Flow-concentration coupling determines features of nonhomogeneous flow and shear banding in entangled polymer solutions

Rheological models can inform optimal processing conditions and provide a basis to predict the onset of flow instabilities in non-Newtonian fluids. Unstable flows are detrimental to the processing of uniform polymeric materials. Numerous complications are well-documented in flows of entangled polymers above a critical flow rate such as elastic turbulence, die-swell, and the so-called “sharkskin” instability [1]. Some complex fluids, such as wormlike micelles, exhibit a shear banding instability, where the fluid separates into two or more regions of different shear rates under an applied shear flow [2]. Shear banding in wormlike micelles is usually explained to result from a nonmonotonic relationship between shear stress and shear rate [3]. By contrast, since it is generally believed that entangled polymers exhibit monotonic constitutive behavior [4–10], the possibility of shear banding in entangled polymer solutions is hotly debated.

An underexamined aspect of this debate is the nearly universal assumption in experimental studies of flowing polymer solutions that the fluid remains compositionally homogeneous. However, it is well-known that the shear flow can enhance local fluctuations in polymer concentration [11–15]. Recent work has suggested that these fluctuations could lead to major modifications of the transient flow profile in the startup of the Taylor–Couette flow [16]. Furthermore, two-fluid models for entangled polymers with a monotonic constitutive equation predict linearly unstable fluctuations in concentration that result in shear-induced demixing [17–19]. This instability leads to steady-state, banded concentration profiles where each region of distinct concentration coincides with a locally distinct shear rate in the corresponding shear banded flow profile.

Over the last decade, conflicting experimental results concerning shear banding in entangled polymer solutions have been reported [20–23]. Numerous studies reported that shear banding occurs for highly entangled polybutadiene liquids, with the “degree of banding” depending on the polymer molecular weight [24] and viscosity of the solvent [25,26]. Conversely, other studies [23,27] reported no significant departure from homogeneous flows for polymeric solutions with similar numbers of entanglements per chain and at similar applied dimensionless shear rates to those in Refs. [24–26]. When banded velocity profiles were observed [20–26], it generally occurred in the presence of significant wall slip. A “state diagram” was reported to describe the observed empirical interplay between wall slip and shear banding, and a mechanism for shear banding that originates from wall slip was proposed [28]. These results seemed to suggest that wall slip is a necessary condition for shear banding to occur in entangled polymer solutions. The investigators who had seen no evidence of shear banding concluded that the observations of shear banded velocity profiles in entangled polymers were likely a consequence of experimental complications [20,22,23,27,29].

In many cases, the conclusions in this prior body of experiments were limited by a poor spatiotemporal resolution of the flow data, imposing great difficulty in distinguishing banded velocity profiles from curved velocity profiles that result from geometry-induced stress gradients. To address this general ambiguity, a model-free statistical method was recently developed to distinguish between shear thinning and shear banded flow profiles by calculating numerical derivatives of dense, time-averaged, velocimetry data [30]. Importantly, it was shown that the interface width of the flow profile in a shear banding fluid is independent of $Wiapp$⁠,

where $τd$ is the reptation time of the fluid and $γ˙$ is the applied shear rate assuming the homogeneous flow. By contrast, the apparent interface width calculated from the velocity profile of a shear thinning but nonbanding fluid increases with $Wiapp$⁠. Until recently [31], no attempt has been made to rationalize the experimental findings in support and/or against banding with direct comparison to model predictions.

Two-fluid model predictions of coupled, nonlocal concentration variations and shear-banded velocity profiles [17–19,32–34] offer one potential explanation for the conflicting reports [23,35] of the flow profiles of entangled polymer solutions. However, macroscopic variations in concentration do not necessarily imply the existence of shear-banded flow profiles. Shear-induced migration of high molecular weight polymers in dilute solutions occurs across curved streamlines and leads to an approximately linear macroscopic concentration gradient across the fluid [36–39], which would suggest minimal changes to the resulting shear rate profile. Previous two-fluid modeling of dilute, unentangled polymer solutions showed that the extent of shear-induced polymer migration is highly sensitive to the aspect ratio of the Taylor–Couette flow geometry and the viscosity ratio of polymer to solvent [40]. Interestingly, the predicted concentration and velocity profiles of dilute, unentangled polymer solutions were found to resemble shear banding for certain aspects and viscosity ratios due to a migration of polymer molecules across curved streamlines [40]. It is, therefore, crucial for experimental studies of shear banding in entangled polymer solutions to measure both concentration and velocity profiles.

According to the two-fluid model, the magnitude of concentration variation in a flow depends on numerous parameters; chiefly among them is the ratio of elastic stresses to osmotic stresses,

where $G(ϕ)$ is the shear modulus, $χ−1$ is the osmotic susceptibility, and $ϕ$ is the polymer concentration [17–19]. However, this ratio and its associated physics have not been considered in rationalizing the inconsistencies of the experiments that either support or deny the existence of shear banding in entangled polymers. Specifically, differences in E are expected for polymeric liquids of differing chemical composition, since $χ−1$ depends on the polymer and solvent chemistries, and this could provide an explanation for why banding is observed in some solutions and not others despite observing fluids with similar levels of entanglement in the same range of $Wiapp$ [23,35].

Until very recently, the suggestion that flow-induced concentration nonuniformities might be a possible source of shear-banded velocity profiles was an untested theoretical prediction. The first step toward experimental verification was reported in a recent paper [31]. Specifically, using a new rheofluorescence imaging technique, the first experimental evidence was reported for steady-state, macroscopic concentration nonuniformities in a narrow-gap Taylor–Couette flow cell for a 10 wt. % solution of 1,4-polybutadiene (⁠$Mw=9.6×105g/mol$⁠, Ð = 1.08) in dioctyl phthalate (corresponding to $Z=38$ entanglements per chain). These nonuniformities were far larger than anything that could be caused by the combination of shear thinning and migration due to the intrinsic curvature of the Taylor–Couette flow velocity profile, thus confirming the existence of demixing instability, as previously predicted using two-fluid theories by our group [17–19] as well as in earlier works by Fielding and Olmsted [32–34]. The nonuniformities of the steady-state concentration profiles were accompanied by corresponding nonuniformities in the steady-state velocity profiles. Furthermore, after estimating a value of E by fitting the predicted velocity profiles of the two-fluid model to the measured data, the measured concentration profiles (that depend on E) were in good quantitative agreement with the resulting predictions of the two-fluid model [17–19,31].

