Equibiaxial elongational deformations are ubiquitous in the processing of polymeric materials. In spite of this, studies on the rheology of entangled polymer liquids in these flows are limited due to the challenges of generating well-controlled equibiaxial elongational deformations in the laboratory. In the present study, we examine the rheological behavior of several well-characterized polystyrene liquids in constant strain rate equibiaxial elongation flows using a novel technique known as continuous lubricated squeezing flow. The linear polymer systems considered here display strain softening behavior. A portion of this new data set is used to demonstrate, in contrast to uniaxial elongational flows, that the nonlinear behavior of entangled polymers in equibiaxial elongation is universal. We also make comparisons of predictions from two molecularly based models with experimental data for one of the polymer melts in shear flow, uniaxial elongational flow, and equibiaxial elongation flow. While both models are able to predict shear flow behavior, neither model is able to quantitatively predict both uniaxial and equibiaxial elongation flows.
I. INTRODUCTION
Equibiaxial elongation is a deformation where a material is stretched equally in two directions and contracts in the third direction. This deformation occurs in numerous industrial polymer processes, such as film blowing, blow molding, and foaming. Generating rheologically controlled equibiaxial deformations represents a significant experimental challenge. Hence, the relation between rheology and processing is poorly understood in these flows. The challenges of generating well-defined equibiaxial deformations in the laboratory also impede the development of molecular models designed to predict the rheological behavior of entangled polymer melts, which should be capable of describing flow in both shear and elongational deformations. For equibiaxial elongational flows, the stress difference is measured as a function of Hencky strain rate , which can be used to define the equibiaxial viscosity . There are several rheological techniques for generating equibiaxial flows [1] with sheet stretching and lubricated squeezing flow (LSF) being the most common.
The sheet stretching technique was developed by Meissner et al. [2] and Meissner [3] resulting in the multiaxiale Dehnung (MAD) rheometer [4]. The MAD rheometer used a series of rotary clamps to stretch a circular sheet of material equally in all directions while measuring the force exerted on the clamps. This device, which no longer exists, was used to measure the equibiaxial viscosity of three polymer melts at several Hencky strain rates to a Hencky strain of [4].
The LSF technique [5] involves squeezing a disk-shaped sample between parallel plates that are coated with a thin layer of a low-viscosity liquid or lubricant. The lubricant film is intended to prevent shear from occurring in the sample so that an essentially equibiaxial deformation is generated. The LSF technique has been used extensively to study polymer melts, foods, and biomaterials [6–22]. However, it has been established [23–26] that lubricant thinning limits constant strain rate LSF experiments to relatively small Hencky strains . This was definitively shown [27] by making direct comparisons of LSF measurements of with those from the MAD rheometer; viscosities measured using LSF show an apparent strain hardening that is an artifact caused by the buildup of stress in the thinning lubricant films. It should also be noted that larger strains can be achieved in step-strain experiments using the LSF technique [6,7,12,16,21,22].
More recently, an experimental technique known as continuous lubricated squeezing flow (CLSF) has been developed to perform equibiaxial elongational experiments on polymer melts and other forms of soft matter [28]. The CLSF technique is an adaptation of LSF [29] where the parallel plates are porous, which allows lubricant to be pumped through them during an experiment thus mitigating the lubricant thinning problem. This technique has been used to obtain measurements of that are consistent with those obtained from the MAD rheometer [28]. The CLSF technique has been used to study the rheological behavior of commercial polymer melts in constant strain rate equibiaxial elongational flows [30] and, more recently, CLSF has been used to conduct constant stress (creep) experiments [31].
Since the introduction of the well-known tube model of Doi and Edwards [32], there has been extensive research focused on the development of alternative molecular models to describe the dynamics of entangled polymer melts. Two of the more successful models are the Graham-Likhtman-Milner-McLeish (GLaMM) tube-based model [33] and the discrete slip-link model (DSM) [34]. Both of these models have been successful in predicting shear flow but do not describe well the uniaxial elongational flow [33,35]. Due to the absence of reliable experimental data, neither GLaMM nor DSM have been evaluated in equibiaxial elongational flows.
