# On the Use of a Non-Constant Non-Affine or Slip Parameter in Polymer Rheology Constitutive Modeling

^{1}

^{2}

^{*}

## Abstract

**:**

_{48}polyethylene melt obtained via direct non-equilibrium molecular dynamics simulations in shear. We find that the conformation tensor data are very well predicted; however, the predictions of the material functions are noted to deviate from the NEMD data, especially at large shear rates.

## 1. Introduction

## 2. Model Modification

**R,**with its center-of-mass at position

**r**, i.e., $C\left(r,t\right)=\langle RR\rangle \left(r,t\right)={{\displaystyle \int}}^{}RR\mathsf{\Psi}\left(R,r,t\right)dR$ with the brackets denoting a configurational average. The evolution equation for the dimensionless conformation tensor $c\left(r,t\right)=3C\left(r,t\right)/{\langle {R}^{2}\rangle}_{eq}$, as derived by Stephanou et al. [29], is:

#### Asymptotic Behavior of the Model for Steady-State and Transient Shear Flow

## 3. Molecular Model and System Studied

_{48}H

_{98}melts. Equilibrium MD simulations were performed in the NPT ensemble to fully relax the initial PE configurations at temperature T = 450 K and pressure P = 1 atm using the united-atom potential model of Siepmann et al. [39]. The simulations were carried out using the LAMMPS simulation engine [40], employing the Nosé–Hoover thermostat [41,42] and the Parrinello–Rahman barostat [43] to preserve the temperature and pressure, respectively, at their prescribed values. Subsequently, several fully relaxed configurations from the MD runs were selected as input for the NEMD simulations under shear. The NEMD runs were performed again with LAMMPS in the NVT ensemble at T = 450 K, using the SLLOD algorithm [44], together with the Nosé–Hoover thermostat to control the temperature. The microscopic set of equations of motion was integrated numerically using the reversible Reference System Propagator Algorithm (r-RESPA) [45], with 2 different time steps: (a) a large one (dt = 4 fs) for the integration of the slowest varying forces arising from non-bonded interactions at long interatomic distances, and (b) a small one (dt = 0.5 fs) for the integration of the fast-varying forces corresponding to the bonded (i.e., bonds, angles, and dihedrals) interactions.

_{48}H

_{98}and were subjected to periodic boundary conditions in all three directions (x, y, and z). In the course of the NEMD simulations, the x- and y-directions were selected as the flow and the shear gradient directions, respectively, whereas z was the neutral direction. The simulation cell had dimensions (462 Å) × (231 Å) × (231 Å) along the x-, y-, and z-directions. The cell was purposefully enlarged in the (x-) flow direction to ensure minimal system size effects, particularly with the NEMD runs at high shear rates, where the polymer chains tend to stretch and orient towards the flow direction. To this end, for the C

_{48}H

_{98}chains, the equilibrium root-mean-square of the chain end-to-end vector $\sqrt{{\langle {R}^{2}\rangle}_{eq}}$ and the theoretical maximum chain extension of ${\left|R\right|}_{\mathrm{max}}$ were calculated to be equal to $\sqrt{{\langle {R}^{2}\rangle}_{eq}}=27.2\pm 0.12\AA $and ${\left|R\right|}_{\mathrm{max}}=63.1\AA $, respectively. Compared to the simulation cell dimensions, the maximum chain length ${\left|R\right|}_{\mathrm{max}}$ is 7.3 times shorter than the dimension in the x-direction and 3.6 times shorter than the dimension in the y-direction. Thus, we can safely expect that the simulation cell is sufficiently large to ensure the absence of system size effects due to chain alignment in the flow direction or tumbling motion in the shear gradient direction.