Under the limited conditions of these initial experiments, however, the velocity profiles were not obviously banded by visual inspection [31], but the profiles were proven to be shear banded by application of the previously developed statistical method [30]. These results showed that the width of the interface between the banded velocity profiles was independent of the applied shear rate. Given the single value of temperature $(T)$ and the number of entanglements per chain $(Z)$ studied, it is not clear whether the flow profiles are sensitive to changes in T or Z, as predicted by the two-fluid model. Another important issue concerns the transient development of the measured velocity profiles, which appeared to reach a steady state long before the measured concentration profiles.

In the present work, we again use entangled polybutadiene (PBD) in dioctyl phthalate (DOP) solutions as a model system and the same experimental procedures that were described in the previous experiments [31]. The propensity for entangled polymer solutions to exhibit nonhomogeneous flows is explored through systematic variations of E (by changing $T$⁠), Z, and the applied dimensionless shear rate (⁠$Wiapp$⁠). The results confirm that the magnitude of nonuniformity in the flow is sensitive to the solvent quality of PBD-DOP solutions as well as the number of entanglements. A higher Z results in greater nonuniformity of the steady state flow, consistent with two-fluid model predictions. Temperature is also found to dictate flow nonuniformity, albeit in a nonmonotonic way.

1,4-trans-Polybutadiene with M_w = 9.6 × 10⁵ g/mol [PBD(1M), Ð = 1.08] was purchased from Polymer Standards Service. 1,4-trans-Polybutadiene with M_w = 4.0 × 10⁶ g/mol [PBD(4M), Ð = unknown] was generously donated by Professor Shi-Qing Wang (University of Akron). DOP (CAS:117-81-7, ≥99.5% purity), anhydrous tetrahydrofuran (THF, CAS: 109-99-9, ≥99.9%), 7-methyl-3-mercaptocoumarin (CAS: 137215-27-1, ≥97.0% purity), and 2,2-dimethoxy-2-phenylacetophenone (DMPA, CAS: 24650-42-8, 99% purity) were purchased from Sigma-Aldrich and used without further purification. Glass tracer particles (∼10 μm in diameter) were donated by TSI, Inc.

Synthesis details for the fluorescent-tagging reaction of PBD-coumarin (PBDC) were described in detail elsewhere [31]. Briefly, 7-methyl-3-mercaptocoumarin was reacted with PBD in anhydrous THF under 365 nm UV light with DMPA functioning as the reaction initiator. Success of the reaction was confirmed with wavelength-specific size-exclusion chromatography experiments.

Entangled PBD-DOP solutions were prepared by adding DOP to PBD solids for a ≈ 10 g sample. The number of entanglements (⁠$Z$⁠) in each solution was estimated by the following relation:

where $Mw$ is the weight-averaged molecular weight of the polymer, $Me$ is the entanglement molecular weight (1650 g/mol for PBD), and $ϕ$ is the concentration of polymer in solution. Polydispersity in the molecular weight of PBD samples affects the precision of Z estimates. In all samples, 0.2 wt. % butylated hydroxytoluene was added as an antioxidant. For rheo-PTV experiments, 300–500 ppm of glass tracer particles were added to the PBD and DOP. For rheofluorescence experiments, 0.2 wt. % of PBDC, synthesized using the same molecular weight of PBD as the rest of the sample, was added to the PBD and DOP. PBD-DOP mixtures were diluted with ≈ 30 mL of toluene to serve as a cosolvent. Samples were placed in an oil bath at $T=50°C$ and gently stirred for several days until they were visually homogeneous. Next, samples were uncapped and left in an oil bath under gentle stirring in the hood to evaporate off the toluene. Once the samples became too viscous for a magnetic stir bar, they were placed in a vacuum oven to remove the residual toluene.

To prepare the PBD-DOP solution with $Z=38$⁠, the required amount of DOP was added to PBD(1M) to yield a 10 wt. % PBD(1M) in a DOP sample. Likewise, to prepare the PBD-DOP solution with $Z=66$⁠, DOP was added to PBD(4M) to achieve a 5 wt. % PBD(4M) in the DOP sample.

All rheo-PTV measurements were performed on a Paar Physica MCR 300 rheometer. Specifics of this custom setup are described in detail elsewhere [16,30,41]. All measurements were performed using a Taylor–Couette flow cell with a gap of 500 μm. A transparent outer cup (⁠$Ro=17.5mm$⁠) and anodized aluminum bob (⁠$Ri=17mm$⁠) permitted flow visualization measurements to be conducted simultaneously with steady shearing. The small gap minimizes the stress gradient across the gap imposed by the geometry (⁠$∼6%$⁠), which correspondingly minimizes the curvature of the velocity profile for fluids with shear thinning rheology. The curvature, $q=(Ro−Ri)/Ri$⁠, of the Taylor–Couette flow cell is 0.029. Glass tracer particles were added to the entangled PBD-DOP solutions and the particle trajectories were tracked between image pairs using a widely known image analysis algorithm [42].

All rheofluorescence measurements were performed on an Anton Paar MCR 702 rheometer. The PBD-DOP solutions with small amounts of PBDC were illuminated with 350 nm UV light under shear to excite the fluorophore, and changes in the fluorescence intensity were recorded with a CCD camera. Spatiotemporal changes in the fluorescence intensity were related to the local effective concentration of the solution by

where $I(r,γ˙,t)$ is the transient measured fluorescence intensity at a particular shear rate and location, $I0(r)$ is the measured fluorescence intensity before shearing, and $Ibg$ is the quiescent image intensity without illumination. Additional details of the custom rheofluorescence instrument are described in further detail in Ref. [31]. All rheofluorescence measurements were performed using the same Taylor–Couette flow cell used for rheo-PTV measurements.

