thickness, and detector sensitivity using IGOR macros available from NIST and following standardized procedures.10 The 2-D scattering data were averaged azimuthally to obtain one-

EXPERIMENTAL SECTION

Materials. A series of PVBP samples were prepared via free radical polymerization (both in bulk and in solution) and by anionic polymerization of 4-vinylbiphenyl (Boron Molecular). Solution free radical polymerization was carried out in 30 wt % toluene solutions at 100 °C, using azobis(isobutyronitrile) (AIBN) as initiator. The polymers were precipitated in cold methanol and collected as white powder. The precipitates were dried at room temperature for 24 h, followed by drying at 120 °C for 3 h under vacuum. A PS sample was also synthesized in solution using styrene (≥99%, Sigma-Aldrich) and AIBN as initiator. Bulk polymerization was performed by heating the monomer to 130 °C for 8 h, with no initiator present. Anionic polymerization was carried out following standard methods. In short, 4-vinylbiphenyl monomer was first purified by recrystallization from ethanol, transferred into a drybox, and dissolved in dry benzene to prepare a 10% solution. Then, a specially designed air-free flask was oven-dried at 110 °C and transferred into the drybox. The monomer solution was then passed through a short activated basic alumina column into the air-free flask. The flask was then closed, taken out of the drybox and freeze−pump−thaw using dry ice in acetone cold bath three times, and then allowed to warm to room temperature. sec-Butyllithium (sec-BuLi) was added in the air-free flask very quickly B


Macromolecules

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dimensional plots of intensity, I, versus scattering wave vector, q. SANS data from the isotope blends were analyzed using the Debye function, D(x)

D(x) =

2(e−x + x − 1) ; x2

x = (qR g)2

Article

RESULTS AND DISCUSSION Molecular Structure of PVBP. Figure 1 shows azimuthally averaged SANS data for the 90/10 PVBP171/dPVBP171

(5)

where Rg is the polymer radius of gyration. The Debye function represents the form factor of a linear polymer chain in the melt or in dilute solution (in a theta solvent).11 The scattering is calculated as I(q) = AD(x)

(6)

where the prefactor A is a function of the volume fraction, the degree of polymerization, and the scattering contrast. Wide-Angle X-ray Scattering (WAXS). 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, λ = 1.54 Å), focusing optics, and Dectris Pilatus 300K detector (487 × 619 pixels, pixel dimension 172 μm). 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. Molecular Dynamic (MD) Simulations. Polymers and amorphous cells were constructed and optimized with Scienomics MAPS 4.0.1 software. MD simulations were performed with Desmond from Schrodinger Materials Science Suite 2016-4 with the Optimized Potentials for Liquids Simulations (OPLS3) force field at ambient pressure.12 Cells were heated to 600 K and simulated under an isobaric−isothermal ensemble (NPT) for 1.2 ns (2 fs integration time, chained Nosé−Hoover thermostat (1 ps relaxation time), and Martyna−Tobias−Klein barostat (2 ps relaxation time)). The cells were then cooled to 463 K (polyvinylbiphenyl) or 378 K (polystyrene) and equilibrated (NPT) for 1.2 ns. Production MD NPT runs followed at the equilibrated temperatures for an additional 1.2 ns. Averages of production frames were used for scattering predictions and distributions. X-ray scattering predictions were performed with BioVia Materials Studio 2017R2. Intra- and interchain radial distribution functions (RDF) were also determined for tertiary and aromatic carbons. Rheological Measurements. Two sets of dynamic mechanical spectroscopy tests were performed to measure the viscoelastic response of the samples using a strain-controlled rheometer ARESG2 (TA Instruments) with a nitrogen environment. Isochronal dynamic temperature ramps (DTR) measurements at a frequency of 1 Hz and strain of 0.1% were performed, using serrated 8 mm parallel plates to determine the glass transition temperature (Tg) of the PS and selected PVBP samples. The DTR data for PVBP507 are shown in Figure S1b, and the Tg values are listed in Table 1. Isothermal dynamic frequency sweeps (DFS) measurements were carried out at temperatures ranging from 150 to 270 °C, with frequencies ranging from 100 to 0.001 Hz. At the highest temperatures (T > Tg + 40 °C) the DFS tests were performed using 8 mm diameter parallel discs, whereas at temperatures < Tg + 40 °C, special 3 mm diameter serrated parallel plates are used along with sample gaps > 2.5 mm, to minimize instrument compliance effects, as suggested by Schröter et al.13 The strain amplitude used at each temperature varied from 0.1% (at the lowest temperature) to 10% (at the higher temperature) which were checked to be within the linear viscoelastic regime of the samples. The DFS data were shifted vertically and horizontally following the time− temperature superposition (tTs) principle to construct master curves of the dynamic moduli at a reference temperature T0 = 190 °C. The time (horizontal) shift factors are fitted to the Williams−Landel−Ferry (WLF) equation14

log aT =

− C1(T − T0) C 2 + (T − T0)

