All the materials were analyzed in linear rheological response and in transient shear, and compared to both the full PDE and RoliePoly constitutive equations. These tube models require just two parameters (plus the weak CCR parameter , which we have set to 0.1) for each choice of local polymer chemistry, reflecting the physics at the coarse-grained level of the tube diameter: the entanglement modulus G_e (defined as 5/4 times the plateau, modulus G), and the Rouse relaxation time of an entanglement segment _e. For PS at 170 °C, these are 2.04×10⁵  Pa and 7.1×10^–4  s, respectively, and for PB at 25 °C they are 1.6×10⁶  Pa and 4.8×10^–7  s.

Figure 2 shows the measured linear rheology of representative materials compared with the full tube model in linear response of Likhtman and McLeish (2002). The terminal, reptation time itself, corrected for contour length fluctuations, is calculated in this formulation of the tube theory from the series:

where the "bare" reptation time ₀(Z) = 3Z³_e. The terminal time, shape of terminal peak, slope of G() at higher frequencies, and the minimum are all captured by the theory using a unique parameter set for each chemistry.

Figure 2.

Figure 3 contains the transient nonlinear data for two representative PS samples in shear, and the higher molecular weight also in extension. Also given are the predictions for these viscometric flows from the full tube model of Eq. 2, setting the constraint release parameter c = 0.1. We predict and observe some extensional hardening when the flow rates exceed the Rouse (stretch) relaxation time of the monodisperse linear polymers. The Rouse time may be predicted from molecular theory directly, from

and this value is used within the full tube-CCR model. The Rouse time affects both the shape of the shear transients and the onset and form of the extensional hardening. By this method, we found values of 0.16 s and 0.6 s for PS250 and PS485, respectively, at 170 °C. In particular, in these monodisperse materials with a well-defined lowest Rouse (stretch) relaxation time, the onset of extensional hardening is sudden, as a function of extension rate. This contrasts with commercial materials where the onset is typically much more gradual.

Figure 3.

The relaxation times as well as the fixed parameters of the model for the polymers in this study are summarized in Table II. Note that we describe the melt moduli in terms of the parameter G_e of Likhtman and McLeish (2002). It is 5/4 of a standard values of G.

The parameters of the RoliePoly equation (amplitude of five modes, reptation, and Rouse times used in the slowest nonlinear mode) were adjusted for each material so that it optimally matched the predictions of the full theory in both extension and in shear. These then became computationally efficient models of the materials to employ in the flow solver. The transient shear flows for the two PS materials matched to the predictions of the RoliePoly models are given in Fig. 4.

Figure 4.

The flow field contains several qualitatively different regions. Slow flow of the Pouseuille type occurs in the broad channels up and downstream. The slot itself is dominated by a much higher wall shear rate, and is preceded by a region of positive extensional deformation upstream, and negative downstream. As we have outlined in Sec. III above, there are two important time scales inherent in each of the monodisperse materials we employ: The reptation time _d, which determines relaxation of tube segment orientation, and the faster Rouse time _R, which determines chain stretch. There are therefore three qualitatively different regimes of flow rate, as measured by the wall shear rate in the channel _w (the extension rate at the constriction will be of the same order of magnitude):

_wd<1   linear response;

_wd > 1,_wR<1   orienting flow without chain stretch; and

_wd > 1,_wR > 1   orienting flow with chain stretch.

We established that the terminal relaxation times of the materials were such that steady states could be achieved at the flow rates employed within a single pass of the MPR pistons. The numerical simulations were also run until a steady state was obtained.

The near symmetry of the observed flow is correctly predicted by the simulations up to the maximum piston speed of 0.5 mm/s (wall shear rate of 29  s^–1). At this rate, the first asymmetries in the flow, arising from nonlinearities in the material response, can be detected at the outflow. The wall shear rate, made dimensionless by the reptation time is about 80 at this flow rate, but only order 4 with respect to the Rouse time. Stress concentrations arise at the re-entrant corners, and the stress contours away from the die are slightly more linear than in the inflow region.

Figure 5 shows a comparison between the experimentally observed birefringence and the simulated contours of principal stress difference (). These are related via the stress-optical law:

where v is the wavelength of the light used (514 nm), d is the depth of the sample (10 mm), and k is the fringe number. A value for the stress-optic coefficient, C, of 5.6×10^–9  Pa^–1 was used for both the PS materials, in agreement with literature values [Janeschitz-Kriegl (1983)]. The immediate finding of both theory and experiment is that, although deep into the first nonlinear flow regime, the stress field is highly symmetric, with asymmetry only detected near the corners. Agreement of the predicted and experimental fringe pattern is good.

Figure 5.