The model used for comparison to the experimental results in this paper is presented in detail elsewhere [17–19]. Here, important equations are included to allow for quick reference by the reader. In the two-fluid formalism, it is assumed that the solvent and the polymer can each be treated as a continuum fluid, with the total velocity vector (⁠$u$⁠) written as the volume-weighted sum of each component,

where the subscripts p and s indicate polymer- and solvent-specific quantities. A momentum balance for each component can be written as

where $ρ$ is the density, $η$ is the viscosity, p is the isotropic pressure, $ζ$ is the frictional coupling between the polymer and the solvent, and $Π$ is the total stress tensor. These equations can be combined to yield

where $ϖ$ is the ratio of the solvent viscosity to the polymer viscosity. The total stress tensor is defined as

The first term on the right-hand side of Eq. (9) represents the stress contribution of the polymer, where the configuration tensor $Q=⟨RR⟩/R02$ relates the second moment of the end-to-end vector to the equilibrium value. $πel$ is the isotropic elastic stress defined as

where $α$ is the concentration scaling exponent of the shear modulus (⁠$α=2.25$⁠), and $π$ is the osmotic stress described by

where $ξ¯0$ is the dimensionless solution correlation length and $H¯$ is the dimensionless gap width of the flow device. $E$ is the elastic to osmotic ratio, which describes the degree of flow-concentration coupling. E is defined as

Here, $G(ϕ)$ is the shear modulus and $χ−1$ is the osmotic susceptibility.

Small angle neutron scattering measurements report solution correlation length values for hydrogenated polybutadiene to be about 4 nm [43]. Thus, an experimentally relevant value of the ratio $ξ¯0/H¯$ is estimated to be approximately $8×10−6$⁠. Computational constraints on the spatial resolution of our two-fluid model calculations caused us to use a larger value of $ξ¯0H¯≥2.5×10−4$ instead (roughly 30 times larger than the experimental estimate). Previous work has shown that the value of $ξ¯0/H¯$ has little effect on the shear-induced demixing instability once the gap width becomes larger than the solution correlation length [19].

The dynamics of the configuration tensor Q is assumed to be described by the Rolie–Poly constitutive equation [44], a single-mode approximation of the GLaMM model [10], for entangled polymeric liquids,

Here, $τd$ is the timescale for reptation, $τR$ is the timescale for Rouse relaxation, and $λ=tr(Q)/3$ represents chain stretch. This form of the Rolie–Poly equation predicts a monotonic shear stress versus shear rate relationship for all polymer concentrations and number of entanglements $(3Z=τd/τR)$⁠. Therefore, the model does not predict steady-state shear banding in the absence of flow-concentration coupling.

In neglecting contributions from inertia in Eq. (2), it can be shown that a difference in species velocities is driven by a divergence in stress as

Additionally, in the absence of flow-concentration coupling (i.e., $E=0$⁠), the right-hand side of Eq. (10) is equal to zero, meaning there is no difference between species velocities. Conversely, there will be differences in the species velocities when a divergence in stress occurs for any non-zero value of E. The concentration evolution is described by

where $ϕ¯$ is the polymer concentration normalized by the initial, uniform polymer concentration.

The two-fluid model predicts steady-state flow and concentration profiles that depend on the degree of flow-concentration coupling, described by the E parameter. As shown in Fig. 1(a), the magnitude of the concentration difference across the fluid and the severity of the resulting nonhomogeneous flow increase with E. Furthermore, as $E$ is increased, the flow profiles appear to be more convincingly shear banded. When $E=0.05$⁠, there is a near linear change in concentration from the inner wall (⁠$r/H=0.0$⁠) to the outer wall (⁠$r/H=1.0$⁠), and no clear distinction between high and low concentration bands [Fig. 1(a)]. The velocity profile for $E=0.05$ [Fig. 1(b)] shows increased curvature compared to the homogenous concentration case (⁠$E=0$⁠), but likewise does not appear banded. However, as E is increased, the concentration and velocity profiles show increasingly definitive visual evidence of shear banding. When the concentration profiles are banded, there is a spatial variation in concentration within each band rather than constant concentration as predicted for the linear shear flow by Cromer et al. [17,18]. The spatial variation of concentration in each band results from the flow geometry curvature and resulting polymer migration across curved streamlines.

Poor spatial resolution of the flow profiles in previous experimental studies of shear banding in entangled polymer solutions has limited robust characterization of the measured flows to facilitate comparison with theory. In particular, the two-fluid model with flow-concentration coupling makes specific predictions of the nature of the flow profiles as $Wiapp$ is increased (Fig. 2). Namely, the interface location between the high and low shear bands is found to propagate from the shearing boundary into the bulk of the fluid as $Wiapp$ increases in a linear fashion [Fig. 2(a)]. Although a similar propagation of the interface location across the fluid (and constant interface width) is also predicted using a “minimal model” that utilizes a scalar nonmonotonic constitutive relationship, this “minimal model” does not account for the important effects of concentration coupling and normal stresses [45]. The change of slopes for interface location versus $Wiapp$ can be rationalized by reference to the $E$-dependence of linear stability boundaries for shear banding [Fig. 2(a)]. For $E=0.10$⁠, a linear stability is not predicted and changes in concentration result from a streamline curvature. The predicted low and high linear instability boundaries for $E=0.15$ are $Wi=3.3$ and $Wi=9$⁠, respectively. The predicted low and high linear instability boundaries for $E=0.20$ are $Wi=2.4$ and $Wi=11.5$⁠, respectively. Hence, shear banding is predicted to result from shear-induced demixing when the applied $Wi$ lies between the upper and lower $Wi$ boundaries of linear instability for each E. Additionally, the width of the interface separating the two shear bands is approximately constant as $Wiapp$ is varied and decreases in magnitude (i.e., the interface sharpens) as the strength of flow-concentration coupling (i.e., E) is increased [Fig. 2(b)].

In flows with a shear stress gradient, such as rotating parallel plates and the Taylor–Couette flow, there is ambiguity in distinguishing truly shear banded flow profiles from shear thinning flow profiles. A previously established approach using numerical derivatives of spatially dense velocimetry data was used to discern between shear thinning and shear banding conditions [30]. One important criterion established by Cheng et al. [30] for identifying flow profiles as shear banded is a constant interface width as a function of $Wiapp$⁠. Using this metric, the flows predicted by the two-fluid model are shear banded at sufficiently high levels of flow-concentration coupling [Fig. 2(b)].

