Predicting a transient stress overshoot for polymer melts under startup shear flow is challenging. In recent, the classical White–Metzner (WM) constitutive equation of nonlinear viscoelastic fluids was potentially extended. For viscoelastic material functions, the minus ratio of the second normal stress difference to the first normal stress difference (*−N*_{2}/*N*_{1}) is important in characterizing a fluid's elasticity related to molecular structures and molecular weight distribution. Using the extended WM model to analyze a dramatic change in stress overshoot with respect to the *−N*_{2}/*N*_{1} ratio at high Weissenberg numbers would be significant. As a validation, numerical predictions of shear stress growth coefficient at different shear rates are in good agreement with experimental data.

Stress overshoot is very beneficial in seeking to understand a variety of nonlinear viscoelastic behaviors of polymer melts and entangled polymer solutions exhibited by undergoing startup shear at sufficient flow strengths.^{1} For Newtonian fluids, the stress instantaneously reaches its steady state. In particular with regard to non-Newtonian fluids, it is common to observe an overshoot, followed by a decrease toward the steady value. If the steady shear rate is high enough, the shear stress can overshoot; then, they can even undershoot. Thus, Fig. 1 depicts the transient shear viscosity or shear stress growth coefficient with respect to time for Newtonian fluids and non-Newtonian fluids.

Generally, the molecular origin of overshoot can be explained: when the material has a structure on the microscopic scale, deforming the material implies that the structure must be re-organized.^{2} From its molecular origin, the overshoot response is found to be primarily due to chain stretching rather than chain orientation.^{3,4} Additionally, the overshoot is a phenomenon stemming from an elastic energy storage mechanism and moving toward a dissipative energy release mechanism after the stress peak.^{5} Most studies on the nonlinear transient shear response include experimental measurements and theoretical predictions to deeply explore the origin of stress overshoot for polymer chains.^{1,3,5–7} Osaki *et al.*^{6} experimentally investigated the stress overshoot and also detected the undershoot trough and double-overshoot in a series of polystyrene (PS) solutions up to high shear rates. Ebrahimi *et al.*^{7} used the K-BKZ (Kaye-Bernstein–Kearsley–Zapas) integral constitutive model^{8–11} to well predict the transient shear response of the high-density polyethylene (HDPE) melt, as compared with experimental data obtained from a rotational rheometer. Saengow *et al.*^{12–15} applied the Oldroyd 8-constant differential constitutive equation^{16} to provide slightly early underpredictions of the overshoots at high shear rates for a high viscoelastic fluid of wormlike micellar solution. It is significant that Pipe *et al.*^{17} performed an important study of *concentrated* wormlike micellar solutions in quantitative comparison between experimental data and scission model predictions. In particular, Moore *et al.*^{18} used the nonequilibrium molecular dynamics (NEMD) simulation^{19,20} and the Doi–Edwards (DE) molecular rheological theoretical model prediction^{21} to exhibit the pronounced overshoot for a polyethylene (PE) melt at high shear rates.

A dramatic change in shear stress overshoot, with respect to viscoelastic material functions of the first and second normal stress differences, thus far, has been barely investigated from the complex multi-mode nonlinear constitutive equations. Among the viscoelastic fluid models, the earlier White–Metzner (WM) model^{22,23} expresses a relatively simple nonlinear viscoelastic fluid of polymer melts. Such a model is the single-mode upper-convected Maxwell (UCM) constitutive equation combined with the generalized-Newtonian-fluids (GNF) model of shear viscosity.^{16,24,25} However, such a differential stress model, substantial with strong hyperbolic and singular problems, has hitherto always obtained unsatisfactory simulations of corner vortex in a typical contraction flow, especially for high Weissenberg numbers. Most recently, Tseng^{26} proposed a modification of the WM constitutive equation, called WMT-X (WM e**X**tended by **T**seng). This model can fit the first normal stress difference for characterizing a fluid's elasticity, as well as shear viscosity and extensional viscosity. It is significant to discuss the vortex formation and growth, with the predicted vortex sizes in good agreement with the experimental data. Note that the numerical calculation of the WMT-X viscoelastic fluids is stable convergent at high shear rates.

For completeness, the WMT-X model^{26} is introduced below:

where $\tau $ is the extra stress tensor; **D** is the rate-of-deformation tensor and the symmetric matrix of the velocity gradient $\u2207v$; $\lambda $ is the relaxation time; $\eta W$ is the weighted viscosity of the GNF-X (eXtended GNF) model; $\lambda $ and $\eta W$ are a function of strain rates $\gamma \u0307$.