The GLaMM model incorporates reptation, constraint release, contour length fluctuations, chain stretch, longitudinal stress relaxation, and convective constraint release. There are five parameters in the GLaMM model [33]. These parameters include the plateau modulus , which can be determined from dynamic modulus experiments , and the average number of entanglements , which is related to , but can be used as an independent fitting parameter. The third is , the characteristic relaxation time of the strand. The fourth is , the empirically determined constraint release factor, and the fifth is , which is the prefactor of the retraction term [33]. These parameters should be independent of molecular weight and deformation [36]. This model has been shown to describe the rheology of linear, narrow molecular weight distribution polymers in the start-up of constant strain rate shear flows [33].
The DSM is more detailed than tube models (including the GLaMM model) where the average number of entanglements fluctuates stochastically [34,35,37]. The chain is treated as a random walk of Kuhn steps; the entanglements are distributed randomly on the chain [34,38]. The entanglements deform affinely and finite extensibility can be included; constant chain friction is also assumed [35]. There are two types of dynamics accounted for in the DSM. Sliding dynamics account for Kuhn steps shuffling through entanglements, and at the ends of the chain, this can create or destroy entanglements. Constraint dynamics account for the creation/destruction of entanglements on the probe chain caused by sliding dynamics of surrounding chains. In this way, the model self-consistently keeps track of the number of entanglements. The probability of creating or destroying an entanglement is calculated through detailed balance [39]. There are two parameters in the DSM. The first is which is related to the entanglement density and depends on chemical composition but not on chain length. It has recently been found [40] that can be found from atomistic simulations. The second is , which is the characteristic timescale for a Kuhn step to slide through an entanglement. Both parameters in the DSM are found from experimental data. The DSM has been shown to be capable of quantitatively predicting the rheological behavior for linear monodisperse and polydisperse systems as well as branched systems [35,41].
In the present study, measurements of the equibiaxial viscosity in constant strain rate flows are obtained on a series of well-characterized polystyrene melts using the CLSF technique. Results are also presented for a diluted polystyrene melt. With these new data, there now exists a complete set of data that includes shear, uniaxial elongation, and equibiaxial elongation for a single polymer melt. This type of data set, which heretofore has been unavailable, is compared to the predictions of GLaMM and DSM molecular models.
II. EXPERIMENTAL CONSIDERATIONS
A. Materials
The polystyrene samples examined in this study have narrow molecular weight distributions with molecular weights () that cover a range of entanglements per chain , where is the average molecular weight between entanglements. is known for many polymers and depends on polymer chain chemistry and is only weakly dependent on temperature [42]. However, the presence of a solvent (or an unentangled oligomer) increases according to , where is the volume fraction of polymer and is the entanglement molecular weight for the undiluted () polymer. For undiluted polystyrene, the entanglement molecular weight has the value [42].
Three undiluted melts were tested: PS105k with (polymer source), PS206k (G. Kisker), and PS392k with (polymer source). The linear viscoelastic (LVE) properties of these samples and their binary blends have recently been compared with slip-link model predictions [43]. The fourth sample is a diluted melt with volume fraction of polystyrene with (polymer source) in styrene oligomer with (polymer source), which is designated as PS425k-47. Additional material properties are shown in Table I. The diluted melt was prepared by dissolving the polystyrene and oligomer in tetrahydrofuran (THF) at a concentration of 60% and stirring for roughly 24 h. The liquid was poured onto glass plates and allowed to dry. After approximately two weeks of drying in a fume hood, the mixture was moved to a vacuum oven and kept at 50 C. The weight of the mixture was continually monitored and once the concentration of THF was below 30%, the mixture was subjected to 4-h bursts at 98 C in a vacuum. After each weighing, the dynamic modulus was measured. If the results were repeatable and there were no bubbles in the sample, the material was deemed dry. Samples for CLSF were compression molded under vacuum into disks having thicknesses 2.5–4 mm and diameters of either 10 or 25 mm. Silicone oils (GE Viscasil) with viscosities Pa s at the test temperatures were used as lubricants.
