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.

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 σB is measured as a function of Hencky strain rate ε˙B, which can be used to define the equibiaxial viscosity ηB=σB/ε˙B. 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 ηB of three polymer melts at several Hencky strain rates ε˙B to a Hencky strain of εB2.5 [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 εB1. This was definitively shown [27] by making direct comparisons of LSF measurements of ηB 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 εB1.6 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 ηB 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 GN0, which can be determined from dynamic modulus experiments G(ω), and the average number of entanglements Z, which is related to GN0, but can be used as an independent fitting parameter. The third is τe, the characteristic relaxation time of the strand. The fourth is cν, the empirically determined constraint release factor, and the fifth is Rs, 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 Z 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 β which is related to the entanglement density and depends on chemical composition but not on chain length. It has recently been found [40] that β can be found from atomistic simulations. The second is τK, which is the characteristic timescale for a Kuhn step to slide through an entanglement. Both parameters in the DSM are found from G(ω) 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 ηB in constant strain rate ε˙B 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.

The polystyrene samples examined in this study have narrow molecular weight distributions with molecular weights (Mw) that cover a range of entanglements per chain Z=Mw/Me, where Me is the average molecular weight between entanglements. Me 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 Me according to Me(ϕ)=Me0/ϕ, where ϕ is the volume fraction of polymer and Me0 is the entanglement molecular weight for the undiluted (ϕ=1) polymer. For undiluted polystyrene, the entanglement molecular weight has the value Me0=13.3kDa [42].

Three undiluted melts were tested: PS105k with Mw=105kDa (polymer source), PS206k Mw=206kDa (G. Kisker), and PS392k with Mw=392kDa (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 ϕ=0.47 volume fraction of polystyrene with Mw=492kDa (polymer source) in styrene oligomer with Mw=2.3kDa (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 G(ω) 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 μ=555 Pa s at the test temperatures were used as lubricants.

TABLE I.

Properties of polystyrene samples.

SampleMw (kDa)Mw/MnZη0 (kPa s)τd (s)T (°C)
PS105k 105 1.03 7.8 192 150 
PS206k 206 1.04 15.5 402 160 
PS392k 392 1.06 29.5 1280 26 170 
PS425k-47 425 1.04 15.4 1012 61 130 
SampleMw (kDa)Mw/MnZη0 (kPa s)τd (s)T (°C)
PS105k 105 1.03 7.8 192 150 
PS206k 206 1.04 15.5 402 160 
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 G(ω)=G(ω)+iG(ω). 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. G(ω) data were used to determine the spectrum of relaxation times (gi,τi), from which the zero-shear viscosity η0=igiτi and mean relaxation time τd=igiτi2/igiτi 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 G(ω) data using Gc and τc, which are the modulus and reciprocal frequency, respectively, defined by Gc=G(ω=1/τc)=G(ω=1/τc). As shown in the inset to Fig. 2, data for PS425k-47 and PS206k, which have the same number of entanglements Z, superimpose when normalized by Gc and τc. These results are an example of the well-established universality of the LVE behavior of entangled polymer liquids.

FIG. 1.

Dynamic modulus G (squares) and G from left to right (circles) for polystyrene melts PS392k (blue), PS206k (black), and PS105k (red) at 170 °C.

FIG. 1.

Dynamic modulus G (squares) and G from left to right (circles) for polystyrene melts PS392k (blue), PS206k (black), and PS105k (red) at 170 °C.

Close modal
FIG. 2.

Dynamic modulus G (squares) and G (circles) for diluted polystyrene melt PS425k-47 (orange) at 130 °C. Inset shows reduced data for both PS206k (black) and PS425k-47 (orange).

FIG. 2.

Dynamic modulus G (squares) and G (circles) for diluted polystyrene melt PS425k-47 (orange) at 130 °C. Inset shows reduced data for both PS206k (black) and PS425k-47 (orange).

Close modal

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 (2Rp) of either 15 or 25 mm. The permeability of the disks κ is in the range 1091010 mm2, and the disk thickness b 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].

FIG. 3.