Figure 1. SANS profiles of PVBP171/dPVBP171 isotope blend. Solid line is the best fit of the data to eq 6.

isotope blend measured at 190 °C. The solid lines are best first to the Debye function (eq 6), using the Rg value of 92.4 Å. The fitted Rg values can be used to calculate other chain dimensions, such as the mean-square end-to-end distance, ⟨h2⟩0, the statistical segment length, b, and the persistence length, lp, using the well-known relationships15 Rg2 =

lpL ⟨h2⟩0 Nb2 = = 6 6 3

(8)

where N is the degree of polymerization and L is the contour length (maximum length of the fully extended chain). The latter can be computed as L = 2.52N, where the prefactor 2.52 accounts for the approximate average length between two monomers in the backbone. The Mn value in Table 1 is used to calculate N = Mn/M0, where M0 = 185.25 g/mol is the monomer MW. Equation 8 is used then to compute b = 7.35 Å, lp = 10.7 Å, and ⟨h2⟩0/M = 0.30. The latter quantity is used to calculate the packing length, using eq 2, whose value is discussed in relation to the rubbery plateau modulus below. The chain dimensions of PVBP, along with values of common polymers reported by Fetters and co-workers,3 are listed in Table 2. Note that although the measurements reported by Fetters et al. were carried out at different temperatures, which slightly affect the distribution of chain conformations, PVBP clearly has the largest persistence length and packing length. This reflects the bulkier nature of the side chains, which makes them stiffer than the rest of the polymers listed in Table 2. This has direct consequences in the viscoelastic response of this polymer, as discussed below. Additional structure information is obtained from the analysis of the azimuthally averaged WAXS profiles shown in Figure 2, which include data for both the PVBP171 and the PS samples. Both samples show three main peaks, which are labeled with their position in the q-axis as q1*, q2*, and q4*. Note that PVBP shows an additional peak at q*3 , not observed in PS, that is discussed below. The two major peaks in PS were first reported by Kats,16 who termed the peak at q1* “the polymerization ring” because it is only observed in the polymer

(7)

where C1 and C2 are empirical constants. C


Article

Macromolecules Table 2. Molecular and Rheological Characteristics of PVBP and Some Common Polymers polymer a

polyethylene polypropylene (atactic)a polyisobutylenea PSa polyvinylcyclohexanea PVBP a

T (°C)

ρ, (kg/m3)

⟨h2⟩0/M (Å2 mol/g)

b (Å)

lp (Å)

p (Å)

G0N (MPa)

Me (kg/mol)

dt (Å)

140 140 140 140 160 190

785 791 849 969 920 981

1.25 0.67 0.57 0.43 0.32 0.30

5.9 5.3 5.6 6.7 5.9 7.35

5.7 4.6 5.2 7.3 6.9 10.7

1.69 3.13 3.43 3.95 5.59 5.65

2.60 0.47 0.32 0.20 0.068 0.065

0.84 4.62 7.29 13.3 39.0 58.3

32.8 60.7 66.4 76.5 108 132

Data reported by Fetters et al.3

order. WAXS analysis of oriented PS (by cold drawing) has been used to determine that the “polymerization” peak, q*1 , arises from interchain interactions, as it intensifies on the direction perpendicular to the stretching direction upon drawing.16−19 Therefore, this peak results from backbone-tobackbone correlations, which, due to the atactic nature of the chains, have no long-range order.20 We now analyze the diffraction pattern of PVBP, shown in Figure 2, in light of the early studies on PS structure. The “polymerization” peak, q*1 , is at much lower scattering angle in PVBP than in PS, which is expected as biphenyl groups in PBVP force the backbones of adjacent chains to be farther apart that in PS. Given the similarity of the peak at q*2 observed in PVBP with that observed in PS, we deduce that it arises from interaction between adjacent phenyl groups from neighbor chains, as illustrated in Figure 3a. Note that Figure 3 depicts characteristic distances computed as di = 2π/q*1 (with i = 1, 2, 3, and 4). The shoulder observed at q*3 in PVBP (but not in PS) also arises from interphenyl interactions with a different configuration, as discussed below. The position of the peak at the wider angle, corresponding to d4 = 2.12 Å, suggests that it results from the correlation between adjacent phenyl groups attached to the backbone, as illustrated in Figure 3b. This is because these phenyl rings are forced to be apart by a distance of ∼2.2 Å, which corresponds to the distance between every other carbon in the backbone. To have a better understanding of the origin of the peaks at q2* and q3*, we need to consider the aromatic π−π interactions between two adjacent phenyl rings. The three main