Properties of polystyrene samples.
Sample . | (kDa) . | . | . | (kPa s) . | (s) . | (C) . |
---|---|---|---|---|---|---|
PS105k | 105 | 1.03 | 7.8 | 192 | 2 | 150 |
PS206k | 206 | 1.04 | 15.5 | 402 | 4 | 160 |
PS392k | 392 | 1.06 | 29.5 | 1280 | 26 | 170 |
PS425k-47 | 425 | 1.04 | 15.4 | 1012 | 61 | 130 |
Sample . | (kDa) . | . | . | (kPa s) . | (s) . | (C) . |
---|---|---|---|---|---|---|
PS105k | 105 | 1.03 | 7.8 | 192 | 2 | 150 |
PS206k | 206 | 1.04 | 15.5 | 402 | 4 | 160 |
PS392k | 392 | 1.06 | 29.5 | 1280 | 26 | 170 |
PS425k-47 | 425 | 1.04 | 15.4 | 1012 | 61 | 130 |
Small amplitude oscillatory shear experiments were performed on the samples to obtain the dynamic modulus . Frequency sweeps were made at multiple temperatures, and the principle of time–temperature superposition was used to obtain master curves. Details of these experiments can be found elsewhere [43]. These measurements are shown in Fig. 1 for samples PS105k, PS206k, and PS392k and in Fig. 2 for sample PS425k-47. data were used to determine the spectrum of relaxation times (), from which the zero-shear viscosity and mean relaxation time were computed (see Table I). The relaxation spectrum was also used to compute LVE predictions, which are a useful reference for nonlinear deformations. In some cases, it is convenient to normalize data using and , which are the modulus and reciprocal frequency, respectively, defined by . As shown in the inset to Fig. 2, data for PS425k-47 and PS206k, which have the same number of entanglements , superimpose when normalized by and . These results are an example of the well-established universality of the LVE behavior of entangled polymer liquids.
Dynamic modulus (squares) and from left to right (circles) for polystyrene melts PS392k (blue), PS206k (black), and PS105k (red) at 170 C.
Dynamic modulus (squares) and from left to right (circles) for polystyrene melts PS392k (blue), PS206k (black), and PS105k (red) at 170 C.
Dynamic modulus (squares) and (circles) for diluted polystyrene melt PS425k-47 (orange) at 130 C. Inset shows reduced data for both PS206k (black) and PS425k-47 (orange).
Dynamic modulus (squares) and (circles) for diluted polystyrene melt PS425k-47 (orange) at 130 C. Inset shows reduced data for both PS206k (black) and PS425k-47 (orange).
B. Methods
The CLSF technique, which is shown schematically in Fig. 3, was implemented on two different rheometer platforms. One is the RSAIII (TA Instruments) as described in [28]; the other is the MCR301 (Anton Paar) as shown in Fig. 4. Samples PS105k, PS206k, and PS392k were tested using the RSAIII with a normal force capacity of 35 N, while sample PS425k-47 and the highest strain rate for sample PS206k were tested using the MCR301 with a normal force capacity of 50 N. Both rheometers were equipped with custom-made porous plates having a diameter () of either 15 or 25 mm. The permeability of the disks is in the range – mm, and the disk thickness was fixed at 7.3 mm. The RSAIII setup is heated by small band heaters attached to the plates, and the MCR301 is equipped with a convection oven for temperature control. The two setups have been shown to give consistent results [30].
Schematic of the CLSF plates. Lubricant with viscosity is delivered to the reservoir at a flow rate of , which then flows through the porous plate. Each plate has a permeability of and thickness of . The thickness of the lubricant layer on each plate is . The sample has thickness , radius , and viscosity .
Schematic of the CLSF plates. Lubricant with viscosity is delivered to the reservoir at a flow rate of , which then flows through the porous plate. Each plate has a permeability of and thickness of . The thickness of the lubricant layer on each plate is . The sample has thickness , radius , and viscosity .