Schematic of the CLSF plates. Lubricant with viscosity μ is delivered to the reservoir at a flow rate of q, which then flows through the porous plate. Each plate has a permeability of κ and thickness of b. The thickness of the lubricant layer on each plate is δ. The sample has thickness h, radius R, and viscosity η.

FIG. 3.

Schematic of the CLSF plates. Lubricant with viscosity μ is delivered to the reservoir at a flow rate of q, which then flows through the porous plate. Each plate has a permeability of κ and thickness of b. The thickness of the lubricant layer on each plate is δ. The sample has thickness h, radius R, and viscosity η.

Close modal
FIG. 4.

CLSF tools mounted to the MCR301. The lower tool has a cup to collect excess lubricant.

FIG. 4.

CLSF tools mounted to the MCR301. The lower tool has a cup to collect excess lubricant.

Close modal

A constant Hencky strain rate ε˙B was imposed for all experiments such that the sample thickness followed h=h0exp(2ε˙Bt) with stain rates in the range from 0.002 to 0.3 s1. 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 h˙/h=2ε˙B, where h˙=dh/dt. In the CLSF technique, the lubricant thickness δ is controlled by supplying a constant lubricant flow rate q from the reservoir to the lubricant film (see Fig. 3). If the lubricant thickness is held constant (δ=δ0), then the normalized flow rate ω=q/πRp2δ0ε˙B 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 ω=0.5. For the MCR301 experiments, ω=1 so the lubricant film thickness remained constant. Results were independent of the initial lubricant layer thickness δ0, which was in the range 1050μm. We note that at early stages of our tests, the sample radius R often was smaller than the plate radius Rp (see Fig. 3); when R<Rp, the sample radius was determined by R=R0h0/h. As discussed in [28], the success of the CLSF technique is based on the criteria κR2/bδ03α(R/δ0)21, where α=μ/η0. For the CLSF results reported here, α105, α(R/δ0)2110, and κR2/bδ030.010.1. Also, accurate stress measurements require ϕ=δ0/h01, which ensures that edge effects are negligible; for the results presented here ϕ0.01. The measured stress difference is determined from the measured force F as σB=F/πR2 and the equibiaxial viscosity by ηB=σB/ε˙B.

We note that the CLSF experiments presented in this study are limited by the force capacity of the transducer Fmax. Using the approximation σBηBε˙B6GNτdε˙B leads to (τdε˙B)maxFmax/(6πR2GN). For polystyrene melts using the current CLSF setup, we find (τdε˙B)max1, 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 μm, 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.

The CLSF technique was used to obtain measurements of the time-dependent equibiaxial viscosity ηB at several Hencky strain rates ε˙B in the range 0.003–0.3 s1 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 G(ω) 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 Wi=ε˙Bτd. 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 ε˙B. Due to limitations of the rheometer (the maximum velocity of the motor), the largest achievable strain rate for this sample corresponds to Wi=0.2. 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 Wi=1.2. From this figure, we see that the measured ηB 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 Wi=0.78 for the highest rate. As a result, the measured viscosity follows the LVE prediction for this sample.

FIG. 5.

Transient equibiaxial viscosity ηB for PS105k at 150 °C for different Hencky strain rates ε˙B(s1): 0.003 (), 0.01 (□), 0.03 (◊), and 0.1 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are α=4.5×105, α(R/δ0)2=1.43.3, β=0.050.43, and ϕ=0.020.03.

FIG. 5.

Transient equibiaxial viscosity ηB for PS105k at 150 °C for different Hencky strain rates ε˙B(s1): 0.003 (), 0.01 (□), 0.03 (◊), and 0.1 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are α=4.5×105, α(R/δ0)2=1.43.3, β=0.050.43, and ϕ=0.020.03.

Close modal
FIG. 6.

Transient equibiaxial viscosity ηB for PS206k at 160 °C for different Hencky strain rates ε˙B(s1): 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 α=8×1069.5×105, α(R/δ0)2=3.27.7, β=0.140.4, and ϕ=0.0080.02.

FIG. 6.