The two-fluid model predicts a flow-induced demixing instability that leads to macroscopic changes to the concentration profile across the fluid [Fig. 3(a)]. Initial changes to the profile are found to develop at the boundaries, and in time, significant concentration heterogeneities arise throughout the bulk of the fluid [Fig. 3(a)]. Simultaneously, notable changes to the velocity profile are detected as the flow-induced concentration heterogeneities develop and coarsen under an applied shear flow. As illustrated in Fig. 3(b), the velocity profile is linear at early times following the onset of shear, coinciding with a uniform concentration profile [Fig. 3(a)]. In time, heterogeneities develop in the concentration profile and lead to minor changes to the local shear rates in the velocity profile. At later times, these concentration heterogeneities coarsen to form larger “bands” that differ appreciably in concentration; likewise, the velocity profile develops large regions of locally distinct shear rates due to the corresponding differences in fluid viscosity. Finally, at steady state, the concentration profile develops two bands of distinct concentration that coincide with a shear banded velocity profile, with the low concentration (low viscosity) band resulting in a high shear rate band and the high concentration (high viscosity) band resulting in a low shear rate band.

The predicted time for significant heterogeneities to develop with infinitesimal perturbations of the concentration profile is very long (⁠$106τd$⁠), but once developed, the concentration profiles reach steady state within $100τd$⁠. Previous work has shown how perturbations to the initial concentration profile of varying wavelength and magnitude influence the transient flow profile evolution but ultimately result in same steady-state profiles [17–19]. Thus, an experimental system with small but finite perturbations to the initial polymer concentration profile could be expected to reach a steady-state banded flow profile on much shorter time scales O(⁠$102τd$⁠) than predicted by theory O(⁠$108τd$⁠). Fluids become more susceptible to a shear-induced demixing instability as either E or Z is increased. Transiently, this increased susceptibility for shear-induced demixing results in a faster onset and coarsening of concentration heterogeneities that are greater in magnitude.

Small amplitude oscillatory shear frequency sweeps were performed in the linear viscoelastic regime to quantity the relaxation times of the two entangled PBD-DOP solutions (⁠$Z=38$ and $Z=66$⁠) at varying temperatures (Fig. 4). As expected, the longest relaxation time (⁠$τd$⁠) is found to decrease with increasing temperature (Fig. 4 insets) due to the influence of favorable polymer-solvent interactions. Furthermore, the plateau modulus is higher in the PBD-DOP solution with $Z=38$ [Fig. 4(a)] than the solution with $Z=66$ [Fig. 4(b)] due to the higher polymer concentration necessary to achieve the targeted number of entanglements for the lower molecular weight PBD. This behavior is expected since the plateau modulus (⁠$GN0$⁠) depends on the number density of entanglement strands as

where v is the number density of entanglement strands, $kB$ is the Boltzmann constant, and T is the temperature.

The PBD-DOP solutions exhibit the expected small amplitude oscillatory shear rheological behavior for entangled polymer solutions for both $Z=38$ and $66$ at all temperatures investigated, as indicated by an observed plateau modulus at high frequencies, terminal flow scaling of $G′∼ω2$ and $G′′∼ω1$ at low frequencies, and a crossover of $G′$ and $G′′$ at an intermediate frequency. Importantly, the fluids do not depart from this expected rheological behavior as the temperature is decreased toward the phase boundary for PBD-DOP [with an upper critical solution temperature, $TUCST≈13°C$] [46]. Thus, the PBD-DOP solutions serve as an ideal model fluid to explore the effects of flow-concentration coupling on the resulting measured flows. One specific postulate is that changes in the temperature of the fluid will result in changes of the solvent quality (as accounted for in $χ−1$⁠) and, therefore, impact the strength of flow-concentration coupling (i.e., E decreases as the temperature is increased or E increases as the temperature is decreased). No enhanced scattering was observed as $TUCST$ was approached. This lack of observable enhanced scattering is attributed to the small refractive index difference between PBD and DOP. Other systems, such as polystyrene and DOP, do indeed show noticeable turbidity (a signature of local concentration fluctuations) as the $TUCST$ is approached [16].

Time-averaged, steady-state velocity profiles were measured to investigate the effect of changes in $Wiapp$ and E (due to changes in temperature) on the observed flow profiles. In our previous study, shear-banded velocity profiles were observed in a PBD-DOP solution with $Z=38$ for the particular case of $T=50°C$ [31], but the temperature sensitivity of bulk flow behavior was not investigated. According to two-fluid model predictions, the bulk flow behavior (and propensity for shear banding) should be dependent on the solution temperature, as implicitly included in E. The resulting flows for the PBD-DOP solution with $Z=38$ at $T=16°C$ are shown in Fig. 5. As can be seen in Figs. 5(a) and 5(b), curvature in the steady-state velocity profiles becomes increasingly pronounced with increasing $Wiapp$⁠. For $Wiapp≥3.2$⁠, the profiles exhibit two distinct regions of different shear rates. At $T=21°C$ [Figs. 5(c) and 5(d)], the steady-state velocity profiles exhibit qualitatively similar behavior with increasing $Wiapp$⁠; however, the distinction between regions of different shear rates at high $Wiapp$ [Fig. 5(d)] is significantly less pronounced. The qualitative differences in the curvature between the high $Wiapp$ profiles of Figs. 5(b) and 5(d) indicate a dependence of the flow kinematics on temperature aside from just a change of relaxation time (as accounted for in $Wiapp$⁠). Further increasing the temperature to $T=50°C$ yields similarly shaped velocity profiles, with enhanced curvature for $Wiapp>1$⁠. Interestingly, the curvature of velocity profiles at $T=50°C$ appears more pronounced than $T=21°C$⁠, suggesting that temperature has a nonmonotonic influence on the flow profile curvature.