CLSF tools mounted to the MCR301. The lower tool has a cup to collect excess lubricant.
CLSF tools mounted to the MCR301. The lower tool has a cup to collect excess lubricant.
A constant Hencky strain rate was imposed for all experiments such that the sample thickness followed ) with stain rates in the range from 0.002 to 0.3 s. For the RSAIII experiments, this was imposed using an arbitrary wave input function; for the MCR301, a constant Hencky strain rate was imposed by setting , where . In the CLSF technique, the lubricant thickness is controlled by supplying a constant lubricant flow rate from the reservoir to the lubricant film (see Fig. 3). If the lubricant thickness is held constant (), then the normalized flow rate is set to a value of one. For the RSAIII, it was found [28] that allowing a controlled thinning of the lubricant films was desirable so that . For the MCR301 experiments, so the lubricant film thickness remained constant. Results were independent of the initial lubricant layer thickness , which was in the range – m. We note that at early stages of our tests, the sample radius often was smaller than the plate radius (see Fig. 3); when , the sample radius was determined by . As discussed in [28], the success of the CLSF technique is based on the criteria , where . For the CLSF results reported here, , , and . Also, accurate stress measurements require , which ensures that edge effects are negligible; for the results presented here . The measured stress difference is determined from the measured force as and the equibiaxial viscosity by .
We note that the CLSF experiments presented in this study are limited by the force capacity of the transducer . Using the approximation leads to . For polystyrene melts using the current CLSF setup, we find , which means attempts to probe well into the nonlinear regime are limited. The situation is improved somewhat for diluted melts.
Before performing experiments, the distance between the porous plates is “zeroed” using the procedure outlined by [28]. This procedure is performed at the test temperature after allowing the rheometer to thermally equilibrate for at least 1 h. Force measurements are taken over a range of gaps (0.1 mm) with a known lubricant flow rate and compared with the predicted force. This procedure yields a zero gap that is accurate to 10 m, which is comparable to the estimated parallelism of the plates. For each test, a fresh sample is loaded between the disk and allowed to thermally equilibrate for at least 15 min before the experiment is initiated.
All reported results are the average of three experiments, and error bars represent the standard deviation of the data set. Uncertainty estimates based on error propagation gave results consistent with the standard deviation obtained from repeat experiments.
III. RESULTS AND DISCUSSION
A. Equibiaxial elongational experimental results
The CLSF technique was used to obtain measurements of the time-dependent equibiaxial viscosity at several Hencky strain rates in the range 0.003–0.3 s for the four samples listed in Table I. These data are compared with the LVE prediction calculated from the discrete relaxation spectrum obtained from the data shown in Figs. 1 and 2. In the discussion that follows, we use the dimensionless strain rate, or Weissenberg number, which is defined as . We report ranges for all CLSF experimental parameters used to obtain these results, but note that as long as the criteria discussed in Sec. II are satisfied, the results are independent of these parameters.
Results for sample PS105k at 150 C are shown in Fig. 5 at four strain rates . Due to limitations of the rheometer (the maximum velocity of the motor), the largest achievable strain rate for this sample corresponds to . Hence, as expected, the data are consistent with the LVE prediction. Figure 6 shows the equibiaxial viscosity data for PS206k at 160 C at five strain rates corresponding to a maximum of . From this figure, we see that the measured falls below the LVE prediction at larger strain rates, which indicates strain softening. This behavior is in contrast to the behavior reported in the literature [44,45] for narrow molecular weight distribution polystyrene melts in uniaxial elongation, which display rather pronounced strain hardening. It is worth noting that the uniaxial elongation data obtained by Bach et al. [44], because they were obtained at much lower temperatures, reached Wi numbers several orders of magnitude larger than those reachable in the present study. Results for sample PS392k are shown in Fig. 7 at 170 C. Due to experimental limitations (in this case the capacity of the force transducer), the applied strain rates were limited such that for the highest rate. As a result, the measured viscosity follows the LVE prediction for this sample.