Transient equibiaxial viscosity ηB for PS206k at 160 °C for different Hencky strain rates ε˙B(s1): 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 α=8×1069.5×105, α(R/δ0)2=3.27.7, β=0.140.4, and ϕ=0.0080.02.

Close modal
FIG. 7.

Transient equibiaxial viscosity ηB for PS392k at 170 °C for different Hencky strain rates ε˙B(s1): 0.003 (), 0.01 (□), and 0.03 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are α=2.6×105, α(R/δ0)2 = 3.3, β = 0.43, and ϕ = 0.02.

FIG. 7.

Transient equibiaxial viscosity ηB for PS392k at 170 °C for different Hencky strain rates ε˙B(s1): 0.003 (), 0.01 (□), and 0.03 (). The solid line is the prediction of LVE theory. The parameters for CLSF experiments are α=2.6×105, α(R/δ0)2 = 3.3, β = 0.43, and ϕ = 0.02.

Close modal

Constant strain rate ε˙B 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 s1. These strain rates correspond to Wi in the range 0.12–18. Of the five rates that were tested, the three where Wi1 display nonlinear behavior and show strain softening.

FIG. 8.

Transient equibiaxial viscosity ηB for PS425k-47 at 130 °C for different Hencky strain rates ε˙B(s1): 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 α=5×106, α(R/δ0)2=2.1, β=0.24, and ϕ=0.008.

FIG. 8.

Transient equibiaxial viscosity ηB for PS425k-47 at 130 °C for different Hencky strain rates ε˙B(s1): 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 α=5×106, α(R/δ0)2=2.1, β=0.24, and ϕ=0.008.

Close modal

These results represent the highest experimentally obtained Wi for polymer melts in equibiaxial flow. While a Wi10 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 Wi1. The strain softening behavior for Wi1 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 GN and τd are taken into account, the average number of entanglements Z 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 Z15. 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.

FIG. 9.

Normalized equibiaxial elongational viscosity at 130°C for PS206k (filled) with τc=954s,Gc=41kPa and PSN492k-47 (open) with τc=45s,Gc=9.2kPa for Wi=ε˙Bτc0.1(°,),0.4(,), 1.2 (□, ■). The solid line is the prediction for LVE theory.

FIG. 9.

Normalized equibiaxial elongational viscosity at 130°C for PS206k (filled) with τc=954s,Gc=41kPa and PSN492k-47 (open) with τc=45s,Gc=9.2kPa for Wi=ε˙Bτc0.1(°,),0.4(,), 1.2 (□, ■). The solid line is the prediction for LVE theory.

Close modal

Huang et al. [46,47] have presented uniaxial elongational viscosity ηE data on several polystyrene melts and diluted melts having the same number of entanglements Z. One diluted polymer (ϕ=0.72,Mw=285kDa) has approximately the same value of Z as the PS206k sample. Comparison of data for melts and equivalent diluted melts (rescaled as in Fig. 9) displayed rather significant differences for Wi10. The comprehensive comparisons presented in these studies [46,47] appear to indicate that universality, when based only on Z, does not hold for large ε˙ 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 ηB normalized by 6η0 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 εB2.5. Nevertheless, it appears that the data for Wi 1 show a power-law dependence on strain rate with ηBε˙B0.5. These data are consistent with the results of Hassager and co-workers ηEε˙0.5 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 ηEε˙1 and ηBε˙B1 but consistent with predictions of a modified tube model where “interchain pressure” is taken into account [48].

FIG. 10.

Normalized steady-state equibiaxial viscosity ηB versus normalized Hencky strain rate Wi=ε˙Bτd for polystyrene systems listed in Table I: PS105k (□), PS206k (◊), PS392k (), and PS425k-47 (). The straight line has a slope of 0.5.

FIG. 10.

Normalized steady-state equibiaxial viscosity ηB versus normalized Hencky strain rate Wi=ε˙Bτd for polystyrene systems listed in Table I: PS105k (□), PS206k (◊), PS392k (), and PS425k-47 (). The straight line has a slope of 0.5.

