Consistent modeling of nonlinear shear and elongational start-up data of entangled polystyrene solutions

The tube model of Doi and Edwards (1978, 1979) has been a decisive step in mesoscopic molecular modeling of the nonlinear rheology of polymer melts and solutions. While being nearly quantitative with the stress relaxation data after nonlinear step-shear strains, the tube model of Doi-Edwards (DE) does not predict overshoot of the first normal stress difference in shear nor does it take into account chain stretch in elongational flow resulting in elongational viscosity thinning exponent of − 1 versus approximately − 0.5, as experimentally observed for polymer melts. Later refinements of the tube model reviewed, e.g., by Narimissa and Wagner (2019), led to improved mesoscopic modeling of the rheology of entangled polymer systems by invoking additional molecular processes requiring occasionally a multitude of nonlinear fitting parameters. In this contribution, we show that the nonlinear start-up data of shear viscosity and apparent first normal stress difference as well as elongational viscosity of three polymer solutions as reported by Cui et al. (2025) can be described by a tube-idea inspired model based exclusively on the linear-viscoelastic characterization of the solutions and a consistent set of Rouse stretch relaxation times. This work confirms earlier analysis of the shear and elongational rheology of three-arm star polymers and corresponding blends of long and short polymers revealing the solution effect of the short arm in asymmetric stars (Wagner et al. 2022, 2024) and extends this work to the shear and elongational rheology of entangled polymer solutions consisting of the same weight fraction of a linear long-chain polystyrene and three different styrene oligomeric solvents.

The three entangled polystyrene (PS) solutions labeled as 600 k-4 k-50%, 600 k-10 k-50%, and 600 k-8a4k-50%, share the same linear long chain polymer (PS-600 k) but use different styrene oligomeric solvents (OS-4 k, OS-10 k, and OS-8a4k), with the weight fraction of PS-600 k fixed at 50%. We note that neglecting the possible influence of end groups, the solutions are athermal, i.e., on the scale of the monomer groups, the polymer–solvent interaction is the same as the polymer–polymer interaction. The molecular weight, dispersity, and glass transition temperatures of the solutions and their components are summarized in Table 1. More detailed information on preparing the solutions is reported in Cui et al. (2025).

The linear viscoelastic (LVE) properties were characterized using the ARES-G2 rheometer (TA Instruments, USA) through small amplitude oscillatory shear (SAOS). Frequency sweeps were performed across a temperature range of 105–190 °C, using parallel plates with either 8 mm or 4 mm diameter depending on the magnitude of the shear modulus. The LVE master curves at specific temperatures were subsequently constructed using time–temperature superposition (TTS) with the TRIOS software.

Start-up shear measurements were performed at 170–175 °C with a cone-partitioned plate (CPP) fixture on the ARES-G2. The CPP fixture comprises an inner cylinder (10 mm diameter) concentrically aligned within an outer collar (25 mm diameter) and a matching 25 mm conical base featuring a 0.098 rad angle and 0.051 mm truncation. The start-up shear rates for each sample were incrementally increased from 0.003 s⁻¹ up to the maximum rates defined by instrument normal force limits. Start-up elongation measurements were performed at 130 °C on a filament stretching rheometer VADER-1000 (Rheo Filament, Denmark). The applied Hencky strain rates for each sample were chosen to achieve similar Weissenberg numbers ensuring direct comparability across datasets.

The linear-viscoelastic characterization of the polymer solutions was obtained via parsimonious relaxation spectra,

The partial moduli g_i and relaxation times τ_i of the samples were obtained from the master curves of storage and loss modulus at the reference temperature T_r = 130 °C and are reported by Cui et al. (2025) together with the factors for time–temperature shift (TTS) and T_g shifting. The zero-shear viscosity \(\eta_{0}\) and the disengagement time \(\tau_{d}\) were calculated from the discrete relaxation spectra by

The Rouse stretch relaxation times \(\tau_{R}\) for PS-600 k in the solution 600 k-4 k-50% at the temperatures T = 130 and 170 °C were calculated by