_{48}H

_{98}PE chains at T = 450 K and P = 1 atm was found to be equal to ${\tau}_{R,\mathrm{eq}}=0.6\pm 0.01\mathrm{ns}$, as estimated by integrating the stretched–exponential curve [46] over time, describing the time autocorrelation function of the chain end-to-end unit vector. The MD simulations were conducted for a total of 6 ns, which is 10 times larger than the chain relaxation time to ensure that the PE chains were fully equilibrated. The NEMD simulations were executed over a broad range of shear rates spanning the range from the linear up to the highly non-linear viscoelastic regime, corresponding to Weissenberg numbers (with ${\tau}_{R,\mathrm{eq}}=0.6\mathrm{ns}$) in the interval [0.1, 285].

## 4. Results and Discussion

#### 4.1. Model Predictions in Steady-State Shear Flow

_{xx}is reported to increase from its equilibrium value after about ≈Wi = 0.3 and eventually reach its limiting value, which, however, differs from the value of the parameter b. On the other hand, c

_{xy}is noted to increase linearly with the shear rate at low shear rates, as dictated by Equation (4b), reaching a maximum value, and then decreasing inversely proportional to the shear rate. Finally, the two remaining diagonal elements of the conformation tensor in the shear gradient direction and neutral direction (c

_{yy}and c

_{zz}, respectively), are observed to be mirrors of the noted behavior of c

_{xx}: they initially decrease from their equilibrium value, eventually reaching a finite asymptotic value at high shear rates. We note, in Figure 2, that the value of $\gamma $ plays only a very modest role for all elements, since the value of ${\xi}_{0}$ is small; note that one would expect such small values to be used since larger values would lead to very intense oscillations in the time-dependent material functions (see the next section). On the other hand, by increasing the parameter ${\xi}_{0}$, we note the predictions at low shear rates to be insensitive. Still, the limiting asymptotic values at high shear rates are pointed out to decrease for c

_{xx}and increase for both c

_{yy}and c

_{zz}, whereas the c

_{xy}curve shifts to lower shear rates at higher shear rates whilst keeping the power-law unaffected. This is a direct result of allowing tumbling to occur sooner, since the slip parameter is larger, thus refraining the flow to deform the chain further. Finally, when we increase the value of $\epsilon $ while keeping the same ${\xi}_{0}$ and $\gamma $ values (Figure 3), we note that the asymptotic values of the diagonal elements remain the same, but the curves are shifted to the right, a direct result of the steeper decrease in the relaxation time, cf. Equation (1g). As the anisotropic (or Giesekus) parameter, $\alpha $, is increased, we observe that the curve of c

_{xx}shifts downwards and the one of c

_{zz}shifts upwards (see Figure 3). In constrast, when the FENE parameter is increased (from 20 to 50, meaning that the chain is now longer or its molecular weight is larger), we observe the reverse behavior, which is the expected outcome. However, note that during these parameter value changes, the predictions of the other elements, c

_{xy}(panel (b)) and c

_{yy}(panel (c)) are only modestly affected. As in Figure 2, the value of $\epsilon $ controls the rate at which the asymptotic values, in the case of the diagonal elements, are reached, whereas the c

_{xy}curve shifts rightwards.

#### 4.2. Model Predictions in Start-Up Shear Flow

_{xx}, c

_{xy}, and c

_{yy}also present a dumping behavior, which is much more intense relative to the one noted in the slip parameter (Figure 6). It intensifies as the ${\xi}_{0}$ parameter increases and becomes less intense when the $\epsilon $ parameter increases. Such a behavior is obtained only when ${\xi}_{0}>0$. On the other hand, c

_{zz}seems not to present such a behavior, although a small saddle point is noted when $\epsilon =0$. The time-dependent behavior of the conformation tensor is seen in Figure 8 to be insensitive to the value of the parameters $\alpha $ and $\epsilon $.