Close modal

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 Z15, 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 (GN0,Z,τe) and DSM (β,τK) were determined from LVE data. Additional parameters for the DSM are based on recommended [33] values (cν = 0.1, Rs=2). 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 (Mw200 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 Z15 in shear flows at several strain rates. Similar DSM predictions have been previously presented [35]; here, we have used slightly larger value for β 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.

FIG. 11.

Experimental shear viscosity η from top to bottom for a polystyrene melt with Z15 at 175 °C for different strain rates γ˙(s1): 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: GN0=270 kPa, Z=12, τe=0.9 ms, cν=0.1, Rs=2 and the DSM model (solid) with parameters: β=17, τK=2μs.

FIG. 11.

Experimental shear viscosity η from top to bottom for a polystyrene melt with Z15 at 175 °C for different strain rates γ˙(s1): 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: GN0=270 kPa, Z=12, τe=0.9 ms, cν=0.1, Rs=2 and the DSM model (solid) with parameters: β=17, τK=2μs.

Close modal

Predictions for GLaMM and DSM in uniaxial elongation for a polystyrene melt with Z15 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 εE2, both models show significant deviations from experiments: DSM displays an overshoot and excessive strain softening; GLaMM predictions diverge for Wi10. 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].

FIG. 12.

Experimental uniaxial viscosity ηE for a polystyrene melt with Z15 at 130 °C for different strain rates ε˙E(s1): 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: GN0=250 kPa, Z=12, τe=1 s, cν=0.1, Rs=2 and the DSM model (solid) with parameters: β=15, τK=2 ms.

FIG. 12.

Experimental uniaxial viscosity ηE for a polystyrene melt with Z15 at 130 °C for different strain rates ε˙E(s1): 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: GN0=250 kPa, Z=12, τe=1 s, cν=0.1, Rs=2 and the DSM model (solid) with parameters: β=15, τK=2 ms.

Close modal

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.

FIG. 13.

Experimental from right to left biaxial viscosity ηB for PS206k at 160 °C for different strain rates ε˙B(s1): 0.003 (□, black), 0.01 (, red), 0.03 (, green), 0.1 (◊, blue), and 0.3 (, purple). Predictions of the GLaMM model (dashed) with parameters: GN0=225 kPa, Z=12, τe=5 ms, cν=0.1, Rs=2 and the DSM model (solid) with parameters: β=19, τK=0.01 ms.

FIG. 13.

Experimental from right to left biaxial viscosity ηB for PS206k at 160 °C for different strain rates ε˙B(s1): 0.003 (□, black), 0.01 (, red), 0.03 (, green), 0.1 (◊, blue), and 0.3 (, purple). Predictions of the GLaMM model (dashed) with parameters: GN0=225 kPa, Z=12, τe=5 ms, cν=0.1, Rs=2 and the DSM model (solid) with parameters: β=19, τK=0.01 ms.

Close modal

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.

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.