Figure 2. WAXS profiles for the PVBP120 and the PS samples.

but not in liquid styrene. In contrast, the brightest peak, at q2*, is very similar to that observed in both styrene monomer and benzene.17 This indicates that that diffraction peak is dominated by phenyl−phenyl correlations between adjacent PS chains, which are very similar to those associated with shortrange packing between aromatic groups in liquid benzene and styrene. Mitchell and Windle proposed a structural model for PS in which the side phenyl groups segregate on a molecular scale to form stacks with narrow cores of low electron density.17 The nature of the stacks is disordered, i.e., with no long-range

Figure 3. Schematic cartoon of two adjacent PVBP chains showing (a) the interchain distance, the π−π phenyl−phenyl distance, and (b) the backbone-adjacent phenyl−phenyl distance. (c) Schematic representation of benzene dimer (π−π) conformers. D


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Macromolecules

Figure 4. Baseline-subtracted WAXS data for the (a) PVBP171 and (b) PS samples. Solid lines are best fits to Gauss functions. Insets show cumulative fit peaks to the data.

Table 3. Diffraction Peak Positions and Characteristic Distances for PS and PVBP PS PVBP

WAXS MD sim WAXS MD sim

q*1 , Å−1 (fwhm)

d1 (Å)

q*2 , Å−1 (fwhm)

d2 (Å)

0.69 (0.22) 0.64 0.40 (0.19) 0.35

9.11 9.82 15.7 17.9

1.38 (0.39) 1.29 1.33 (0.55) 1.22

4.55 4.87 4.73 5.15

q*3 , Å−1 (fwhm)

1.94 (0.42) 2.01

d3 (Å)

q*4 , Å−1 (fwhm)

d4 (Å)

3.23 3.13

2.89 (0.40) 2.76 2.97 (0.49) 2.86

2.17 2.27 2.12 2.20

Figure 5. Left panel: MD simulations of PVBP and PS. PVBP simulation were carried out at 463 K and 1 atm (average density 0.97 g/cm3), using 41 632 atoms (16 chains of 100 monomers). PS simulation was carried out at 378 K and 1 atm (average density 0.96 g/cm3), using 25 632 atoms (16 chains of 100 monomers). Right panel: X-ray diffraction predictions from MD simulations for PVBP and PS (solid lines) and comparison with WAXS measurements (dashed lines) measured at 190 °C, for PVBP, and 105 °C, for PS.

carbon bond of the adjacent ring. Additionally, the distance between phenyl rings is less in the SD configuration than in either the S or the T configurations, which is reflected as a stabilizing effect of the SD conformers due to interpenetration of the electron clouds.21 To facilitate the diffraction peak analysis, we performed baseline subtraction of the WAXS profiles, followed by multiple peak fit using Gauss functions. The results are shown in Figure 4, where the symbols are the baseline-subtracted data and the solid lines are the fitting results. Also shown (in the insets) are the cumulative peaks resulting from the sum of the individual peak fits. The center (peak position) and the full width at halfmaximum (fwhm) of each peak are listed in Table 3. Note that

configuration resulting from this type of interaction, namely, sandwich (S), T-shaped (T), and parallel displacement (PD), are illustrated in Figure 3c for the case of the benzene dimer (the simplest prototype of aromatic π−π interactions). Equilibrium interphenyl distances, obtained via electronic structure theory,21 are Rs = 3.9 Å for the S configuration, RT = 5.0 Å for the T configuration, and RPD1 = 3.6 Å and RPD2 = 1.6 Å for the PD configuration. Because of quadrupole− quadrupole interaction, the T-shaped conformer has a more stabilizing electrostatic interaction than the S configuration. The PD configuration has the most favorable electrostatic interaction because some of the partially positive hydrogens on each aromatic ring are located on top of the partially negative E


