Simultaneous stress and birefringence measurements during uniaxial elongation of polystyrene melts with narrow molecular weight distribution
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DOI: 10.1007/s00397-005-0452-5
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- Luap, C., Müller, C., Schweizer, T. et al. Rheol Acta (2005) 45: 83. doi:10.1007/s00397-005-0452-5
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Abstract
Tensile stress and flow-induced birefringence have been measured during uniaxial elongation at a constant strain rate of two polystyrene melts with narrow molecular weight distribution. For both melts, the stress- optical rule (SOR) is found to be fulfilled upto a critical stress of 2.7 MPa, independent of strain rate and temperature. Estimation of the Rouse times of the melts, from both the zero-shear viscosity and the dynamic-shear moduli at high frequency, shows that the violation of the SOR occurs when the strain rate multiplied by the Rouse time of the melt exceeds by approximately 3. The presented results indicate that in contrast to current predictions of molecular theories, the regime of extensional thinning observed by Bach et al. (2003) extends well beyond the onset of failure of the SOR, and therefore the onset of chain stretch in the non-Gaussian regime.
Keywords
Polymer meltBirefringenceElongational flowPolystyreneRheo-opticsNonlinear viscoelasticityIntroduction
Flows encountered during polymer processing can involve extremely large velocity gradients. To describe properties of entangled polymer melts in these fast flow situations, reptation-based molecular theories include the concept of chain stretch. Allowing chain retraction to be incomplete, the occupied tube length can exceed its equilibrium value. The retraction process is assumed to take place on a time scale given by the Rouse time \( \tau _R \) of the chain, and therefore chain stretch can occur when the velocity gradient exceeds \( {1 \mathord{\left/ {\vphantom {1 {\tau _R }}} \right. \kern-\nulldelimiterspace} {\tau _R }} \). First introduced by Marrucci and Grizzuti (1988) into the original reptation model, the significance of the chain stretch mechanism is now well established as it explains two main particularities of the transient response of polymers upon start-up of fast steady flow: the appearance of an overshoot in the first normal stress difference in shear, and the phenomenon of strain hardening observed in elongational flow.
Concerning steady-state properties, current reptation-based theories predict that the onset of significant chain stretch gives rise to an upturn of the steady elongational viscosity at a strain rate of the order of the reciprocal Rouse time of the chain. At larger rates, provided that chain finite extensibility is taken into account, the viscosity grows with strain rate until it saturates when chains reach their full extension (Marrucci et al. 2004). While the predicted upturn of the elongational viscosity was confirmed experimentally for monodisperse entangled PS solutions (Bhattacharjee et al. 2002; Ye et al. 2003), the situation for melts is unclear. Early data by Li (1988) (see also Takahashi et al. 1993) on a polystyrene (PS) melt with a relatively narrow molecular weight distribution (NMWD) indicated the onset of an extension thickening regime, though this conclusion relies on a single data point. The recent data on two PS melts with NMWD published by Bach et al. (2003) show, however, that the steady-state elongational viscosity decreases monotonically with the strain rate without any sign of an upturn in the vicinity of \(\dot \varepsilon \tau _R \approx 1.\) Though their transient data presented a substantial strain hardening, the lack of additional information about the degree of chain stretch apparently led to ambiguities in possible explanations. Whereas Bach et al. discussed the high rate scaling of the viscosity considering that chains nearly reach full extension, Marrucci and Ianniruberto (2004) suggested that the absence of upturn might be explained by an overestimation of the Rouse time of the melts. According to the latter argument, the onset of significant steady chain stretch and the resulting upturn in viscosity would occur simply at a higher strain rate, beyond the experimentally explored range.