In particular, $\tau $ is called the Gordon–Schowalter hybrid time derivative^{24} with a slip factor $\xi $ is a linear combination of the Oldroyd upper-convected time derivative $\tau \u2207$ and the Jaumann corotational time derivative^{27} $\tau \u25cb$:

where **W** is the vorticity tensor, which is the anti-symmetric matrix of the velocity gradient $\u2207v$; $D\tau Dt$ is the well-known material derivative convective term. In recent times, it is significant that Ramlawi *et al.*^{28} derived new analytical solutions for the non-affine Johnson–Segalman/Gordon–Schowalter constitutive equation with a general relaxation kernel in medium-amplitude oscillatory shear deformation. The slip parameter $\xi $ can also be interpreted through the framework of the Oldroyd 8-constant model of Saengow *et al.*^{14,15}

The Weissenberg number $Wi$ is the product of the shear rate $\gamma \u0307$ and the longest relaxation time $\lambda $:

At a large Weissenberg number, $Wi\u226b1$, the fluid will respond with elastic behavior like that of a solid. For $Wi\u226a1$, a liquid-like viscous state is expected.

In addition, the Weissenberg number was originally related to the ratio of the first normal stress difference $N1$ to the shear stress $\tau 12$:

where $\tau 11$, $\tau 22$, and $\tau 33$ are flow-directional, gradient-directional, and neutral-direction normal (diagonal) stress components, respectively. In other words, this number also indicates the dimensionless magnitude of *N*_{1}. The second normal stress difference $N2$ is expressed as:

Generally, the second normal stress difference *N*_{2} is more difficult to measure than the first normal stress difference *N*_{1}. In most cases, the negative *N*_{2} value is typically 10% of *N*_{1} in its magnitude.^{16} Thus, the ratio −*N*_{2}/*N*_{1} is especially used to indicate the dimensionless magnitude of *N*_{2}, reasonably existing in a region between 0 and 0.5. Note that the normal stress differences are important in characterizing a fluid's elasticity with respect to molecular structures (entanglement and orientation) and molecular weight distribution (sharp and broad).^{29–31}

Recently, Tseng^{32} derived the weighted shear/extensional viscosity $\eta W(\gamma \u0307)$, called the eXtended GNF (GNF-X) viscous model, as expressed below:

where *W* is the weighting function, and also called the extension fraction; $\eta S$ and $\eta E$ are the GNF shear viscosity and extensional viscosity with respect to strain rates, respectively. $\gamma \u0307S$ and $\gamma \u0307E$ are the principal shear rate and principal extensional rate, respectively; and $\gamma \u0307T$ is the total strain rate. For a pure shear flow, the GNF-X weighted viscosity returns to the GNF shear viscosity, namely, $\eta W=\eta S$. The details of the GNF-X weighted viscosity are available elsewhere.^{32}

In the shear flow, the complete form of the WMT-X model,^{26} therefore, is unfolded:

where **D*** and **W*** are dimensionless tensors of the rate-of-deformation tensor **D** and the vorticity tensor **W**, respectively. Note that $\u2202\tau \u2202t$ = 0 and $\u2207\tau $ = 0 indicate the steady state and homogenous flows, respectively. The normal stress parameter $CN$ is limited between 0 and 1.

In the homogenous, steady-state, simple shear flow, the viscoelastic material functions are further derived:

It is critically important to indicate that both *N*_{1} and *N*_{2} are steady-state values, and not the transient ones. Thereby, it is important to find $\xi $ and $CN$ related to −*N*_{2}/*N*_{1}, which are a function of shear rate.

In recent times, Saengow *et al.*^{14,15} provided an interpretation of the slip factor based on the Oldroyd 8-constant model.

For a homogenous simple shear flow with an *x-*axis flow direction, a *y*-axis in a gradient direction, and a z-axis in a neutral direction, the flow strength of velocity-gradient tensor $\u2207v$ is given by shear rate $\gamma \u0307$ to determine the rate-of-deformation tensor **D** and the vorticity tensor **W**, which are the symmetric matrix and anti-symmetric matrix of the velocity gradient tensor, respectively.

These tensors are inputted to the WMT-X of Eq. (14).

Here, one can perform the startup calculation of shear flow under homogenous conditions with the WMT-X viscoelastic constitutive equation of Eq. (14) and adopt the fourth-order Runge–Kutta method^{33} to solve the time-evolution. Figure 2 shows the transient shear stress responses with respect to time at different values of −*N*_{2}/*N*_{1} between 0.01 and 0.5 under transient shear with shear rate $\gamma \u0307$ = 1.0 s^{−1} and shear viscosity $\eta S=1\xd7105$ Pa·s. Note that the ratio −*N*_{2}/*N*_{1} = 0.5 implies the viscoelastic fluid reflecting in the correlational Maxwell model.