M_w is the molecular weight of PS-600 k, M_cm the critical molecular weight (taken as 30 kg/mol), and ρ is the density (taken as 1000 kg/m³). φ is the volume fraction of the polymer and is equal to 0.5 for all the solutions in this work. The Rouse times of PS-600 k in the solutions 600 k-10 k-50% and 600 k-8a4k-50% were then calculated from the values of the Rouse time of PS-600 k in the solvent OS-4 k by taking into account the differences in the glass transition temperature T_g of the solutions (Table 1), i.e., by considering iso-T_g conditions (Wagner 2014). The zero-shear viscosity \(\eta_{0}\), the disengagement time \(\tau_{d}\), and the Rouse time \(\tau_{R}\) of all samples investigated are summarized in Table 2.

In Fig. 1, experimental data (symbols) of the start-up shear viscosity \(\eta^{ + } (t,\dot{\gamma })\) for solutions 600 k-4 k-50%, 600 k-10 k-50%, and 600 k-8a4k-50% comprising 4 orders of magnitude in the shear rate \(\dot{\gamma }\) are compared to predictions of the RZS model of Eq. (7). Within experimental accuracy, the agreement of data and model can be rated as excellent. In particular, the shear viscosity overshoot is correctly described, and as already observed by Cui et al. (2025), the steady-state shear viscosity is in agreement with the Doi-Edwards model as expressed by Eq. (8). We note that the maximal values of the Weissenberg number \(Wi_{R} = \dot{\gamma }\tau_{R}\) reached in the experiments are 6.0 (600 k-4 k-50%), 3.7 (600 k-10 k-50%), and 10.8 (600 k-8a4k-50%), i.e., the shear rates investigated extend considerably into the nonlinear regime. The shear viscosity minimum after the maximum seen in the data of the two highest shear rates of 600 k-10 k-50% (Fig. 1b) is most likely due to minor flow instabilities causing an increasing shear viscosity at large strains, rather than to true transient shear stress undershoots as observed in polymer melts (see, e.g., Wagner 2024).

The data of the apparent first normal stress growth \(N_{1app}^{ + }\) as a function of shear strain \(\gamma\) of the three solutions considered are presented here for the first time (Fig. 2). \(N_{1app}^{ + }\) is defined in terms of the measured normal force F of the inner partition with diameter D_stem and can be expressed in terms of the first normal stress difference \(N_{1}^{ + }\), the second normal stress difference \(N_{2}^{ + }\), and the sample diameter D_s (for more details, see, e.g., Costanzo et al. 2018),

As \(\left| {N_{2}^{ + } } \right| < < N_{1}^{ + }\), \(N_{1app}^{ + }\) is mainly determined by \(N_{1}^{ + }\). For the start-up shear experiments considered, D_stem = 10 mm and sample diameter D_s = 12.5 mm as calculated from the geometry of the CPP tool and the sample mass of 50 mg. The experimental delay of the normal stress growth data (symbols) is caused by the limited stiffness of the normal force transducer, which leads to a minute opening of the gap between cone and plate, and consequently to radial inflow of the melt [see, e.g., Narimissa et al. 2020]. In addition, while the sensitivity of the normal force tranducer does not allow accurate mesurements at very low shear rates, the maximal normal force limit of 20 N was reached in the case of solutions 600 k-4 k-50% at the highest shear rate of \(\dot{\gamma } = 30s^{ - 1}\) and for solution 600 k-8a4k-50% at two highest shear rates of \(\dot{\gamma } = 10s^{ - 1}\) and \(\dot{\gamma } = 30s^{ - 1}\). Considering these experimental limitations, the agreement of data measured at intermediate shear rates and the RZS model predictions (lines) of Eq. (7) can be rated as encouraging for solutions 600 k-4 k-50% and 600 k-10 k-50%. In particular, for solution 600 k-10–50%, the overshoot of the apparent first normal stress growth \(N_{1app}^{ + }\) at higher shear rates is well predicted by the RZS model. We recall that the Doi-Edwards model does not predict overshoot of the first normal stress difference and that overshoot of the first normal stress difference is caused by stretch relaxation [Narimissa et al. 2020] as expressed by the first term on the right-hand side of Eq. (7). For solution 600 k-8a4k-50%, while the maximal stress level of \(N_{1app}^{ + }\) is reasonably well predicted by the model, the experimental data at the higher shear rates are affected by flow instability and decrease continuously after the maximal value of \(N_{1app}^{ + }\).