Yet another experimental manifestation of strong chain stretching is the failure of the stress-optical rule (SOR) that occurs at large deformation when melts are stretched far above the glass transition temperature. The stress-optical rule states that the anisotropic parts of the refractive index and the stress tensors are proportional to each other. For uniaxial elongational flow it means that the birefringence Δn is related to the tensile stress σ through a constant stress-optical coefficient C, i.e., Δn=C σ. If chains stretch and approach their full extension, the SOR obviously fails since the stress continues to grow whereas the birefringence tends to saturate. More precisely, departure from the linear relation will occur when the degree of chain stretch is such that significant non-Gaussian effects set in (Mead and Leal 1995). For melts, such failure has been observed many times for uniaxial elongational flow (Matsumoto and Bogue 1977; Muller and Froelich 1985; Muller and Pesce 1994; Venerus et al. 1999). These studies concerned polydisperse systems only, and, to our knowledge, the unique stress-optical data published on a PS melt with a NMWD (Kotaka et al. 1997) reported violations from the SOR that are inconsistent with the occurrence of chain stretch beyond the Gaussian regime. It therefore remains unclear, above which stress level the degree of chain stretch is such that effect of finite extensibility becomes measurable, and above which stress level chains reach their full extension.
The goal of this study is to fill this lack of stress-optical data on PS melts with NMWD. We report simultaneous measurements of tensile stress and birefringence during the uniaxial elongation of two nearly monodisperse PS melts with molecular weights comparable to those used by Bach et al. (2003). Tests were performed within a temperature-strain rate suitable to evidence the breakdown of the SOR. We will show in particular that the regime of extensional thinning, observed by Bach et al. (2003), extends well beyond the onset of failure of the SOR and therefore the onset of significant chain stretch in the non-Gaussian regime.
Experimental details
Materials and samples preparation
Characteristics of the polystyrenes as determined by GPC
Polymer | M_{w} (g mol^{−1}) | M_{n} (g mol^{−1}) | I=M_{w}/M_{n} |
---|---|---|---|
PS206k | 206,000 | 198,000 | 1.04 |
PS465k | 465,000 | 430,000 | 1.08 |
The polymers, were delivered in powder form and were kept for several weeks at 70 °C under vacuum before usage in order to eliminate eventual residual solvent. Samples were prepared by the following procedure. The polymer powder was first pressed into tablets at room temperature. The tablets were then compression moulded under vacuum for 30 min under 22 bar at 190 and 200 °C for PS206k and PS465k. Samples for the rheo-optical elongation experiments required a smooth surface and were moulded between glass plates. Parallelepipedic specimens were machined with the following dimensions: a total length of 56 mm, a width of 10 mm, and a thickness of 1 or 0.4 mm. All samples were stored under vacuum at 70°C prior to the measurements.
Linear viscoelastic measurements
The linear viscoelastic properties of the sample were determined by small amplitude oscillatory shear-flow experiments with a UDS 200 mechanical spectrometer from Paar Physica, in parallel-plate geometry (diameter 25 mm). The storage and loss moduli were measured over a certain frequency range at 135, 140, 150, 160, and 180°C (190 °C for PS465k) under nitrogen atmosphere. Master curves at a reference temperature of 150 °C were obtained, via the time–temperature superposition principle, using both vertical and horizontal shift factors. The horizontal shift factors were found to follow the WLF equation (Ferry 1980) with c_{1}^{0}=6.8 and c_{2}^{0}=98 for T_{0} = 150 °C.
Stress and birefringence measurements
Experimental set-up and procedure
It should be noted that the cross-polarizers system used here gives access only to the absolute value of the birefringence. The latter is negative for PS stretched above the glass transition temperature (Janeschitz-Kriegl 1983).
Compared to the original set-up of Venerus et al. (1999), a stiffer leaf spring has been mounted, extending the maximal measurable force to 250 g. A boroscope connected to a digital camera has been incorporated in the RME housing and positioned near the sample, at an angle to the laser beam, such that it does not block the laser beam. Visualisation of the middle part of the sample allowed us to check if the laser beam was passing entirely through the sample during the whole test. In addition, measurement of the variation of sample width with time allowed us to check the consistency of the strain rate and detect eventual inhomogeneous deformation.