Transient equibiaxial viscosity for PS105k at 150 C for different Hencky strain rates : 0.003 (), 0.01 (□), 0.03 (◊), and 0.1 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are , –, –, and –.
Transient equibiaxial viscosity for PS105k at 150 C for different Hencky strain rates : 0.003 (), 0.01 (□), 0.03 (◊), and 0.1 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are , –, –, and –.
Transient equibiaxial viscosity for PS206k at 160 C for different Hencky strain rates : 0.003 (), 0.01 (□), 0.03 (◊), 0.1 (), and 0.3 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are –, –, –, and –.
Transient equibiaxial viscosity for PS206k at 160 C for different Hencky strain rates : 0.003 (), 0.01 (□), 0.03 (◊), 0.1 (), and 0.3 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are –, –, –, and –.
Transient equibiaxial viscosity for PS392k at 170 C for different Hencky strain rates : 0.003 (), 0.01 (□), and 0.03 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are , = 3.3, = 0.43, and = 0.02.
Transient equibiaxial viscosity for PS392k at 170 C for different Hencky strain rates : 0.003 (), 0.01 (□), and 0.03 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are , = 3.3, = 0.43, and = 0.02.
Constant strain rate results for PS425k-47 at 130 C are shown in Fig. 8 at rates of 0.002, 0.01, 0.03, 0.1, and 0.3 s. These strain rates correspond to Wi in the range 0.12–18. Of the five rates that were tested, the three where display nonlinear behavior and show strain softening.
Transient equibiaxial viscosity for PS425k-47 at 130 C for different Hencky strain rates : 0.002 (), 0.01 (□), 0.03 (◊), 0.1 (), and 0.3 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are , , , and .
Transient equibiaxial viscosity for PS425k-47 at 130 C for different Hencky strain rates : 0.002 (), 0.01 (□), 0.03 (◊), 0.1 (), and 0.3 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are , , , and .
These results represent the highest experimentally obtained Wi for polymer melts in equibiaxial flow. While a is likely to be smaller than that encountered in industrial processes such as blow molding, it should be noted that data obtained from the MAD rheometer [4] for a commercial polystyrene melt achieved a maximum . The strain softening behavior for is expected because the polystyrene chains have a linear structure; however, the data reported here appear to be the first to confirm such behavior.
As discussed in Sec. II (see Fig. 2), universality in the LVE behavior of entangled polymer liquids is well established. By universality, we mean that when the temperature and/or concentration dependence of and are taken into account, the average number of entanglements dictates (linear and nonlinear) rheological behavior within the range spanned by viscous and rubbery regimes. We now examine the question of universality in the nonlinear rheological behavior of entangled polymer melts in equibiaxial elongational flows. To do so, we use the data in Figs. 6 and 8 for the two systems having the same number of entanglements . Using the same parameters used to rescale the LVE data in Fig. 2, we compare the rescaled equibiaxial elongational data in Fig. 9. From this figure, we see evidence that the nonlinear rheological behavior of entangled polymer melts in equibiaxial elongational flow is indeed universal.
Normalized equibiaxial elongational viscosity at C for PS206k (filled) with and PSN492k-47 (open) with for 1.2 (□, ■). The solid line is the prediction for LVE theory.
Normalized equibiaxial elongational viscosity at C for PS206k (filled) with and PSN492k-47 (open) with for 1.2 (□, ■). The solid line is the prediction for LVE theory.
Huang et al. [46,47] have presented uniaxial elongational viscosity data on several polystyrene melts and diluted melts having the same number of entanglements . One diluted polymer () has approximately the same value of as the PS206k sample. Comparison of data for melts and equivalent diluted melts (rescaled as in Fig. 9) displayed rather significant differences for . The comprehensive comparisons presented in these studies [46,47] appear to indicate that universality, when based only on , does not hold for large uniaxial elongational flows of entangled polymers. In particular, diluted melts display more pronounced strain hardening than undiluted melts. These results are at odds with shear flow data and with the results presented in Fig. 9. Clearly, further investigation of this unresolved issue is critical.