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Macromolecules the brightest peak at q*2 for both PS and PVBP correspond to similar distances (4.6 and 4.7, respectively), which indicates that the majority of the phenyl−phenyl interactions are in the T-shaped configuration for both polymers. Interestingly, this is not the most stable conformer for the benzene dimer, as discussed above. The predominance of this conformer in the polymers could result from the fact that attachment of the phenyl rings to the backbones of PS and PVBP considerably hinder their freedom to acquire the PD configuration. Figure 4a shows that the brightest peak and the adjacent shoulder for PVBP can be deconvoluted in two distinct peaks, q*2 and q*3 . The characteristic distance corresponding to peak 3 is 3.23 Å, which suggests that it arises from π−π stacking with PD geometry. The area ratio between the two peaks 3 and peak 2 is 0.084, which implies that less than 10% of the phenyl π−π stacking is in the PD configuration and the rest is predominantly in the T configuration. Note that peak 3 is absent in PS (Figure 4b), indicating that PD conformers are not allowed in this polymer. Let us consider now the characteristic distances corresponding to the interchain (“polymerization”) peaks, d1, which are 8.7 Å for PS and 15.4 Å for PVBP. The average overlap distance between phenyl rings from adjacent chains is computed as doverlap = 2Lphenyl − d1, where Lphenyl is the total length of the single phenyl rings in PS (4.3 Å) or the biphenyl groups in PVBP (8.7 Å), including the C−C distance from the backbone to the adjacent phenyl ring and the distance between the two phenyl rings in the biphenyl groups. Therefore, on average, there is no overlap between phenyl rings from adjacent chains in PS, which explains the absence of PD conformers in PS. Although the phenyl rings do not overlap, they still interact in a T-shaped π−π configuration, as shown by WAXS. In PVBP, there is an average overlap distance of 2 Å (illustrated schematically in Figure 3a), which explains the small population of PD π−π conformers. The equilibrium conformations of PVBP and PS (at 463 and 378 K, respectively), obtained by MD simulations, are shown in Figure 5, along with predictions of the wide-angle diffraction. The simulations predict very well the diffraction peaks in both the PS and the PVBP samples with some quantitative discrepancies. The interphenyl peaks predicted by the simulations are slightly wider and centered at lower q-values compared the measured WAXS. However, the differences between the characteristic distances predicted by MD simulations are within 10% from those measured by WAXS (see Table 3). The radial distribution functions (RDFs) shown in Figure 6 validate the experimental conclusions about the nature of the scattering peaks. The low q polymerization peaks correlate with the interchain RDF distance, 10 and 18 Å for PS and PVBP, respectively. The interchain distance distribution is wider for PVBP and is reflected in the less intense, broad nature of the peak at q1. The low r oscillations of the interphenyl RDF are indicative of “order” between the phenyl side chains, as illustrated in the snapshots in Figure 6 (lower panel). Two distinct maxima are observed at r of ∼5 and ∼6 Å. The former is responsible for the scattering peak at q2. It is of interest to analyze the temperature effect on the microstructure of the PVBP samples. Figure 7a shows WAXS profiles of PVBP171 measured at different temperatures. MD simulated profiles, as a function of temperature, can be found in the Supporting Information. A slight shift in the position of both the interchain and the interphenyl peaks to lower q-values

Figure 6. Left: interchain radial distribution functions (RDFs) based on tertiary carbon (top) and aromatic carbon (bottom): Right: snapshot of final trajectory frame highlighting interphenyl distances for PVBP and PS.