Experiments were conducted at 140 and 150 °C for PS206k, and at 150 °C and 160 °C for PS465k. The strain rate was varied between 0.1 s^{−1} and 1 s^{−1} . Sample failure or inhomogeneous deformation limited the maximum Hencky strain to the range 3–3.5.
We note that shortly before failure, a systematic whitening of the samples was observed, which we attribute to the formation of crazes or micro-voids in the sample. Whitening was not observed for technical PS melts stretched under similar conditions.
Data evaluation and experimental errors
The force was calibrated with a set of weights before each measurement. The spring response was checked to be linear in the range 1–240 g. For each temperature, the same force calibration constant value was used for all measurements. The maximal error in the measured force was estimated to be 2.5%.
Measurements of the time variation of the sample width have shown that the set and true strain rate values are consistent within 3%, in agreement with independent particle tracking measurements. Data were disregarded as soon as inhomogeneous deformation was suspected.
In practice, an instantaneous constant strain rate deformation of the sample, as assumed in Eq. 4, is difficult to realize. Sagging of the sample, delayed motor response, and transducer compliance lead to a deviation from constant strain rate at the beginning of the test. This “imperfect start-up” can be detected on the calculated transient growth of the viscosity, as a departure at small strains from the linear viscoelastic response (Meissner and Hostettler 1994).
Assuming that the major source of error in the birefringence values stems from the uncertainty in the thickness of the sample, one obtains:\( \Delta {{(\Delta n)} \mathord{\left/ {\vphantom {{(\Delta n)} {(\Delta n)}}} \right. \kern-\nulldelimiterspace} {(\Delta n)}} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\Delta \varepsilon \). This error reaches 6% for a Hencky strain of 3.
We note that beyond the “imperfect” start-up period, the time-dependent relative error in the stress-optical coefficient (SOC), C(t)=Δn(t)/σ(t) reduces to the time-dependent relative error in the sample width and therefore remains below 5% for a Hencky strain of 3.
Results and discussion
Linear viscoelastic properties and estimation of the Rouse time
Characteristic time scales for the studied polystyrenes at a temperature T=150 °C
Polymer | η_{0} (Pa s) | \(J_{e}^{0}\) (Pa^{−1}) | τ _{w} (s) | A (Pa s^{1/2}) | \( \tau _R^{G'} \)^{ (a)} (s) | \({\tau _{{\rm R}}^{\eta}}\)^{ (a)} (s) |
---|---|---|---|---|---|---|
PS206K | 2.33·10^{6} | 1.3·10^{−5} | 30 | 2.07·10^{4} | 2.38 | 2.35 |
PS465K | 2.70·10^{7} | 1.5·10^{−5} | 400 | 2.00·10^{4} | 10.55 | 8.74 |
Transient and steady mechanical data/comparison with previous results
All plots include the linear viscoelastic prediction \( 3\eta ^0 (t) \) deduced from the complex dynamic moduli.
Except for the short imperfect start-up period, the qualitative behaviour is similar to that reported by Bach et al. (2003) for PS melts with narrow MWD. The transient viscosity first follows and then rises above the linear viscoelastic predictions. At a later time, it saturates approaching a limiting value that is below 3η_{0}. As noticed by Schweizer (2000), this last regime where the tensile stress attains a constant value corresponds to a “highly unstable situation for the sample and the faintest irregularity in the cross-section leads immediately to necking and rupture”. Control of neck location and of the strain rate within the neck region is a major advantage of the filament stretching technique used by Bach et al. (2003) and allowed them to obtain a steady state of over 1 to 2 Hencky strain units. In our case, a regime of constant tensile stress could be observed only within a very narrow time window, allowing us to extract only estimates of the steady-state stress or viscosity.
