Stress and neutron scattering measurements on linear polymer melts undergoing steady elongational flow
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DOI: 10.1007/s00397-012-0622-1
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- Hassager, O., Mortensen, K., Bach, A. et al. Rheol Acta (2012) 51: 385. doi:10.1007/s00397-012-0622-1
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Abstract
We use small-angle neutron scattering to measure the molecular stretching in polystyrene melts undergoing steady elongational flow at large stretch rates. The radius of gyration of the central segment of a partly deuterated polystyrene molecule is, in the stretching direction, increasing with the steady stretch rate to a power of about 0.25. This value is about half of the exponent observed for the increase in stress value σ, in agreement with Gaussian behavior. Thus, finite chain extensibility does not seem to play an important role in the strongly non-linear extensional stress behavior exhibited by the linear polystyrene melt.
Keywords
ScatteringPolymer meltUniaxial extensionPolystyreneIntroduction
Elongational flow has long been and continues to be a topic of significant rheological interest (Petrie 1979; McKinley and Sridhar 2002). For dilute polymer solutions, it is known from visual observations (Perkins et al. 1997) that the polymer molecules may become fully extended in strong elongational flows. Our understanding of the fully extended state continues to improve as a result of accurate theoretical modeling (Underhill and Doyle 2007). Moreover, the transition from near equilibrium to the fully stretched state is a rather sharp transition that occurs at a specific elongation rate (De Gennes 1974).
For entangled polymer melts, our understanding of the molecular behavior in strong elongational flow is less satisfactory. Simple continuum models such as the Oldroyd B model or the Lodge rubberlike liquid (Bird et al. 1987a) predict a stress singularity at a critical elongation rate. This singularity may be suppressed by invoking finite extensibility in the associated structural models (Bird et al. 1987b), but for most models, finite extensibility merely replaces the singularity with a sharp transition to an asymptotic behavior that may be associated with a fully stretched state.
While one would not expect such simple models such as the Oldroyd B model to provide more than just a first-order approximation for non-linear dynamics, it is more serious that the situation is not much better for most current structural models. This is true in particular for the tube theory framework for modeling the dynamics of entangled polymer systems. Going back to the pioneering development by Doi and Edwards (1986), the basic model predicts a limiting stress in strong extensional flow. The molecular origin of the limiting stress is the assumption of instantaneous chain retraction so that the stress is due to orientation only. Perfect orientation of the tubes but with no chain stretching gives a stress that is five times the plateau modulus. The corresponding steady elongational viscosity then decreases with elongation rate \(\dot{\epsilon}\) as \(\dot{\epsilon}^{-1}\). However, it was recognized early that chain stretching will occur at deformation rates faster than the inverse Rouse time (Marrucci and Grizzutti 1988; Pearson et al. 1989). The concept of chain stretching was consequently included in several models based on the Doi and Edwards framework (Fang et al. 2000; Mead et al. 1998; Schieber et al. 2003). The result of including chain stretching alone is to introduce a stress singularity at an elongation rate of the order the inverse Rouse time. This singulary may again be relieved by invoking finite extensibility. The resulting models exhibit four flow regimes: a linear viscoelastic regime dominated by reptation wherein the elongational viscosity is constant, a second regime wherein the elongational viscosity decreases as \(\dot{\epsilon}^{-1}\), and a third regime with a sharp increase in the elongational viscosity to an asymptotic behavior in the fourth regime. There was at the time no data for entangled monodisperse polymer melts to compare with (Mead et al. 1998). However, subsequent data on steady elongational viscosity of narrow molar mass distribution (NMMD) polystyrene (Bach et al. 2003a; Luap et al. 2005) demonstrated a rather different nonlinear behavior with an elongational viscosity scaling approximately as \(\dot{\epsilon}^{-0.5}\). The conflict was recognized immediately by Marrucci and coworkers who introduced the concept of interchain pressure in polymer dynamics (Marrucci and Ianniruberto 2004) whereby chain stretching is balanced by lateral forces between the chain and the tube wall. In this way, they were able to explain the \(\dot{\epsilon}^{-0.5}\) scaling. Moreover, they predicted a maximum in the steady elongational viscosity for smaller molar masses that was subsequently observed (Nielsen et al. 2006). Wagner and coworkers introduced the interchain pressure concept into their molecular stress function theory whereby they obtained a constitutive equation with the correct \(\dot{\epsilon}^{-0.5}\) scaling (Wagner et al. 2005). Later refinements have allowed quantitative predictions of available non-linear rheological measurements from linear viscoelastic data alone (Wagner and Rolón-Garrido 2009, 2010). The interchain pressure concept has also been introduced into the Rolie–Poly (Likhtman and Graham 2003) constitutive equation, whereby likewise a quantitative description of the elongational viscosity is obtained (Kabanemi and Hétu 2009). While there are currently two tube-based models capable of describing the non-linear stress behavior in elongational flow, the situation is not entirely satisfactory for two reasons. First the number of modifications needed on the basic model is in itself worrying. Second the very fact that two different tube-based models both describe the stress data quantitatively does not leave a clear picture.