The steady-state viscosity normalized by 6 versus dimensionless strain rate Wi for all four samples is shown in Fig. 10. From this figure, we see self-consistency among all four samples with strain softening observed for Wi 1. It should also be noted that it is difficult to confirm that a steady-state has been achieved at the higher strain rates since the maximum achievable . Nevertheless, it appears that the data for Wi 1 show a power-law dependence on strain rate with . These data are consistent with the results of Hassager and co-workers for polymer melts in uniaxial elongational deformations [44,46,47]. However, for both uniaxial and equibiaxial elongational flows, these observations are at odds with the prediction of the original tube model and but consistent with predictions of a modified tube model where “interchain pressure” is taken into account [48].
Normalized steady-state equibiaxial viscosity versus normalized Hencky strain rate for polystyrene systems listed in Table I: PS105k (□), PS206k (◊), PS392k (), and PS425k-47 (). The straight line has a slope of 0.5.
Normalized steady-state equibiaxial viscosity versus normalized Hencky strain rate for polystyrene systems listed in Table I: PS105k (□), PS206k (◊), PS392k (), and PS425k-47 (). The straight line has a slope of 0.5.
B. Evaluation of molecular models
The goal of all molecularly based models for entangled polymer liquids is to predict, with a single set of parameters that are small in number, the nonlinear rheological behavior in multiple deformation modes including shear, uniaxial elongation, and equibiaxial elongation. Ideally, the model parameters are determined using LVE data. While the ultimate goal is to predict rheological behavior of commercial polymers having broad molecular weight distributions, it is desirable to develop models for monodisperse polymers. Until now, a set of nonlinear rheological data that includes shear and uniaxial and equibiaxial elongation deformations for an entangled polymer liquid having narrow molecular weight distribution has been unavailable. For a polystyrene melt with , there now exists such a data set that consists of the equibiaxial elongation data (see Fig. 6), uniaxial elongation data [44], and shear data [49].
Parameters for GLaMM () and DSM () were determined from LVE data. Additional parameters for the DSM are based on recommended [33] values ( = 0.1, ). Differences in the LVE parameters reflect the fact that the experiments were performed at different temperatures and that the polystyrene samples used in the other studies [44,49] have slightly smaller molecular weights ( kDa). Predictions for GLaMM were obtained using a previously developed code [50] while those for DSM using a code described elsewhere [51].
Figure 11 shows a comparison of GLaMM and DSM predictions with experimental data [49] for a polystyrene melt with in shear flows at several strain rates. Similar DSM predictions have been previously presented [35]; here, we have used slightly larger value for determined using a fitting procedure described elsewhere [52]. Predictions for both GLaMM and DSM are in good agreement with experiments. However, the DSM prediction is clearly superior to those for GLaMM at the highest strain rate shown in this figure.
Experimental shear viscosity from top to bottom for a polystyrene melt with at 175 C for different strain rates : 0.1 (□, black), 0.3 (, red), 1.0 (, green), 3.0 (◊, blue), 10 (, purple), 30 (, pink) from [49]. Predictions of the GLaMM model (dashed) with parameters: kPa, , ms, , and the DSM model (solid) with parameters: , s.
Experimental shear viscosity from top to bottom for a polystyrene melt with at 175 C for different strain rates : 0.1 (□, black), 0.3 (, red), 1.0 (, green), 3.0 (◊, blue), 10 (, purple), 30 (, pink) from [49]. Predictions of the GLaMM model (dashed) with parameters: kPa, , ms, , and the DSM model (solid) with parameters: , s.
Predictions for GLaMM and DSM in uniaxial elongation for a polystyrene melt with 15 at 130 C are compared with experiments [44] shown in Fig. 12. Both DSM and GLaMM capture the onset of strain hardening. However, at larger strains , both models show significant deviations from experiments: DSM displays an overshoot and excessive strain softening; GLaMM predictions diverge for . The latter has been reported previously [33] and was claimed to be a consequence of finite extensibility, which is not included in GLaMM. We note, however, that inclusion of finite extensibility in DSM has rather minor effects on the degree of stress overshoot and strain softening [35].