For the purpose of comparison, the figure includes steady-state data extracted from previous investigations on quasi-monodisperse PS melts: the study of Li et al. (1988) and the study by Bach et al. (2003). For all literature data sets, Rouse time was estimated, as described in the previous section, from the reported weight average molecular weight and zero-shear viscosity values. To illustrate typical predictions of reptation-based models including chain stretch and finite extensibility, simulation results of the model of Öttinger (1999) are shown for three different\(Z ={{\tau _d}/{3\tau _R}}\)ratios. \( \tau _d \) and \( \tau _R \)are the chain reptation and Rouse time, the other model parameters, maximum stretch λ _{max} and plateau modulus G_{N}^{0}, being set to values that are appropriate for PS melts: λ _{max}= 4.5, G_{N}^{0}=170 kPa (for more details see Fang et al. 2000; van Meerveld 2004a).
The following points can be underlined. The model predictions for different\({{\tau _d}/{3\tau _R}}\)values, e.g., different molecular weights, fall onto a unique curve for \( De_R > 1 \). As soon as chain stretch sets in (\( De_R > 1 \)), the rate dependence of the steady-state stress shows a steep increase that corresponds to the up-turn in the steady-state viscosity, mentioned in the introduction. Although they cannot be considered as a validation, our estimated steady-state mechanical data for PS206k and PS465k, appear to be in close agreement with that of Bach et al. (2003). They do not reveal any abrupt change in the rate dependence of the steady-state stress in the vicinity of \(De_R \approx 1\), in contrast to the trend shown by the early data of Li et al. (1988), and to the current model predictions. Within the whole investigated range, the steady-state stress scales approximately as \( \sigma _{ss} \propto (De_R)^\alpha \) with \(\alpha \approx {\text{0}}{\text{.6}}\pm {\text{0}}{\text{.1}}\). In the next section, we will show that this scaling, that corresponds to a continuous decrease of the steady elongational viscosity with the strain rate, is observed even in the regime where the stress-optical rule is violated, e.g., where significant chain stretching is present.
Stress-optical behavior
To analyse the stress-optical behaviour and determine the regime of validity of the SOR, time-dependent birefringence data are plotted directly as function of tensile stress.
Increasing the strain rate to 1 s^{−1}, e.g.\(De_R \approx{\text{2}}{\text{.4}}\), small departures from linearity appear at large stresses (see Fig. 7b). Though systematic, the observed decrease in the apparent SOC \(C(\sigma) = {{\Delta n(t)}/{\sigma (t)}}\) only slightly exceeds the estimated experimental error of 5% at the steady state.
Data collected for different strain rates appear to fall into a unique curve indicating that the stress-optical behaviour is essentially independent of the elongational rate even in the nonlinear region. The same observation was previously made by Muller and Froelich in 1985 and by Muller and Pesce in 1994 on polydisperse PS melts stretched at relatively low temperatures (120–130 °C). In the low-stress region, all data follow a straight line consistent with the SOC value obtained at 150 °C\(\left| {C_0} \right| = {\text{(4}}{\text{.6}}\pm {\text{0}}{\text{.1)}} \cdot {\text{10}}^{- {\text{9}}}\,{\text{Pa}}^{- {\text{1}}} \).