Steady-state velocity profiles with significant curvature are also observed in the more entangled PBD-DOP solution with $Z=66$ (Fig. 6). For $T=16°C$ [Figs. 6(a) and 6(b)], the measured flow profiles undergo several transformations with increasing $Wiapp$⁠. At $Wiapp=4.8$⁠, the flow profile is curved with a high shear rate region near the inner wall (⁠$r/H=0.0$⁠) and a low shear rate region adjacent to the outer wall (⁠$r/H=1.0$⁠). The curvature is less apparent as $Wiapp$ is increased to 7.1, with slip occurring at the outer wall. As $Wiapp$ is further increased to 9.8, the flow profile appears to develop three regions of distinct shear rates. A high shear rate region exists between $r/H=0.1$ and $0.3$ and a low shear rate region manifests from $r/H=0.3to1.0$⁠. It seems that a second low shear rate region occurs adjacent to the inner wall (⁠$r/H=0.0to1.0$⁠). At this time, the cause of this “three-banded” profile is still unclear. The flow behavior adjacent to the boundary surface could be governed by wall slip, but it is generally assumed that wall slip is an interfacial phenomenon, confined to a distance on the order of the radius of gyration of the polymer [47,48]. The low shear rate region adjacent to the inner wall for $Wiapp=9.8$ is $50–100μm$ thick and is, thus, orders of magnitude greater than the anticipated length scale of slip. Furthermore, three-banded profiles due to flow-concentration coupling are predicted in the low curvature limit for the TC flow [18]. With increasing $Wiapp$⁠, the velocity profile remains banded, and the interface separating the bands of different shear rates moves toward the outer wall [Fig. 6(b)]. It can be seen that for $Wiapp=39$⁠, the velocity profile has a high shear rate band spanning $r/H=0.0to0.8$ and a low shear rate band from $r/H=0.8to1.0$⁠. At $Wiapp=47$⁠, the velocity profile remains curved, but distinct bands are not apparent.

Similar trends with increasing $Wiapp$ are observed for $T=21°C$ [Figs. 6(c) and 6(d)]. Significant curvature in the velocity profiles is observed for $Wiapp≥4.4$⁠. The interface between the regions of different shear rates is found to move toward the outer wall with increasing $Wiapp$⁠. Again, as was observed for $Z=38$⁠, the velocity profiles do not appear as sharply banded for $T=21°C$ compared to $T=16°C$⁠.

For $T=50°C$⁠, the velocity profiles of the PBD-DOP solution with $Z=66$ develop shear bands for $Wiapp>1$ [Figs. 6(e) and 6(f)]. At $Wiapp=9.4$⁠, the high shear rate band has a lower slope than $Wiapp=12$ and the location of the interface separating the two bands appears to be further from the inner wall (⁠$r/H=0.0$⁠). At $Wiapp=35$⁠, the velocity profile is most significantly banded with a high shear-rate region that spans $r/H=0.0to0.2$⁠, and a low shear-rate region with a slope that is commensurate with the lower $Wiapp$ profiles. It is important to note that wall slip is present at both the inner and outer surfaces.

The steady-state velocity profiles measured for varying Z and T indicate the importance of the number of entanglements and solvent quality on the bulk flow of entangled polymer solutions (Figs. 5 and 6). From visual inspection, the measured velocity profiles have the greatest spatial variation of the local shear rate at the lowest T studied. This temperature sensitivity is consistent with the speculation that flow-concentration coupling is strongest close to the $TUCST$⁠.

The time-averaged steady-state velocity profiles for various values of $Wiapp$⁠, Z, and T were compared with the two-fluid R-P model predictions. Figures 7(a) and 7(b) directly compare the measured velocity profiles at particular values of $Wiapp$⁠, Z, and T to the corresponding two-fluid R-P model predictions for varying values of E. As indicated in the Theory section, E represents the balance of elastic stresses, which store stress under the flow, to osmotic stresses, which act to homogenize gradients in fluid composition. Therefore, $E=0$ corresponds to the case where the polymer concentration remains homogeneous across the fluid due to the absence of flow-concentration coupling. As seen in both Figs. 7(a) and 7(b), the $E=0$ prediction poorly captures the local shear rate variation across the PBD-DOP solution that is measured experimentally. With the increasing magnitude of E, an agreement is improved between the measured velocity profiles and the two-fluid R-P model predictions.

There is excellent agreement between the measured velocity profile and two-fluid R-P model prediction with $E=0.14$ for the fluid with $Z=38$ [Fig. 7(a)]. For $Z=66$ [Fig. 7(b)], there is a disagreement between the model predictions and the measured velocity profile from $r/H=0.0to0.2$ for all E. Notably, there is a significant slip occurring at the inner wall (⁠$r/H=0.0$⁠). In the region adjacent to the inner wall (⁠$r/H<0.1$⁠), the measured velocity is smaller than that predicted by the two-fluid R-P model. Additionally, between $r/H=0.15and0.25$⁠, the measured velocity exceeds the model predictions and appears to broaden the “sharpness” of the interface that is predicted by the model. For $r/H>0.25$⁠, the model predictions for $E=0.18$ agree very strongly with the measured values.

We define the best fit of the model to the experimental data as achieved when the following objective function is minimized:

where $vm(r)$ is the model predicted velocity and $vi(r)$ is the experimentally measured velocity. The results of this optimization are included in Fig. 7(c). The best fit of the model predictions to experimentally measured profiles occurs when $E=0.14$ and $E=0.18$ for $Z=38$ and $Z=66$⁠, respectively.

The values of E determined from the best-fit criteria at each Z were found to be independent of $Wiapp$ across the range studied. This is illustrated in Fig. 8(a) for $Z=66$⁠. At $Wiapp=4.7and9.8$⁠, the disagreements between the measured velocities and model predictions persist in the region adjacent to the inner wall. At $Wiapp=39$⁠, there is excellent agreement between the model predictions and measured values across the entire flow domain. The E values determined from optimization at each $Z$and T are shown in Fig. 8(b). E is higher for $Z=66$ PBD-DOP solution than the $Z=38$ solution at all T. The highest E value is found at $T=16°C$ for both Z. This intuitively makes sense as PBD-DOP is known to exhibit temperature-dependent phase behavior, with an upper critical solution temperature $TUCST≈13°C$ [46]. As a phase boundary is approached, the solution osmotic susceptibility is expected to tend toward zero, meaning that E would approach infinity. Interestingly, the change in E with increasing T is nonmonotonic for both Z values. The nonmonotonicity could stem from different $T$-dependencies of the elastic modulus and the osmotic susceptibility.