Experimental uniaxial viscosity for a polystyrene melt with at 130 C for different strain rates : 0.001 (□, black), 0.003 (, red), 0.01 (, green), 0.03 (◊, blue), and 0.1 (, purple) from [44]. Predictions of the GLaMM model (dashed) with parameters: kPa, , s, , and the DSM model (solid) with parameters: , ms.
Experimental uniaxial viscosity for a polystyrene melt with at 130 C for different strain rates : 0.001 (□, black), 0.003 (, red), 0.01 (, green), 0.03 (◊, blue), and 0.1 (, purple) from [44]. Predictions of the GLaMM model (dashed) with parameters: kPa, , s, , and the DSM model (solid) with parameters: , ms.
Figure 13 shows a comparison of the DSM and GLaMM predictions and experimental data in equibiaxial elongational flows for PS206k. From this figure, we see that both models over-predict the degree of strain softening at higher strain rates. Similar to the results for uniaxial elongation (see Fig. 12), Fig. 13 shows that DSM predicts a stress overshoot at the highest strain rate. In this case, however, the existence (or absence) of an overshoot in the experiments cannot be determined.
Experimental from right to left biaxial viscosity for PS206k at 160 C for different strain rates : 0.003 (□, black), 0.01 (, red), 0.03 (, green), 0.1 (◊, blue), and 0.3 (, purple). Predictions of the GLaMM model (dashed) with parameters: kPa, , ms, , and the DSM model (solid) with parameters: , ms.
Experimental from right to left biaxial viscosity for PS206k at 160 C for different strain rates : 0.003 (□, black), 0.01 (, red), 0.03 (, green), 0.1 (◊, blue), and 0.3 (, purple). Predictions of the GLaMM model (dashed) with parameters: kPa, , ms, , and the DSM model (solid) with parameters: , ms.
IV. SUMMARY AND CONCLUSIONS
The novel rheological technique known as CLSF has been used to perform constant strain rate equibiaxial elongational experiments on four entangled polystyrene liquids having narrow molecular weight distributions. The nonlinear behavior of these linear polymer systems shows, in contrast to uniaxial elongational tests on similar polymer systems, strain softening. Steady-state viscosities obtained from these data show self-consistency between the different systems when plotted as a function of dimensionless strain rate (Wi) with a power-law dependence. The new data set, which includes a diluted melt with the same number of entanglements as one of the undiluted melts, has also been used to demonstrate universality in entangled polymer melts in equibiaxial elongational flows. Experimental limitations of the current CLSF setup prevented more extensive studies in the highly nonlinear regime.
The data reported in this study, when combined with data from the literature, result in a data set for a well-characterized (i.e., narrow molecular weight distribution) entangled polymer liquid in constant strain rate flows in shear, uniaxial elongation and biaxial elongation. This unique data set was used to carry out comparisons with predictions from two well-established molecular models for entangled polymer liquids. Predictions from the GLaMM model and the DSM were made using model parameters obtained from LVE data. A single set of model parameters (adjusted only to account for changes in temperature) were used to predict the nonlinear behavior of this polymer liquid in the three deformation modes considered. We find that predictions from both DSM and GLaMM are in good agreement with shear flow data. For uniaxial elongational flows, neither model is able to describe experiments: The DSM predicts an overshoot that is not observed, and GLaMM gives unphysical predictions at larger strain rates. Predictions from both DSM and GLaMM give reasonable predictions for equibiaxial elongation although both over-predict the degree of strain softening. Further comparisons of these and other molecular models with more extensive data sets will be essential to the development of robust models for describing the rheological behavior of entangled polymer liquids.
ACKNOWLEDGMENTS
The financial support provided by the National Science Foundation (NSF) (Grant Nos. CTS-0327955 and CBET-1236576) for this study is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the NSF. The authors are grateful for the assistance of Maria Katzarova and Marat Andreev with the DSM and GLaMM predictions.