Summary of the tests performed and occurrence of deviations from the SOR, based on a criterion of 5% deviations
Sample | T (°C) | \(\dot \varepsilon \)(s^{−1}) | \(\dot \varepsilon \tau _R \) | Deviation SOR |
---|---|---|---|---|
PS206k | 150 | 0.1 | 0.24 | < 5% |
PS206k | 150 | 0.3 | 0.71 | < 5% |
PS206k | 150 | 0.7 | 1.7 | < 5% |
PS206k | 150 | 1 | 2.4 | ~ 5% |
PS206k | 140 | 0.1 | 1.4 | <5% |
PS206k | 140 | 0.3 | 4.2 | ~ 5% |
PS206k | 140 | 0.7 | 9.8 | >> 5% |
PS206k | 140 | 1 | 14 | >> 5% |
PS465k | 160 | 0.3 | 0.7 | < 5% |
PS465k | 160 | 0.7 | 1.6 | ~ 5% |
PS465k | 160 | 1 | 2.3 | ~ 5% |
PS465k | 150 | 0.3 | 2.9 | ~ 5% |
PS465k | 150 | 0.7 | 6.8 | >> 5% ^{a} |
PS465k | 150 | 1 | 9.6 | >> 5% |
Concerning the degree of chain stretching, one can notice in Fig. 9 that the birefringence does not show any sign of saturation at large stresses, suggesting that in the regime investigated here, the chains did not reach their full extension. The maximum birefringence reached, of about −0.025 at a stress of 7 MPa, remains significantly smaller than the maximal birefringence that is expected when the PS chains are fully extended. For the latter, a value close to −0.1 (Jasse and Koenig 1979; Neuert et al. 1985) is inferred from the combination of birefringence with, eg., IR-dichroism measurements that give access to the second moment of the orientational distribution of the polymer segments, under the assumption that the transition dipole moment associated with the vibration mode under consideration makes a specific angle with the chain axis (see Ward 1975). It cannot be excluded that the actual maximum birefringence of the melt is limited to a lower value, either due to an actual saturation as a function of stress, or due to systematic sample failure. This issue could not be addressed here, since our limited measurable force range did not allow us to investigate the stress-optical behaviour for larger Deborah numbers.
Summary and conclusions
Simultaneous measurements of tensile stress and flow-induced birefringence have been carried out during the uniaxial elongation at constant strain rate of two PS melts with a NMWD. The investigated temperature/strain-rate range corresponds to a variation of the Rouse time-based Deborah number from approximately 1 to 10. Inhomogeneous sample deformation and sample rupture limited the maximum reachable Hencky strain within 3–3.5 and thereby the possibility to extract accurate steady-state data. Approximate steady-state values were found, however, to be in good correspondence with the recent mechanical data obtained by Bach et al. (2003) on similar systems with a filament stretching rheometer, and in particular consistent, within the whole investigated Deborah number range, with the reported continuous decrease of steady-state viscosity with strain rate. For both melts, the SOR was shown to be valid upto a critical stress level of 2.7±0.5 MPa. Above this stress level, which can be reached for a Rouse time-based-Deborah number De_{R} of about 3, departure from linearity occurs, which corresponds to a decrease of the apparent SOC. The detected decrease of the SOC attains 25% for the largest \( De_R \) explored in this study and provides a clear experimental signature of a significant chain stretch beyond the Gaussian regime.
The following conclusions can be drawn:
- 1.
The continuous decrease of the steady elongational viscosity with the strain rate, evidenced by Bach et al. (2003), extends well beyond the onset of failure of the SOR, and therefore the onset of chain stretch in the non-Gaussian regime.
- 2.
The one-to-one relationship between the tensile stress and the birefringence confirms the picture that, in this regime, the stress can be determined knowing the stretch and orientation of the chains. This of course is not valid for extremely large Rouse time-based Deborah numbers, where phenomena related to the glass transition may occur.
- 3.
The question whether chains reach their maximal extension could not be answered definitively. The absence of birefringence saturation at large stresses as well as the maximum birefringence level reached indicates that at least upto a stress level of about 7 MPa, chains do not reach full extension.
In the original article, the expression is given for the longest stress relaxation time of the Rouse model whose value is half the value of \( \tau _R \)
Acknowledgements
The authors are grateful to Martin Colussi for performing the GPC measurements, to Marina Karlina, Werner Schmidheiny, and Fredy Mettler for their assistance in conducting the experiments. Clarisse Luap thanks Martin Kröger and Jan van Meerveld for helpful discussions.