To gain insight into the physics of strong elongational flows from the experimental side, Luap et al. (2005) measured birefringence together with the stress. For their two NMMD polystyrene melts, they found the stress optical rule fulfilled up to a critical stress of order 2.7 MPa or about ten times the plateau modulus. The fact that the \(\dot{\epsilon}^{-0.5}\) scaling sets in when the steady stress is of order the plateau modulus therefore suggests that the physics behind the \(\dot{\epsilon}^{-0.5}\) scaling is not related to finite extensibility of the chains. However, their measurements also show that the \(\dot{\epsilon}^{-0.5}\) scaling extends unchanged well beyond the onset of chain stretch in the non-Gaussian regime (Luap et al. 2005) evidenced by failure of the stress-optical rule. In other words, finite extensibility does come into play, but it is not the prime reason for the observed scaling nor does it lead to a drastic transition to an asymptotic behavior.
The purpose of this study is to gain further insight into the physics of strong elongational flows by application of small-angle neutron scattering (SANS). In pioneering work (Boué et al. 1982, 1983) on the application of SANS to understand dynamics in stretched polystyrene melts, the melts were rapidly extended by a stretch ratio of 3 at temperatures not far above the glass transition. The subsequent relaxation of the radius of gyration in the direction transverse to the stretching direction was subsequently followed as function of time from a sequence of quenched samples. The early reduction in the transverse direction predicted by the basic tube theory was not observed possibly due to Rouse relaxation during the quenching process. The minimum was, however, clearly observed later by Blanchard et al. (2005) for PI utilizing an in situ SANS device.
SANS measurements have also been performed on NMMD polystyrene samples quenched from steady elongational flow (Muller et al. 1990). While the measurements were performed in the linear viscoelastic regime and therefore do not cast light on the \(\dot{\epsilon}^{-0.5}\) scaling, the study does give credence to a straightforward analysis in terms of radii of gyration in the parallel and transverse direction, respectively. More recently, neutron scattering has been reported on the steady flow in a 4:1 planar contraction/expansion of NMMD polystyrene samples (Bent et al. 2003). However, while the inhomogeneous flow is steady in a Eulerian sense, it is not steady in a Lagrangian sense so it is not straightforward to compare with steady viscosity data. In fact while scattering from deformed polymer melts has been analyzed in considerable detail both experimentally and theoretically (Read 2004), we do not know of SANS measurements that may cast light on the molecular behavior behind the \(\dot{\epsilon}^{-0.5}\) scaling.
Experimental
Synthesis
To eliminate ambiguities due to polydispersity, we used anionic polymerization (Ndoni et al. 1995) to prepare polymer samples of narrow molar mass distribution. The key polymer is a symmetric triblock copolymer PS(H)–PS(D)–PS(H). The middle block (PS(D)) is polystyrene with hydrogen substituted by deuterium, while the two identical end-blocks are ordinary polystyrene (PS(H)). This polymer is mixed into a matrix of ordinary polystyrene of the same molar mass. In this way, we obtain a solution of deuterium labeled middle segments in a sea of ordinary polystyrene. We have labeled only the middle block with deuterium for two reasons. Firstly, it is anticipated that the molecular relaxation of stretch that inevitably will occur during quenching primarily will take place near the ends of the chain. Secondly, if we labeled the entire chain, the molecular length scale in the direction parallel to the macroscopic elongation would be too large to be measured accurately in the SANS facility used.