For nonphase separating viscoelastic fluids, the shear modulus is expected to increase linearly with temperature [49]. However, as the phase boundary is approached in an entangled polymer solution, phase separation of polymer and solvent is expected to disrupt the entangled network structure, resulting in a strong decrease in the shear modulus. A nonmonotonic dependence of shear modulus on temperature in the vicinity of a phase boundary has been previously reported for a PBD-DOP polymer solution [46], underscoring the interplay of rheological properties and polymer solution phase behavior. Additionally, the osmotic susceptibility of polymer solutions is expected to decrease to zero at the phase boundary. The slope of osmotic susceptibility versus temperature has been shown to increase as the critical point is approached [50]. Taken together, we believe that nonmonotonic variations of E (of up to $∼50%$⁠) with T are not surprising, though this will depend on the chemistry of the polymer solution. The agreement between the model predictions and measured velocity profiles for nonzero values of E is strongly suggestive of flow-concentration coupling but is not direct proof of the existence of concentration variations.

The apparent polymer concentration profiles measured using rheofluorescence microscopy for a particular $Wiapp$ at different temperatures are shown in Fig. 9. The most significant changes in concentration are measured for $T=16°C$⁠, which corresponds to the lowest temperature and the highest degree of flow-concentration coupling as determined from $E$-fits to the measured velocity profiles [Fig. 9(a)]. The variations in concentration across the flow device as predicted by the two-fluid R-P model underestimate the magnitude of the concentration changes measured by rheofluorescence. The smallest variation in concentration is observed for $T=21°C$ [Fig. 9(b)], consistent with the estimates of E with temperature [Fig. 8(b)]. At $T=50°C$ [Fig. 9(c)], measured concentration changes are larger than $T=21°C$ but less than $T=16°C$⁠. The agreement between the experiment and model-predicted concentration profiles is quite strong at $T=50°C$⁠, where both the magnitude of concentration variation and interface location are captured by the model. The measured $T$-dependence of the concentration variations in Fig. 9 gives further support to the nonmonotonicity of flow-concentration coupling (⁠$E$⁠) as quantified in Fig. 8(b). As mentioned previously, this nonmonotonicity likely results from a complex $T$-dependence of $χ−1(T)$ [46,50].

Slight discrepancies are observed when comparing the concentration profiles determined by rheofluorescence to the model predictions for the $Z=66$ PBD-DOP solution (Fig. 10). Although the magnitude of concentration variations measured via rheofluorescence and predicted by the two-fluid R-P model are in agreement, the spatial variation disagrees in some regions. Perhaps, the most striking is the disagreement in the high shear rate region (predicted low concentration) from $r/H=0.00$ to about $0.15$ in Fig. 10(a). The measured concentration is much higher in this region compared to the two-fluid R-P model prediction. As discussed previously, there is also a disparity in the measured and predicted velocity profiles in this region.

A higher concentration, as measured by rheofluorescence in Fig. 10(a), would give rise to a lower shear rate as measured in Figs. 7(b) and 8(a) due to an increase in fluid viscosity. Additionally, the measured concentration is less than the model predictions for $r/H=0.2to0.4$⁠. This discrepancy could potentially explain the “broadening” of the interface observed in Fig. 7(b), where the measured velocity gradient is lower than model predictions for $r/H=0.00to0.15$ and higher than model predictions for $r/H=0.20to0.40$⁠. A stronger agreement is found between the rheofluorescence measurements and model predictions for $T=50°C$ [Fig. 10(b)]. The only discrepancies between the measured and predicted concentration values exist near the boundaries (⁠$r/H=0.00and1.00$⁠). At $Wiapp=39$⁠, the interface location has moved to $r/H=0.90$ as reflected in both measured and model predicted concentration profiles [Fig. 10(c)].

Wall slip is known to occur in entangled polymer solutions, and the magnitude of slip depends on many variables, including polymer chemistry, solvent, surface chemistry, concentration, and the number of entanglements [51]. Thus, it is important to quantify the magnitude of slip present in flows of entangled polymer solutions to understand whether the presence of slip impacts the measured nonuniform flows. For example, slip at solid boundaries could influence compositional heterogeneity in the bulk fluid and potentially give rise to three-banded flow profiles observed for PBD-DOP solutions with $Z=66$⁠.

Slip velocities at the boundaries for the $Z=38$ PBD-DOP solution for varying $Wiapp$ are shown for the inner and outer surfaces in Figs. 11(a) and 11(b), respectively. The slip velocities at the inner (⁠$vi,slip$⁠) and outer (⁠$vo,slip$⁠) walls were determined according to

For all temperatures investigated, the resulting slip velocities at the inner surface remain effectively negligible [Fig. 11(a)]. Likewise, in Fig. 11(b), the outer slip velocities are effectively zero across the range of $Wiapp$ investigated. Thus, it can be concluded that the curvature of the steady-state velocity profiles in Fig. 5 results from the bulk behavior of the fluid and is not significantly influenced by the presence of wall slip for the case where $Z=38$⁠.

Unlike the $Z=38$ fluid, appreciable wall slip is observed at both bounding surfaces for the $Z=66$ PBD-DOP solution (Fig. 12). This finding is consistent with other reports of an increase in the slip as the polymer molecular weight increases [51,52]. Although higher $Wiapp$ were explored for the PBD-DOP fluid with $Z=66$ compared to $Z=38$⁠, it is clear that wall slip at the inner [Fig. 12(a)] and outer [Fig. 12(b)] surfaces increases with $Wiapp$⁠. Additionally, higher temperatures are found to result in a more prominent slip at both the inner and outer walls. This temperature dependence of wall slip can be rationalized in the context of a Navier slip condition, where the magnitude of slip is proportional to the shear stress, as higher stresses are measured in the $Z=66$ fluid at higher temperatures for similar $Wiapp$⁠. Reliable measurements of the $Z=38$ solution for $Wiapp>10$ were infeasible because of rod climbing and edge fracture complications due to the higher plateau modulus of the $Z=38$ fluid [Fig. 4(a)].

The nonuniform, steady-state velocity profiles (Figs. 5 and 6) have been shown to coincide with spatial changes in concentration (Figs. 9 and 10) as predicted by a two-fluid model for entangled polymer solutions [19]. However, from visual observation, it is unclear whether these flows are shear banded or merely curved due to the combination of shear thinning rheological behavior and the shear stress gradient of a small curvature Taylor–Couette flow cell. To distinguish between these two possible flow behaviors, the statistical method mentioned previously was employed to calculate numerical derivatives of the high spatial density velocity profiles [30]. Third derivatives of the velocimetry data were used to determine both the interface locations and widths across a range of $Wiapp$ for $Z=38$ (Fig. 13) and $Z=66$ (Fig. 14) PBD-DOP solutions.