At lower rates, this higher molecular weight melt, approaching 30 entanglements per chain, exhibits symmetric flows like the less entangled materials. But above piston speeds of (0.05 mm/s, wall shear rate = 29  s^–1) significant asymmetries appeared, leading eventually to unstable flow above 0.2 mm/s. Results of both experiment and simulation are given in Fig. 6. Both the asymmetric stress fields and the instability seem to be hallmarks of the stretching flow Regime (3) of these monodisperse materials. For example, early indications of a three-lobed appearance of the outflow stress region begin to appear at a wall shear rate of 12  s^–1 [Fig. 6(b)], when _wR7, and become fully formed at higher rates. These are accessible for this material by simulation only, because the imposition of a symmetry plane along the flow centerline suppresses the instability that occurs in experiment [Fig. 6(c) comes from the simulated steady-state flow at a piston speed of 5 mm/s, far into Regime (3) when _wR40]. Simulations of the whole channel width (rather than a half-width with imposed symmetry) did indeed show the onset of instabilities above 0.5 mm/s, but this subject will require further study. Yet the simulations at these higher rates are instructive: In Regime (3) the characteristic "fanglike" features of stretched material in the outflow, seen hitherto only in transient flows of branched melts [Lee et al. (2001)] are predicted to occur even for linear melts. In fact, our calculations predict these features not only in the transient but for steady-state flow as well for monodisperse materials. It will be a challenge to create stable flows in which this prediction can be borne out.

Figure 6.

We may anticipate from studies on the PS material, that the more entangled PB melts are likely to exhibit highly symmetric stress fields in even very nonlinear flows, since the span of Regime (2) for the melts extends to much greater rates. Indeed, at all flow rates accessible experimentally, the flow field was symmetric even though these extend into the first nonlinear Regime (2). We show a representative result for the highest molecular weight material PB210 in Fig. 7 when _wd3.6. The comparison used a stress-optical coefficient of 1.0×10^–8  Pa^–1, which is rather larger than one literature value of between 2.0×10^–9  Pa^–1 and 3.0×10^–9  Pa^–1 [Janeschitz-Kriegl (1983)]. The discrepancy may be related to the far greater tendency for PB to slip in viscometric nonlinear flows than PS. However, both experiment and numerical calculation agree that the only places where any asymmetry is apparent at all (in either experiment or calculation) are the vicinities of the reentrant corners. These are places where, locally, the nonlinearity of the flow is much higher than the mean value.

Figure 7.

The advantage of both the experimental and numerical protocols we have followed is that fully transient flows may be measured and modeled. Here, we present data on pressure-drop transients, as well as stress-birefringence fields during the flow start up. Naïve comparison of pressure-drop transients lead to a strong disagreement between simulation and experiment: The simulation predicts a very rapid rise of the pressure to a value close to its steady state, yet the experiments show a well-resolved rise time (see Fig. 8). This is due to the compressibility of the upstream reservoir, as shown by a simple calculation. For, in order for the melt to transmit stresses into the constriction of the order of the shear modulus G, it must be compressed by the piston by relative volume V/V so that the pressure generated by the bulk modulus B is of order G:

But, for PS, B/G~10³, and, since the upstream piston length is 10 cm, the piston displacement corresponding to this compression is of the order of 0.1 mm. At a piston speed of 0.2 mm/s (that of Fig. 8), the time scale for achieving the steady-state flow at the constriction end of the reservoir is therefore about 0.5 s, as seen in the experiment. We also observe by comparing the characteristic transient time scales with different piston speeds that they follow a constant bulk strain of the upstream melt, as predicted by Eq. 6, rather than any constant intrinsic time scale.

Figure 8.

In order to permit a realistic comparison of transients from experiment and simulation, the compressibility effect must be accounted for. Simulation of the entire upstream reservoir is unfeasible, so instead we modulate the 1D upstream velocity boundary condition with an exponential decay onto the steady-state value that matches just the time constant of the observed pressure transient (dashed curve in Fig. 8). Under such time dependent forcing, the full stress field in time may be computed and compared to the observed field.

A series of results is shown in Fig. 9 for the PS262 melt in Regime (2). The overall shape and magnitude of the stress field is captured throughout the transient flow. Furthermore, there are detailed features of the transient flow that are not present in the steady flow that appear in both the experiment and simulation. In particular, lobes of high principal stress from the upstream and downstream re-entrant corners grow, then retreat, during the transient. In spite of the allowance for the delay in flow at the constriction itself, however, these features still occur slightly earlier in the simulation than in experiment. For example, the lobe at the upstream re-entrant corner attains maximum extent in the experiment at 1.2 s, while the simulations find this maximum at 0.5 s after start up of flow. This is unlikely to be a constitutive flaw simply related to the viscometric transient response, since the shear overshoots in viscometric flows are well captured by the RoliePoly model.

Figure 9.

Figure 10 shows the comparison of predicted and steady-state pressure drops for the high molecular weight PS485 material. The agreement is good well into Regime (2) of nonlinear flow, but as the experiments become increasingly unstable (while the simulations remain artificially stable), an increasingly large overprediction of the stress is observed. This observation of reduction in expected dissipation suggests that the manifestly unstable flow [Fig. 6(b) is a snapshot of an unsteady flow] may also be accompanied by wall slip. This is also consistent with, but not conclusively determined by, the time dependent birefringence images.

Figure 10.