Chromatography
Components in test material
Components | Block copolymer | Homopolymer |
---|---|---|
PS(H)–PS(D)–PS(H) | PS(H) | |
M_{w} | 145 kg/mol | 150 kg/mol |
M_{w}/M_{n} | 1.13 | 1.07 |
Mechanical spectroscopy and filament stretching
To establish a well-defined steady elongational flow, we use a filament stretching rheometer (McKinley and Sridhar 2002) equipped with a thermostat (Bach et al. 2003b). Prior to flow, the sample is allowed to equilibrate for a time longer than the estimated longest relaxation time. Then at times t > 0, the plates are separated from one another in such a fashion that the elongation rate at the plane of observation (the plane of symmetry) is kept constant at a given value \(\dot{\epsilon}\). The force at the end-plate is measured online, as is the filament diameter in the plane of observation. The plate motion is controlled to obtain a constant value of \(\dot{\epsilon}\). At time \(t = 3/\dot{\epsilon}\), the fluid in the plane of observation has undergone locally a Hencky strain of \(\epsilon = \dot{\epsilon}t = 3\). This means that the fluid in the plane of observation has been stretched by a factor e^{3} ≈ 20. The fluid in the plane of observation is the only part of the sample that experiences ideal elongational flow.
Quenching and scattering
Assuming that steady flow conditions are established around Hencky strain 2.5, a sequence of samples obtained by quenching the polystyrene bridge at Hencky strain 3 should contain molecular configurations representative of steady elongational flow. Thus, when the polymers in the plane of observation have reached Hencky strain ε = 3, the liquid bridges were quenched by a series of cooled nitrogen jets thereby providing samples for the SANS experiments. Due to the atactic nature of the anionically synthesized polystyrene, this polymer does not crystallize but undergoes a glass transition around 100°C whereby the molecular configurations are frozen. All cooling equipment was made of stainless steel. Consequently, the cooling equipment acted as a heat sink in the oven even when the gas flow was turned off. The influence on filament temperature was investigated by placing a thermocouple in a filament and measuring the temperature. When the oven temperature was set to 130.0°C, the measured filament temperature was 126.7±0.5°C. We use this temperature as the filament temperature for all elongational measurements, but the uncertainty in the temperature is a concern especially since the polymer time constants are highly dependent on the temperature.
Physical properties of probe melt at experimental conditions (126.7°C)
ρC_{p} | k | τ_{w} | η_{0} | τ_{R} |
---|---|---|---|---|
2 10^{6} J/m^{2} K | 0.17 W/mK | 970 s | 70 MPa s | 160 s |
Elongational stress and SANS data for hot-stretched and quenched samples
Sample | \(\dot{\epsilon}\) (s^{ − 1}) | Wi | Wi_{R} | R_{g ∥ } (Å) | R_{g} ⊥ (Å) | R_{m} (Å) | λ_{ ∥ } | λ_{ ⊥ } | σ (kPa) | σ/G_{0} |
---|---|---|---|---|---|---|---|---|---|---|
epsd 0001al3.0 | 0.001 | 1.0 | 0.16 | 66 | 41 | 48 | 1.4 | 0.89 | 210 | 0.95 |
epsd0001al5.7 | 0.001 | 1.0 | 0.16 | 77 | 43 | 52 | 1.6 | 0.93 | 210 | 0.95 |
epsd001bl3.0 | 0.01 | 10 | 1.6 | 147 | 32 | 53 | 3.2 | 0.69 | 1,380 | 6.3 |
epsd001bl5.7 | 0.01 | 10 | 1.6 | 155 | 31 | 53 | 3.4 | 0.67 | 1,380 | 6.3 |
epsd001cl3.0 | 0.01 | 10 | 1.6 | 149 | 33 | 54 | 3.3 | 0.72 | 1,370 | 6.2 |
epsd001cl5.7 | 0.01 | 10 | 1.6 | 157 | 32 | 54 | 3.4 | 0.69 | 1,370 | 6.2 |
epsd001d10l3.0 | 0.01 | 10 | 1.6 | 152 | 34 | 56 | 3.3 | 0.74 | 1,330 | 6.0 |
epsd001d10l5.7 | 0.01 | 10 | 1.6 | 147 | 35 | 56 | 3.2 | 0.76 | 1,330 | 6.0 |
epsd01cl3.0 | 0.1 | 100 | 16 | 228 | 23 | 49 | 4.9 | 0.50 | 5,700 | 26 |
epsd01cl5.7 | 0.1 | 100 | 16 | 259 | 24 | 53 | 5.6 | 0.52 | 5,700 | 26 |
epsd01a+b+cl5.7^{a} | 0.1 | 100 | 16 | 269 | 22 | 51 | 5.8 | 0.48 | 6,100 | 28 |
Analysis and results
Polymer relaxation times and stresses
The longest relaxation time is used to define a non-dimensional elongation rate in terms of the Weissenberg number \(\mbox{Wi} = \dot{\epsilon}\tau_{\rm w}\). It appears from Table 3 that the experiments cover the range from Wi = 1.0 to Wi = 100. For low values of Wi, one would estimate the steady stress to be \(\sigma_{\rm zz}-\sigma_{\rm rr} \approx 3\, \eta_0 \dot{\epsilon}\). Indeed for \({\rm Wi} = 1.0 (\dot{\epsilon} = 0.001\, \mbox{s}^{-1}\)), this gives a prediction of 210 kPa in agreement with the observation in Table 3. However, for Wi = 10 and 100, the dynamics in steady flow will be outside of linear dynamics. We here make the stresses non-dimensional with the modulus, G_{0} = 220 kPa.