1.
Macosko
,
C. W.
,
Rheology: Principles, Measurements and Applications
(
VCH
,
New York
,
1994
).
2.
Meissner
,
J.
,
T.
Raible
, and
S. E.
Stephenson
, “
Rotary clamp in uniaxial and biaxial extensional rheometry of polymer melts
,”
J. Rheol.
25
,
1
28
(
1981
).
3.
Meissner
,
J.
, “
Polymer melt elongation- methods, results, and recent developments
,”
Polym. Eng. Sci.
27
,
537
546
(
1987
).
4.
Hachmann
,
P.
, and
J.
Meissner
, “
Rheometer for equibiaxial and planar elongations of polymer melts
,”
J. Rheol.
47
,
989
1010
(
2003
).
5.
Chatraei
,
S.
,
C. W.
Macosko
, and
H. H.
Winter
, “
Lubricated squeezing flow: a new biaxial extensional rheometer
,”
J. Rheol.
25
,
433
443
(
1981
).
6.
Soskey
,
P. R.
, and
H. H.
Winter
, “
Equibiaxial extension of two polymer melts: polystyrene and low density polyethylene
,”
J. Rheol.
29
,
493
517
(
1984
).
7.
Khan
,
S. A.
,
R. K.
Prud’homme
, and
R. G.
Larson
, “
Comparison of the rheology of polymer melts in shear, and biaxial and uniaxial extensions
,”
Rheol. Acta
26
,
144
151
(
1987
).
8.
Hsu
,
T. C.
, and
I. R.
Harrison
, “
Measurement of the equibiaxial elongation viscosity of polystyrene using lubricated squeezing
,”
Polym. Eng. Sci.
31
,
223
230
(
1991
).
9.
Takahashi
,
M.
,
T.
Isaki
,
T.
Takigawa
, and
T.
Masuda
, “
Measurement of biaxial and uniaxial extensional flow behavior of polymer melts at constant strain rates
,”
J. Rheol.
37
,
827
846
(
1993
).
10.
Ebrahimi
,
N. G.
,
M.
Takahashi
,
O.
Araki
, and
T.
Masuda
, “
Biaxial extensional flow behavior of monodisperse polystyrene melts
,”
Acta Polym.
46
,
267
271
(
1995
).
11.
Urakawa
,
O.
,
M.
Takahashi
,
T.
Masuda
, and
N. G.
Ebrahimi
, “
Damping functions and chain relaxation in uniaxial and biaxial extensions: Comparison with the Doi-Edwards theory
,”
Macromolecules
28
,
7196
7201
(
1995
).
12.
Nishioka
,
A.
,
T.
Takahashi
,
Y. M. J.
Takimoto
, and
K.
Koyama
, “
Description of uniaxial, biaxial, and planar elongational viscosities of polystyrene melt by the k-bkz model
,”
J. Non-Newton. Fluid Mech.
89
,
287
301
(
2000
).
13.
Sugimoto
,
M.
,
Y.
Masubuchi
,
J.
Takimoto
, and
K.
Koyama
, “
Melt rheology of polypropylene containing small amounts of high-molecular-weight chain. 2. uniaxial and biaxial extensional flow
,”
Macromolecules
34
,
6056
6063
(
2001
).
14.
Campanella
,
O. H.
, and
M.
Peleg
, “
Squeezing flow viscometry for nonelastic semiliquid foods- theory and applications
,”
Crit. Rev. Food Sci. Nutr.
42
,
241
264
(
2002
).
15.
Nasseri
,
S. L.
,
L. E.
Bilston
, and
R. I.
Tanner
, “
Lubricated squeezing flow: A useful method for measuring the viscoelastic properties of soft tissues
,”
Biorheology
40
,
545
551
(
2003
).
16.
Isaki
,
T.
, and
M. T. O.
Urakawa
, “
Biaxial damping function of entangled monodisperse polystyrene melts: Comparison with the mead-larson-doi model
,”
J. Rheol.
47
,
1201
1210
(
2003
).
17.
Nasseri
,
S. L.
,
L. E.
Bilston
,
B.
Fasheun
, and
R. I.
Tanner
, “
Modeling the biaxial elongational deformation of soft solids
,”
Rheol. Acta
43
,
68
79
(
2004
).
18.
Yamane
,
H.
,
K.
Sasai
,
M.
Takano
, and
M.
Takahashi
, “
Poly(d-lactic acid) as a rheological modifier of poly(l-lactic acid): shear and biaxial extensional flow behavior