results from the expected thermal expansion of the melt near and above Tg. Figure 7b shows the change in d-spacing corresponding to both peaks. Interestingly, both spacing values show the same linear increase with temperature, which indicates that the increase in distance between phenyl rings is proportional to the increase in distance between adjacent backbones. Note that the increase in d2 is due to (1) pulling apart of phenyl rings from adjacent chains by their corresponding backbones and (2) perpendicular separation of phenyl rings due to π−π interaction weakening with increasing temperature. Above Tg, the intensity of the interphenyl peak (at q*2 ) slightly decreases and the peak width increases, which are the expected trends for any material (crystalline or amorphous) that undergoes an increase in thermal disorder as the temperature is raised. The fact that the shape of both peaks remain unchanged indicates that the expansion occurs without any overall structural transition. The only behavior that is unusual is the pronounced increase in intensity of the interchain peak as a function of temperature. Similar behavior was previously reported in polystyrene by Mitchel and Windle.17 They postulated that this behavior is a consequence of the increase of the interphenyl ring distances during thermal expansion, which produces areas with electron deficiency (with respect to their surroundings). This results in an increase in Xray contrast between those “void” areas and the electron-rich regions near the chain backbones, which, in turn, manifests as the observed increase in WAXS intensity. We believe the same mechanism operates in the PVBP samples studied here. Note that the intensity increase with temperature is also predicted by the MD simulations, as illustrated in Figure S4 for both PS and PVBP. Linear Rheology. We now discuss the linear viscoelastic response of the PVBP samples. Figures 8a and 9a show the dynamic moduli (G′ and G″, or elastic and viscous modulus) of the PVBP sets with low and high PDI, respectively, constructed using the tTs principle over the whole relaxation spectrum (from the glassy to the terminal regimes). The time scale multiplicative shift factor, aT(T), with reference temperature T0 = 190 °C is plotted as a function of temperature in Figures 8 F


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Macromolecules

Figure 7. (a) WAXS profiles for the PVBP120 measured at the indicated temperatures. (b) Temperature dependences of d-spacing and intensity changes with respect to T = 25 °C.

Figure 8. Master curves: (a) the dynamic moduli and (b) the complex viscosity of the PVBP samples prepared by anionic polymerization constructed via the tTs principle using a reference temperature of T0 = 190 °C. Inset in (a) shows the time shifts factor as a function of temperature used to construct the master curves.

Figure 9. Master curves: (a) the dynamic moduli and (b) the complex viscosity of the PVBP samples prepared by free radical polymerization constructed via the tTs principle using a reference temperature of T0 = 190 °C. Data of the anionically made PVBP857 are shown for comparison. Inset in (a) shows the time shifts factor as a function of temperature used to construct the master curves.

and 9 (insets). The successful superposition of DFS data at different temperatures, after shifting, indicates that all the PVBP samples behave as thermorheologically simple materials22 in the

temperature range studied here. Note that the failure of time− temperature superposition (tTS) in the rubbery-to-glass G


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Macromolecules transition zone has been reported for many polymers.23−26 Therefore, to further verify the validity of the tTs principle near the glass transition temperature for the PVBP samples, we have included plots of the tan δ and complex modulus (|G*|) master curves (magnified in the glass-to-rubber transition zone) in the Supporting Information (Figure S6). We currently do not have an explanation of why the tTs principle is obeyed for PVBP in the rubber to glass transition, despite the observed failure in other polymers. Therefore, further research is warranted to investigate this phenomenon. The exponential decay of aT with temperature is well described by the WLF equation (eq 7), as evidenced in the insets in Figures 8 and 9, where best fits to the WLF equation are plotted as solid lines. The WLF constants, C1 and C2, are listed in Table 1. The similarity on the value of the constants between all the samples suggests that they are independent of MW or PDI, and they take average values of C1 = 8.60 ± 0.4 and C2 = 107 ± 6. Viscoelastic and Thermal Behavior near Tg. All the PVBP samples are optically transparent at any temperature, indicating that they are fully amorphous, which is confirmed by the absence of Bragg diffraction peaks in the WAXS measurements (Figure 2). The Tg values, listed in Table 1, were measured by DSC (Figure S1a) and by DTR as the position of the maximum peak in G″ (Figure S1b). Tg is independent of the MW for the high MW PVBP samples, whereas a slight decrease in Tg is observed for the two samples with lower MW (PVBP47 and PVBP27). This is expected, as these two samples are polydisperse and therefore contain a considerable amount of very low MW (unentangled) PVBP chains. We now discuss the linear viscoelastic response of the PVBP samples in the glassy and segmental regions. Figures 8a and 9a show that at very high frequencies ωτS > 1 (where τS is the segmental relaxation time) the moduli data for all the samples overlap, and the glassy modulus, Gglass, has a value of ∼1 GPa. The latter can be obtained by extrapolating the van Gurp− Palmen curves (plots of the loss angle, δ = arctan(G″/G′), versus the magnitude of the complex modulus,

calculate Me = 58.3 kg/mol for PVBP. Note that this value correspond to a critical degree of polymerization for entanglement of Ne = 324, which is ∼2.5 times larger than the corresponding value for polystyrene, for which the reported Me = 13.3 kg/mol,3 and therefore Ne = 128. This is not surprising, as it is well-known that Me increases as a function of the “bulkiness” of the side groups attached to the polymer chain backbone. This is also reflected on the value of the unperturbed reptation tube diameter, which can be computed as3 dt =