The location of the interface in the flow profiles is found to move from the inner wall (⁠$r/H=0.0$⁠) to the outer wall (⁠$r/H=1.0$⁠) as $Wiapp$ is increased above $Wiapp≈2$ [Fig. 13(a)], in qualitative agreement with two-fluid model predictions (Fig. 2). The determined interface width for the $Z=38$ PBD-DOP solution is nearly constant at $w/H=0.40$ across the examined $Wiapp$ range for $T=16$ and $50°C$ [Fig. 13(b)]. The interface width for these two cases is similar to other shear banding fluids [41,53–57] and is consistent with the value of a shear banding wormlike micellar solution determined from the same method [30]. The broadness of this interface width compared to the two-fluid model predictions in Fig. 2(b) could be partly due to a wavelike instability of the interface [58,59] or due to instabilities in the high shear band resulting in a time varying interface location as predicted by the criterion proposed by Fardin et al. [60]. Such an instability of the interface cannot be predicted by the 1D form of the two-fluid model [19], and predicting whether it occurs using the previously proposed phenomenological criterion is complicated by the fact that the polymer concentration and, therefore, the relaxation time of the fluid (and thus the local value of $Wi$⁠) are spatially nonuniform. Conversely, the interface width is found to increase with $Wiapp$ when $T=21°C$⁠. Thus, it is concluded that whether shear banding occurs in entangled polymer solutions depends critically on the value of E.

Based on the changes of interface width with $Wiapp$⁠, it is concluded that $E=0.08$ (⁠$T=21°C$⁠) is an insufficient degree of flow-concentration coupling to induce shear-banded flows despite a measured macroscopic variation in concentration across the fluid [Fig. 9(b)]. Here, the measured concentration variation appears linear rather than banded. Linear concentration variations have previously been found to occur in dilute polymer solutions due to polymer migration across curved streamlines [36,37]. In this work, differences in the qualitative shape of the measured concentration profiles for different values of E are consistent with the predictions of the two-fluid model. As such, the different shapes of concentration profiles (i.e., linear versus banded) are explained by the relative strength of flow-concentration coupling, where low flow-concentration coupling results in effective shear thinning rather than shear banding.

In $Z=66$ fluid, the apparent interface location moves into the bulk of the fluid with different sensitivities to $Wiapp$ depending on the temperature [Fig. 14(a)]. At $T=16°C$⁠, the interface location moves from $r∗/H=0.20$ at the lowest $Wiapp$ to $r∗/H=0.80$ at the highest $Wiapp$ in a roughly linear fashion. At higher temperatures, the interface location is found to increase only partially into the bulk of the fluid, with an apparent plateau at sufficiently large $Wiapp$⁠.

The computed interface widths for the $Z=66$ fluid are fairly constant across the range of $Wiapp$ [Fig. 14(b)]. The $w/H$ values for the $Z=66$ fluid at $T=16°C$ are the lowest of any temperature examined, ranging from $w/H=0.20to0.35$⁠. These values of $w/H$ are similar to the interface widths determined using this same statistical method for a shear banding wormlike micellar solution [30]. The calculated interface widths for $T=21°C$and $T=50°C$ range from $w/H=0.40to0.50$ and $w/H=0.30to0.40$⁠, respectively.

Previous explanations for the occurrence of shear banded flows in entangled polymer solutions have asserted that the nonhomogeneous flow arises from disentanglements of the entangled polymer chains that initiate at the solid boundary in the form of wall slip and propagate into the bulk of the fluid [35,61]. In the following, it is shown that the wall slip is not necessary to observe a nonhomogeneous flow.

Wall slip can occur transiently during the startup of the shear flow for entangled polymer solutions without resulting in a nonhomogeneous flow. The transient shear stress measured for shearing the $Z=38$ PBD-DOP solution follows the expected behavior for nonlinear shear of an entangled polymer solution as evidenced by a large shear stress overshoot at early times, followed by a plateau to steady state at long times [Fig. 15(a)]. As shown in Fig. 15(b), the wall slip occurs at the moving boundary (⁠$r/H=0.0$⁠) shortly after the startup of the shear flow but subsides by the time the velocity profile achieves a steady state [Fig. 15(c)]. At the steady state, the velocity profile is nearly linear, which shows that the wall slip observed at early times [Fig. 15(b)] does not result in a nonhomogeneous flow.

Under different flow conditions, the $Z=38$ PBD-DOP fluid exhibits simultaneous transient wall slip and the development of a nonhomogeneous flow (Fig. 16). Like the flow in Fig. 15, the wall slip at the moving boundary is observed at early times [Fig. 16(b)] but not at longer times [Fig. 16(c)]. However, unlike the flow measured in Fig. 15, the flow appears nonhomogeneous after the wall slip occurs (Fig. 16). Following the shear stress overshoot, the flow profiles appear to have two regions of different shear rates beginning at $t/τd=1.6$ [Fig. 16(b)]. At steady state [Fig. 16(c)], the fluid experiences a high shear rate region between $r/H=0.0and0.2$ and a low shear rate region from $r/H=0.2to1.0$⁠.

The occurrence of the wall slip is unnecessary for the development of nonhomogeneous flows. The measured shear stress at $Wiapp=5$ and $T=50°C$ [Fig. 17(a)] exhibits a smaller overshoot relative to lower temperatures [Figs. 15(a) and 16(a)]. Interestingly, the velocity profiles in Figs. 17(b) and 17(c) show no clear evidence of wall slip at the moving boundary at all times. Despite the lack of wall slip, nonhomogeneous flows with pronounced curvature develop shortly after the start of the shear flow [Fig. 17(b)]. This curvature persists even at long times, where the low shear band (from $r/H=0.4to1.0$⁠) remains constant, whereas the high shear band exhibits some scatter before reaching a steady shape by $t/τd=21.3$⁠. Ultimately, the flow profiles for the $Z=38$ PBD-DOP solution show a relatively quick development of the nonhomogeneous flow, without clear changes such as “stair-stepped” velocity profiles resulting from a coarsening of concentration heterogeneities as shown in Fig. 3. As discussed in Burroughs et al. [31], the nonhomogeneous flows observed at short times relative to the formation of concentration heterogeneity could result from “transient” shear banding in the startup of shear as suggested by others [62,63]. Moorcroft et al. identified a universal criterion for the onset of transient shear banding in shear startup experiments in which it commences in the vicinity of the shear stress overshoot [64,65]. Here (Figs. 15–17), nonhomogeneous velocity profiles and/or wall slip are observed to occur following the shear stress overshoot, consistent with the criterion for transient shear banding [64,65], and also with previous molecular dynamics simulations [66–69] and experiments [6,57,61,70,71].