,”
J. Rheol.
48
,
599
609
(
2004
).
19.
Stadler
,
F. J.
,
A.
Nishioka
,
J.
Stange
,
K.
Koyama
, and
H.
Münstedt
, “
Comparison of the elongational behavior of various polyolefins in uniaxial and equibiaxial flows
,”
Rheol. Acta
46
,
1003
1012
(
2007
).
20.
Song
,
Y.
,
Q.
Zheng
, and
Z.
Wang
, “
Equibiaxial extensional flow of wheat gluten plasticized with glycerol
,”
Food Hydrocoll.
21
,
1290
1295
(
2007
).
21.
Okamoto
,
K.
, and
M.
Yamaguchi
, “
Stress relaxation under large step equibiaxial elongation for low-density polyethylene
,”
J. Polym. Sci. B Polym. Phys.
47
,
1275
1284
(
2009
).
22.
Kashyap
,
T.
, and
D. C.
Venerus
, “
Stress relaxation in polymer melts following equibiaxial step strain
,”
Macromolecules
43
,
5874
5880
(
2010
).
23.
Kompani
,
M.
, and
D. C.
Venerus
, “
Equibiaxial extensional flow of polymer melts via lubricated squeezing flow. i. experimental analysis
,”
Rheol. Acta
39
,
444
451
(
2000
).
24.
Venerus
,
D. C.
,
M.
Kompani
, and
B.
Bernstein
, “
Equibiaxial extensional flow of polymer melts via lubricated squeezing flow. ii. flow modeling
,”
Rheol. Acta
39
,
574
582
(
2000
).
25.
Burbidge
,
A. S.
, and
C.
Servais
, “
Squeeze flows of apparently lubricated thin films
,”
J. Non-Newton. Fluid Mech.
124
,
115
127
(
2004
).
26.
Engmann
,
J.
,
C.
Servais
, and
A. S.
Burbidge
, “
Squeeze flow theory and applications to rheometry: a review
,”
J. Non-Newton. Fluid Mech.
132
,
1
27
(
2005
).
27.
Guadarrama-Medina
,
T.
,
T.-Y.
Shiu
, and
D. C.
Venerus
, “
Direct comparison of equibiaxial elongational viscosity measurements from lubricated squeezing flow and the multiaxiales dehnrheometer
,”
Rheol. Acta
48
,
11
17
(
2009
).
28.
Venerus
,
D. C.
,
T.-Y.
Shiu
,
T.
Kashyap
, and
J.
Hosttetler
, “
Continuous lubricated squeezing flow: a novel technique for equibiaxial elongational viscosity measurements on polymer melts
,”
J. Rheol.
54
,
1083
1095
(
2010
).
29.
Venerus
,
D. C.
, and
M.
Kompani
, US patent 5,916,599 (
1999
).
30.
Mick
,
R. M.
,
T.-Y.
Shiu
, and
D. C.
Venerus
, “
Equibiaxial elongational viscosity measurements of commercial polymer melts
,”
Polym. Eng. Sci.
55
,
1012
1017
(
2015
).
31.
Mick
,
R. M.
, and
D. C.
Venerus
, “
A novel technique for conducting creep experiments in equibiaxial elongation
,”
Rheol. Acta
56
,
591
596
(
2017
).
32.
Doi
,
M.
, and
S. F.
Edwards
,
The Theory of Polymer Dynamics
(
Oxford University
,
Oxford
,
1986
).
33.
Graham
,
R. S.
,
A. E.
Likhtman
, and
T. C. B.
McLeish
, “
Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release
,”
J. Rheol.
47
,
1171
1200
(
2003
).
34.
Schieber
,
J. D.
, “
Fluctuations in entanglements of polymer liquids
,”
J. Chem. Phys.
118
,
5162
5166
(
2003
).
35.
Andreev
,
M.
,
R. N.
Khaliullin
,
R. J. A.
Steenbakkers
, and
J. D.
Schieber
, “
Approximations of the discrete slip-link model and their effect on nonlinear rheology predictions
,”
J. Rheol.
57
,
535
557
(
2013
).
36.
Likhtman
,
A. E.
, and
T. C. B.
McLeish
, “
Quantitative theory for linear dynamics of linear entangled polymers
,”
Macromolecules
35
,
6332
6343
(
2002
).
37.
Steenbakkers
,
R. J. A.
, and
J. D.
Schieber
, “
Derivation of free energy expressions for tube models from course-grained slip-link models