Me

⟨h2⟩0 M

(9)

As shown in Table 2, PVBP has a tube diameter 1.7 times larger than PS and 4 times larger than polyethylene (PE). Those differences are simply a result of the difference in side groups bulkiness, which makes PVBP a fatter chain than PE or PS. Another quantitative parameter that describes the polymer chain thickness is the packing length, which tends to have smaller values for thinner chains. This is evident in Table 2, which lists the p values (computed with eq 2) of a series of polymers in increasing order of side chain length. It is of interest to determine whether PVPB obeys the universal correlation between plateau modulus and packing length, described in eq 4. Figure 10 shows the linear dependence of G0N

|G*| = G′2 + G″2 ) to δ → 0 in the segmental regime (Figure S5). The segmental relaxation time is obtained from the reciprocal of the frequency value at the moduli crossover in the segmental regime, as indicated in Figure 8a. The τS values, listed in Table 1 for all the samples, are nearly identical for all the samples with Mn > 47 kg/mol, except for PVBP140, which has a higher PDI than the rest. The increase in PDI is also reflected in the stronger frequency dependence of complex viscosity (that reflects broader relaxation mode distribution noted for|G*|), as evidenced in Figure 9b. The two samples with Mn ≤ 47 kg/mol have a slightly shorter segmental relaxation time than the rest of the samples (Table 1), which may be because these two samples are not entangled, as discussed below. Additionally, the transition regime (between the segmental and the rubbery plateau modulus regimes) is also different on these two samples compared to the samples with higher MW, for which the moduli overlap perfectly. Plateau Modulus and Entanglement Molecular Weight. The dynamic moduli data of the PVBP sample with the highest MW (PVBP857), given in Figure 8a, were used to calculate the rubbery plateau modulus as the value of G′ at which G″ reaches a minimum (G0N = G′|min G″) in the rubbery plateau region of the relaxation spectrum.14,27,28 This method yields a G0N value of 0.065 MPa, which is used in eq 1 to

Figure 10. Log−log plot of the rubbery plateau modulus versus the packing length for a series of polymers (data taken from Table 1 in ref 3), including PVBP (this work).

on p−3 using experimental data from an extensive number of polymers compiled by Fetters and co-workers.3 The data for PVBP align very well with the rest of the data, which indicates that this polymer follows the universal property of flexible entangled chains described in eqs 3 and 4, which asserts that entanglements spacing is only determined by density and flexibility. Furthermore, using data in Table 2 and eq 3, the computed value of nt for PVBP is 23.4, which is within reported error, the universal value for entangled polymers.5,7 Viscosity Dependence on MW. As is typical in polymer systems, both the extent of the rubbery plateau and the melt viscosity are highly dependent on MW. Figures 8b and 9b show the complex viscosity, |η*| = |G*|/ω, as a function of frequency for both sets of PVBP samples. Clearly, the zero-shear viscosity, η0, measured as the viscosity plateau at the lowest frequency values, is an increasing function of MW (see Table 1). Figure H


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Macromolecules

length (G0N ∝ p−3) reported by Fetters et al.3 Finally, PVBP shows the typical power law dependences on MW, with exponents 1.1 and 3.2 for MW < Me and MW > Me, respectively.

11 shows the molecular weight dependence of η0 at 190 °C. For the shorter PVBP samples, for which Mn ≤ Me, the weak power

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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.7b01564. Figures S1−S6 (PDF)

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AUTHOR INFORMATION

Corresponding Author

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

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

The authors declare no competing financial interest.

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Figure 11. Zero-shear viscosity (measured at 190 °C) as a function of number-average molecular weight for PVBPs prepared by anionic and free radical polymerization.

ACKNOWLEDGMENTS The authors thank Maksim Shivokhin and Alexander Norman for useful discussions. We also thank Joseph Throckmorton for his assistance with the WAXS measurements.

law dependence (with exponent of 1.1) is consistent with the Rouse model for nonentangled polymers, which predicts that η ∼ M. For the samples with Mn > Me, the viscosity follows the universal strong dependence with MW, namely η ∼ M3.4±0.2. Note that the solid line describing the dependence η0 ∼ M3.2 was obtained by fitting the data from the low PDI samples (anionically prepared) only. Moreover, the crossover MW between the two regimes, Mc, has a value of ∼100 kg/mol, which is nearly double the value of Me, as typical for polymer melts.