To quantify the flow kinematics of entangled PBD-DOP solutions from shear startup to steady state, rheo-PTV measurements were performed to allow for simultaneous determination of rheology and velocity profiles. Figures 18–20 show the evolution of the shear stress and measured velocity profiles under shearing at different temperatures and $Wiapp$ for the PBD-DOP solution with $Z=66$⁠.

Upon the startup of the steady shear flow, the shear stress initially increases in time up to a maximum of $≈0.3τd$ [Fig. 18(a)], during which the velocity profile [Fig. 18(b)] is linear (uniform shear rate) with some wall slip at the inner wall (⁠$r/H=0.0$⁠). Following a maximum at early times, the shear stress first decreases rapidly before decreasing at a slower rate beginning around $2τd$⁠. At the start of this more gradual decrease in shear stress, the velocity profile remains linear with additional slip at the outer wall (⁠$r/H=1.0$⁠). At later times, the velocity profile begins to show a departure from the uniform shear rate. As shown in the velocity profiles at $6.8τd$ and $13τd$⁠, a curvature in the velocity profile becomes significant, presumably due to the onset of shear thinning following the initial elastic response of the fluid to shear startup. The curvature of the velocity profile reduces slightly from 13 to 50.0 $τd$⁠, as expected from model predictions. As the shear stress approaches a steady state, a high shear rate region develops adjacent to the inner wall (⁠$57τd$⁠) and propagates into the bulk of the fluid (⁠$86τd$⁠) [Fig. 18(c)]. This progression of the nonhomogeneous flow is consistent with the predictions of the two-fluid model at long time scales [Fig. 3(a)]. At the steady state, the measured velocity profile appears banded with a high shear rate region adjacent to the inner wall and a low shear rate region adjacent to the outer wall. This banded profile is notably distinct from the curved profiles measured at earlier times (⁠$13τd$ and $50.0τd$⁠) as evidenced by a sharp transition between the bands of different shear rates at $r/H≈0.4$ [Fig. 18(c)].

The measured rheology and flow profiles exhibit qualitative differences at a higher temperature, as shown in Fig. 19. First, the ratio of the maximum in shear stress relative to the steady-state shear stress is clearly lower in Fig. 19(a) than in Fig. 18(a). In addition, while the initial evolution of the shear stress is similar to Fig. 18(a), with an overshoot at early times, there is a slight increase in the shear stress after $50τd$ [Fig. 19(a)]. This increase in the shear stress at long times coincides with detectable changes to the velocity profile. As shown in Fig. 19(b), the velocity profile develops increasing curvature and slip at the inner wall during shear startup, as the stress increases and goes through an overshoot. At $1.1τd$⁠, the curvature of the velocity profile is most significant following the decrease in shear stress after the overshoot. At $15τd$⁠, the velocity profile becomes “stair-stepped,” where multiple regions of different shear rates are observed [Fig. 19(b)]. An attempt to quantify the shear rate within each region was not made due to the low spatial resolution at a single time point. From $23τd$ to $90τd$ [Fig. 19(c)], the regions of differing shear rates appear to merge, forming two distinct regions of different shear rates at steady state (⁠$90τd$⁠).

A combination of wall slip at both surfaces and a “stair-stepped” transient velocity profile is also observed under the flow conditions in Fig. 20. Slip becomes increasingly prominent at the inner wall during the shear stress increase to a maximum [Fig. 20(b)] as with Fig. 19. Following the overshoot in shear stress, the wall slip at the inner wall is reduced and the velocity profile exhibits pronounced curvature for a long time [$28τd$⁠, Fig. 20(c)]. At long times, this pronounced curvature subsides with the velocity profile seemingly returning to a linear form [$69τd$⁠, Fig. 20(c)]. A “stair-stepped” profile then develops (⁠$82τd$⁠) and coarsens into a two-banded profile at steady state (⁠$180τd$⁠). This coarsening of “stair-stepped” velocity profiles at intermediate times to two-banded profiles at steady state matches the transient shear-induced demixing predictions of the two-fluid model (Fig. 3).

The interplay of flow-concentration coupling and its potential to cause shear banding in entangled polymer solutions was investigated through simultaneous measurements of the velocity and concentration profiles of entangled PBD-DOP solutions and compared to the predictions of a two-fluid Rolie–Poly model. Polymer solutions under flow do not necessarily remain homogeneous, depending on both the flow strength and the magnitude of E, which characterizes the ratio of elastic to osmotic stresses and is sensitive to the polymer-solvent interaction (varied here through changes in temperature). When the degree of flow-concentration coupling is low (small $E$⁠), banded velocity profiles are not observed, and a linear variation in polymer concentration is measured. A measured linear variation in concentration has been explained for dilute solutions by polymer migration across curved streamlines [37], but the results in this work still agree favorably with predictions of the two-fluid model for particular values of E [17–19]. Conversely, for large values of E, we show that flow-concentration coupling leads to a shear-induced demixing instability, which results in banded steady-state profiles for both polymer concentration and shear rate. Strong wall slip is observed at the inner and outer boundaries at high levels of entanglement and for large $Wiapp$⁠, but the presence of wall slip is found to be unneccessary for shear banded flows. Measurements of the transient velocity profile evolution reveal evidence of a “stair-stepped” flow profile that coarsens into two bands, in agreement with the two-fluid model predictions of shear-induced demixing. Importantly, this work provides evidence for a mechanism that explains the development of steady-state shear banding in entangled polymer solutions depending on the polymer-solvent chemistry. Future experiments should further investigate the role of polymer-solvent chemistry by screening a variety of polymer solutions bearing different phase behaviors to quantify the extent of flow-concentration coupling in rheological measurements. Future rheological models should account for solution thermodynamics and the potential for flow-induced nonhomogeneous polymer concentration profiles in describing the flows of complex fluids.

The authors have no conflicts to disclose.