For a relatively small Weissenberg number, $Wi$ = 0.1, the transient stress curves are almost kept constant and are a linear viscoelastic response without overshoot. When a large $Wi$ = 1.0 and −*N*_{2}/*N*_{1} > 0.1 are given, the stress increases linearly with time, passes through a maximum (i.e., overshoot), and then decreases to a convergent plateau. At −*N*_{2}/*N*_{1} = 0.5, the overshoot response is obvious. According to rheological treatises,^{16} the negative *N*_{2} value is typically 10% of *N*_{1} in its magnitude for polymer melts. Also, the *overshoot–undershoot–overshoot* peak is found for a very high $Wi$ = 3.0 and reasonable −*N*_{2}/*N*_{1} = 0.1, as shown in Fig. 3. This finding is the same as the previous experimental observation of Osaki *et al.*^{6} for PS solutions at high shear rates, namely, a peculiar stronger nonlinear viscoelastic behavior with an oscillator damping. In the constitutive analysis of the WMT-X model, one therefore can conclude that the stress overshoot is evidently related to the value of −*N*_{2}/*N*_{1}, but hardly occurs at the weak elastic effect or the small Weissenberg number condition of −*N*_{2}/*N*_{1} ≪ 0.1. From a mathematical perspective, the occurrence of overshoot is due to an interdependence relationship of shear stress ($\tau 12$) and normal stresses ($\tau 11$ and $\tau 22$).

To validate the accuracy of the WMT-X model, one can refer to Ebrahimi *et al.*,^{7} who used the rotational rheometer to measure the experimental data of the shear stress growth coefficient for a HDPE melt taken at 180 °C and performed excellent predictions of the K-BKZ integral constitutive model. At different shear rates between 0.5 and 5.0, Fig. 4 shows that the WMT-X model predictions match the experimental data well. For the overshoot at the higher shear rate $\gamma \u0307$ = 5.0 s^{−1}, the overshoot degree of the WMT-X model is somewhat small than that of the K-BKZ model. At low shear rates $\gamma \u0307$ < 1.0 s^{−1}, the response of the shear stress growth coefficient attached without overshoot is almost linear viscoelastic. Hence, the WMT-X model for describing the nonlinear viscoelastic behavior of the overshoot is satisfied for polymer melts. In addition, all the parameters are addressed in Table I. With increasing shear rate, one can find that the shear viscosity and relaxation time are decreased, whereas the elastic parameters $Wi$ and −*N*_{2}/*N*_{1} are increased. According to Eq. (21), the slip factor can be obtained. Thus, Fig. 5 shows that the slip factor is increased with shear rates; this implies the chain entanglement predominates at low shear rates and the chain orientation predominates at high shear rates. From the previous NEMD simulation of Tseng *et al.*,^{34} the −*N*_{2}/*N*_{1} trend is the same as in the present WMT-X work.

$\gamma \u0307$ (s^{−1})
. | $\eta S$ (Pa·s) . | $Wi$ . | $\lambda $ (s) . | −N_{2}/N_{1}
. | $\xi $ . |
---|---|---|---|---|---|

0.5 | $1.0\xd7105$ | 0.8 | 1.6 | 0.02 | 0.04 |

1.0 | $5.5\xd7104$ | 0.8 | 0.8 | 0.08 | 0.16 |

5.0 | $3.0\xd7104$ | 1.0 | 0.2 | 0.3 | 0.6 |

$\gamma \u0307$ (s^{−1})
. | $\eta S$ (Pa·s) . | $Wi$ . | $\lambda $ (s) . | −N_{2}/N_{1}
. | $\xi $ . |
---|---|---|---|---|---|

0.5 | $1.0\xd7105$ | 0.8 | 1.6 | 0.02 | 0.04 |

1.0 | $5.5\xd7104$ | 0.8 | 0.8 | 0.08 | 0.16 |

5.0 | $3.0\xd7104$ | 1.0 | 0.2 | 0.3 | 0.6 |

Eventually, this work has achieved the constitutive analysis of the stress overshoot via the WMT-X viscoelastic model. One can understand the relationship between the overshoot and the elastic parameters of $Wi$ and −*N*_{2}/*N*_{1}. The overshoot easily occurs at high $Wi$ values and −*N*_{2}/*N*_{1} > 0.1. At different shear rates, the predicted transient stress responses are in agreement with the experimental data. The values of the elastic parameters imply a fluid's elasticity depending on entangled/oriented chains and sharp/broad weight distributions of polymer structures.^{29–31} In future work, it will be important to explore the difference of overshoot between complex structure characteristics of polymer melts, such as the linear low-density and branched high-density PE melts.

The data that support the findings of this study are available from the corresponding author upon reasonable request.