Experimental data (symbols) of the start-up elongational viscosity \(\eta_{E}^{ + } (t,\dot{\varepsilon })\) for solutions 600 k-4 k-50%, 600 k-10 k-50%, and 600 k-8a4k-50% measured at T = 130 °C are presented in Fig. 3. The elongation rates \(\dot{\varepsilon }\) range from the LVE regime up to maximal Weissenberg numbers \(Wi_{R} = \dot{\varepsilon }\tau_{R}\) of 16.8 (600 k-4 k-50%), 15.8 (600 k-10 k-50%), and 10.8 (600 k-8a4k-50%), i.e., again well into the nonlinear regime. We recall that for elongational flow, the RZS model reduces to the ERS model. With the same consistent set of Rouse times as used for shear flow, good agreement of data and predictions (lines) is found within experimental accuracy. In particular we note that for solution 600 k-10 k-50%, no strain hardening is observed and predicted for the lowest elongation rate of \(\dot{\varepsilon } = 0.0008s^{ - 1}\) corresponding to a Weissenberg number \(Wi_{R} = 0.25\), while strong strain hardening is found at \(\dot{\varepsilon } = 0.008s^{ - 1}\) and \(Wi_{R} = 2.5\). Similarly, for solution 600 k-8a4k-50%, considerable strain hardening is found at \(\dot{\varepsilon } = 0.006s^{ - 1}\) and \(Wi_{R} = 1.6\), while no strain hardening is observed and predicted for the lowest elongation rate of \(\dot{\varepsilon } = 0.001s^{ - 1}\) corresponding to a Weissenberg number \(Wi_{R} = 0.27\). These observations confirm the values of the Rouse stretch relaxation times given in Table 2.

The nonlinear shear and elongational start-up data of three entangled PS solutions with the same weight fraction (50%) of the same linear long-chain polystyrene (PS-600 k) but different styrene oligomeric solvents reported by Cui et al. (2025) can consistently be described by the RZS model. The solvents are linear styrene oligomers of different molecular weights (OS-4 k and OS-10 k), whereas the solution 600 k-8a4k-50% contains star styrene oligomers (OS-8a4k). From our analysis, we can conclude that the specific structure of the oligomers (short/long linear versus star) is of no consequence for the rheology of the three solutions other than having different effects on the glass transition temperature T_g (Table 1) and therefore on the Rouse time of PS-600 k (Table 2). The RZS model is based on the Doi-Edwards tube model and a flow-strength sensitive evolution equation of stretch. In extensional flows, the RZS model reduces to the ERS model. The modeling of start-up of shear viscosity and apparent first normal stress difference as well as elongational viscosity is based exclusively on the linear-viscoelastic characterization of the solutions and a consistent set of Rouse stretch relaxation times for PS-600 k at iso-T_g conditions. Here we have determined the relaxation time spectrum of the solutions from SAOS measurements, but instead, any theoretical mesoscopic or molecular LVE model that allows quantitative description of the relaxation modulus could be used. While the chain dynamics is accounted for by the relaxation modulus, the RZS/ERS model is based on a minimum number of concepts beyond the Doi-Edwards IA model, which are essential for modeling nonlinear viscoelastic start-up flow in shear and extension on the coarse-grained level of the tube, in particular:

Thus, the stress tensor Eq. (1) together with the evolution of stretch Eq. (5) represent a very concise description of nonlinear viscoelastic shear and extensional start-up flows in agreement with available experimental evidence for polymer melts and entangled polymer solutions of low-dispersity linear and star-shaped polymers.