Scattering
Discussion
\(\dot{\epsilon}=0.001~\mbox{s}^{-1}\), Wi = 1, and Wi_{R} = 0.16: The stress is equal to \(3 \eta_0 \dot{\epsilon}\) which is an indication that the dynamics is within the linear range or at least not far into the non-linear range. This also agrees with expectations at a Weissenberg number equal to 1, although \(\dot{\epsilon}= 0.001~\mbox{s}^{-1}\) may represent the upper limit for the application of linear viscoelasticity. The stretch factors indicate that significant deviation from isotropy occurs in the linear viscoelastic range.
\(\dot{\epsilon}=0.01~\mbox{s}^{-1}\), Wi = 10, and Wi_{R} = 1.6. The stress is significantly smaller than \(3 \eta_0 \dot{\epsilon}\) which is an indication that the dynamics is outside of the linear range as would be expected at Wi = 10. Specifically the stress is larger than five times the modulus, which is the upper limit that can be predicted by the Doi–Edwards model without stretching. Consequently, according to the Doi–Edwards picture, the chains must be stretched relative to their equilibrium length. This agrees also with the Rouse time obtained from the Likhtman method. According to Rouse theory, the coil stretch transition occurs at Wi_{R} = 0.5. Hence, some degree of chain stretching at Wi_{R} = 1.6 is not unreasonable.
\(\dot{\epsilon}=0.1~\mbox{s}^{-1}\), Wi = 100, and Wi_{R} = 16. These data are in the fully non-linear region, with significant chain stretching taking place indicated by σ/G_{0} ≈ 27 as well as the stretch ratios. The value of λ_{ ∥ } may be questioned as discussed below.
Although both λ_{ ∥ } and λ_{ ⊥ } are given separately in Table 3 (using R_{g,0} = 46 Å), in view of the limited q-range which was accessed in the experiment, the perpendicular data are to be trusted more than in the parallel direction. The max q-range considered in Fig. 4 is between qR_{g} = 0.01×46 and 0.07×46, i.e. 0.5R_{g} to 4R_{g}. Hence, the parallel direction is fully out of range, shifted to un-measurable intensities in the beam stop area, and tabulated values are strongly co-determined by the perpendicular strain axis. For the latter axis, since the chain size is reduced due to incompressibility, the radius of gyration shifts to higher q and the simple Debye equation which is then Guinier-like fulfills still better the valid Guinier regime within the measured q-range. The microscopic deformation ratios λ_{ ∥ } are then ~1.5, 3.2, and 5.4. As an alternative to the direct determination of λ_{ ∥ }, we may also estimate the effective microscopic deformation ratios for the middle segment of the triblock chain as \(\lambda_\parallel^{e} = 1/\lambda_{\perp}^2\). This would give values ca. 1.3, 2, and 4 which are lower than those in Table 3. The power law for the parallel direction is virtually unchanged, however. The data and line in Fig. 6 are just shifted upward with a factor of about 1.2.
It remains to address the question whether the chain stretching approaches the fully stretched non-Gaussian limit. For Gaussian behavior, one would expect \(\Delta \sigma \sim \lambda_{\parallel}^2 - \lambda_{\perp}^2\) which for large λ_{ ∥ } would correspond to β = α/2 as observed within experimental accuracy. This may be taken as indirect evidence in support of the suggestion that finite extensibility is not the main mechanism behind the scaling observed in non-linear elongational flow (Luap et al. 2005). The maximum stretching one would anticipate can be estimated from the number of Kuhn segments in the deuterated middle segment as \(\sqrt{60}= 7.7\). Hence while the measurements are probably within the Gaussian range, the values at Wi = 90 may be probing the upper limit Gaussian statistics.
Acknowledgments
We are grateful to the Danish Technical Research Council for financial support to the Danish Polymer Centre, to the Danish Natural Science Council for financial support in the DANSCATT grant and to the Paul Scherrer Institut for use of their neutron beam facility, and the SoftComp EU-Networks of Excellence.