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REFERENCES

(1) Dealy, J. M.; Larson, R. G. Structure and Rheology of Molten Polymers: From Structure to Flow Behavior and Back Again; Hanser Publishers: Munich, 2006. (2) Graessley, W. W. Polymeric Liquids and Networks: Dynamics and Rheology; Garland Science: London, 2008. (3) Fetters, L. J.; Lohse, D. J.; Richter, D.; Witten, T. A.; Zirkel, A. Connection between Polymer Molecular Weight, Density, Chain Dimensions, and Melt Viscoelastic Properties. Macromolecules 1994, 27, 4639−4647. (4) Witten, T. A.; Milner, S. T.; Wang, Z. In Multiphase Macromolecular Systems; Culbertson, B. M., Ed.; Plenum: New York, 1989. (5) Fetters, L. J.; Lohse, D. J.; Graessley, W. W. Chain dimensions and entanglement spacings in dense macromolecular systems. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 1023−1033. (6) Fetters, L. J.; Lohse, D. J.; Milner, S. T.; Graessley, W. W. Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights. Macromolecules 1999, 32, 6847−6851. (7) Lin, Y. H. Number of entanglement strands per cubed tube diameter, a fundamental aspect of topological universality in polymer viscoelasticity. Macromolecules 1987, 20, 3080−3083. (8) Grubisic, Z.; Rempp, P.; Benoit, H. A universal calibration for gel permeation chromatography. J. Polym. Sci., Part B: Polym. Phys. 1996, 34, 1707−1713. (9) Willenberg, B. Proton-deuteron exchange in polystyrene − A new simple method for rapid deuteration of polymers with aromatic substituents. Makromol. Chem. 1976, 177, 3625−3628. (10) Kline, S. R. Reduction and analysis of SANS and USANS data using IGOR Pro. J. Appl. Crystallogr. 2006, 39, 895−900. (11) Roe, R. J. Methods of X-ray and Neutron Scattering in Polymer Science; Oxford University Press: New York, 2000; p 331. (12) Shivakumar, D.; Harder, E.; Damm, W.; Friesner, R. A.; Sherman, W. Improving the Prediction of Absolute Solvation Free Energies Using the Next Generation OPLS Force Field. J. Chem. Theory Comput. 2012, 8, 2553−2558. (13) Schröter, K.; Hutcheson, S. A.; Shi, X.; Mandanici, A.; McKenna, G. B. Dynamic shear modulus of glycerol: Corrections due to instrument compliance. J. Chem. Phys. 2006, 125, 214507.

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CONCLUSIONS In this work, we present a detailed study of the microstructure, chain dimensions, and rheological response of PVBP using a combination of small-angle scattering (SANS and WAXS), MD simulations, and rheological measurements. To our knowledge, this is the first systematic study of a linear polymer having strong π−π interactions. We determined the chain dimensions in the melt using SANS measurements of an entangled isotope blend, which revealed that PVBP is a semiflexible polymer with persistence length and packing length ∼1.5 times greater than PS. The relatively low flexibility is due to the bulkier biphenyl groups in PVBP compared to the less-voluminous single-phenyl group in PS. This is also reflected in the difference in positions of the interchain peak (also known as “polymerization peak16,20), measured by WAXS. Besides the interchain peak, an additional high-q “interphenyl” peak is observed in PVBP, which, similar to that reported in PS,16,17,20 is due to phenyl− phenyl correlations which have no long-range order. Unlike PS, the interphenyl peak in PVBP consist of two convoluted broad peaks, which reveal a major population of π−π stacking in the T-shaped configuration and only ∼10% in the PD configuration. Rheological measurements reveal that PVBP is thermorheologically simple, with glassy modulus Gglass ∼ 1 GPa and rubbery plateau modulus G0N = 0.065 MPa. The low G0N value is another consequence of the low flexibility of the PVBP chains, which results in relatively large Me value (58.3 kg/mol). Nevertheless, it was found that PVBP follows the universal correlation between plateau modulus and packing I


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Molecular Structure, Chain Dimensions, and Linear Rheology of Poly

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