,”
J. Chem. Phys.
137
,
034901
(
2012
).
38.
Khaliullin
,
R. N.
, and
J. D.
Schieber
, “
Analytic expressions for the statistics of the primitive-path length in entangled polymers
,”
Phys. Rev. Lett.
100
,
188302
(
2008
).
39.
Khaliullin
,
R. N.
, and
J. D.
Schieber
, “
Self-consistent modeling of constraint release in a single-chain mean-field slip-link model
,”
Macromolecules
42
,
7504
7517
(
2009
).
40.
Steenbakkers
,
R. J. A.
,
C.
Tzoumanekas
,
Y.
Li
,
W. K.
Liu
,
M.
Kröger
, and
J. D.
Schieber
, “
Primitive-path statistics of entangled polymers: mapping multi-chain simulations onto single-chain mean-field models
,”
New J. Phys.
16
,
015027
(
2014
).
41.
Katzarova
,
M.
,
M.
Andreev
,
Y. R.
Sliozberg
,
R. A.
Mrozek
,
J. L.
Lenhart
,
J. W.
Andzelm
, and
J. D.
Schieber
, “
Rheological predictions of network systems swollen with entangled solvent
,”
AIChE J.
60
,
1372
1380
(
2014
).
42.
Fetters
,
L. J.
,
D. J.
Lohse
, and
R. H.
Colby
, in
Physical Properties of Polymers
, edited by
J. E.
Mark
(
Springer
,
New York
,
2007
), pp.
447
454
.
43.
Katzarova
,
M.
,
T.
Kashyap
,
J. D.
Schieber
, and
D. C.
Venerus
, “
Linear viscoelastic behavior of bidisperse polystyrene blends: experiments and slip-link predictions
,”
Rheol. Acta
57
,
327
338
(
2018
).
44.
Bach
,
A.
,
K.
Almadal
,
H. K.
Rasmussen
, and
O.
Hassager
, “
Elongational viscosity of narrow molar mass distribution polystyrene
,”
Macromolecules
36
,
5174
5179
(
2003
).
45.
Luap
,
C.
,
C.
Müller
,
T.
Schweizer
, and
D. C.
Venerus
, “
Simultaneous stress and birefringence measurements during uniaxial elongation of polystyrene melts with narrow molecular weight distribution
,”
Rheol. Acta
45
,
83
91
(
2005
).
46.
Huang
,
Q.
,
O.
Mednova
,
H. K.
Rasmussen
,
N. J.
Alvarez
,
A. L.
Skov
,
K.
Almdal
, and
O.
Hassager
, “
Concentrated polymer solutions are different from melts: Role of entanglement molecular weight
,”
Macromolecules
46
,
5026
5035
(
2013
).
47.
Huang
,
Q.
,
N. J.
Alvarez
,
Y.
Matumiya
,
H. K.
Rasmussen
,
H.
Watanabe
, and
O.
Hassager
, “
Extensional rheology of entangled polystyrene solutions suggests importance of nematic interactions
,”
Macro Lett.
2
,
741
744
(
2013
).
48.
Marrucci
,
G.
, and
G.
Ianniruberto
, “
Interchain pressure effect in extensional flows of entangled polymer liquids
,”
Macromolecules
37
,
3934
3942
(
2004
).
49.
Schweizer
,
T.
,
J.
van Meerveld
, and
H. C.
Öttinger
, “
Non-linear shear rheology of polystyrene melt with narrow molecular weight distribution- experiment and theory
,”
J. Rheol.
48
,
1345
1363
(
2004
).
50.
Schieber
,
J. D.
,
D. C.
Venerus
, and
H. L.
Scott
, Development of multi-scale modeling software for entangled soft matter in advanced soldier protection, Technical Report, DTIC Document, 2011.
51.
Andreev
,
M.
, and
J. D.
Schieber
, “
Accessible and quantitative entangled polymer rheology predictions, suitable for complex flow calculations
,”
Macromolecules
48
,
1606
1613
(
2015
).
52.
Katzarova
,
M.
,
L.
Ying
,
M.
Andreev
,
A.
Córdoba
, and
J. D.
Schieber
, “
Analytic slip-link expressions for universal dynamic modulus predictions of linear monodisperse polymer melts
,”
Rheol. Acta
54
,
169
183
(
2015
).