#### 4.3. Comparison with NEMD Simulation Data for an Unentangled PE Melt

_{48}PE melt over a broad spectrum of shear rates. We only analyze the steady-state data of these NEMD simulations. To fit the simulation data, following Stephanou et al. [29], we first identify the asymptotes ${c}_{ii}^{\infty}$ of the diagonal elements of the conformation tensor in the limit of high shear rates. Additionally, the equilibrium relaxation (Rouse) time, as mentioned in Section 3, is equal to ${\tau}_{R,\mathrm{eq}}=0.6\mathrm{ns}$, whereas the zero-shear-rate viscosity can be obtained from the shear viscosity NEMD data, which is equal to ${\eta}_{0}=5\mathrm{mPa}.\mathrm{s}$. Note that as small shear rates, the NEMD data of the viscometric functions come with large error bars, and it is difficult, particularly for the normal stress coefficients, to accurately estimate their zero-shear-rate values. Then, the value of ${\xi}_{0}\approx 0.104$ is obtained by using the first equation of Equation (45) of Stephanou et al. [29]. Next, the value of the Giesekus parameter $\alpha \approx 0.2$ can be obtained by fitting the ${\mathsf{\Psi}}_{2}$ NEMD data at low shear rates and using Equation (4c) (the value of ${\mathsf{\Psi}}_{1,0}$ is easily calculated from Equation (4b)). Note that this value differs from the value $\alpha \approx 0.06$ obtained using the second equation of Equation (45) of Stephanou et al. [29]; however, the fitting of ${\mathsf{\Psi}}_{2}$ is much improved when using the former value, and the comparison against the conformation tensor data is only mildly worsened. Next, the value of b

_{eff}=5.78 can be obtained from Equation (48) of Stephanou et al. [29], which differs from the value ${3}^{2}/{\langle {R}^{2}\rangle}_{eq}=16.15$. The remaining two parameters, $\epsilon $ and $\gamma $, can be obtained by simply fitting the NEMD data, since they do not affect the ${c}_{\mathrm{ii}}^{\infty}$ values; we obtain $\epsilon =0.4$ and $\gamma =0.01$. Figure 11 shows how well the new model can fit the simulation data for the ${c}_{\mathrm{xx}}$,${c}_{\mathrm{xy}},{c}_{\mathrm{yy}},$ and ${c}_{\mathrm{zz}}$ elements of the dimensionless conformation tensor for the C

_{48}PE system in steady shear. We observe that the predictions of the revised model are in remarkable agreement with the NEMD extracted simulation results over the entire wide range of shear rates considered, especially for the diagonal elements. To quantify how well the model predicts the simulation data, we calculated the sum of the squares of the residuals (i.e., the residual between the simulation value and the one obtained by the model); this turns out to be ≈0.2474. The corresponding comparison for the material functions $\eta ,{\mathsf{\Psi}}_{1}$, and $-{\mathsf{\Psi}}_{2}$ is presented in Figure 12. Contrary to the very good agreement between the refined model predictions and the NEMD simulation data for the dimensionless conformation tensor, the comparison against the viscometric functions is less satisfactory. Deviations from the NEMD data are mainly observed at large shear rates in the case of the shear viscosity (panel (a)) and the second normal stress coefficient (panel (c)). As also mentioned by Stephanou et al. [29], this disaccord should be related to the postulated relation between the stress and conformation tensors, Equation (2), which stems from the assumption of purely entropic elasticity [48]. As such, a more accurate expression for the free energy needs to be invoked in the future [29].

## 5. Conclusions

_{48}. Although the predictions at large shear rates were not significantly modified, the revision amended the problems associated with having $\xi $-dependent zero-shear-rate viscometric functions [29,38], cf. Equations (4), and linear-viscoelastic properties [38], cf. Equations (5). It should be emphasized that although the revised model was not derived through the use of a non-equilibrium thermodynamics formalism [28,30], its thermodynamic admissibility still holds, since $0\le \xi \le 1$ (provided $0\le {\xi}_{0}\le 1$). Additionally, it is a straightforward exercise to extend the model to entangled systems by following our recent work [38]. We expect that the future use of the refined model will allow for more reliable prediction of macroscopic viscoelastic behavior and, therefore, for the development of more reliable computational tools, aiming to tailor-design large-molecular-weight polymeric systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Pedro, J.; Ramôa, B.; Nóbrega, J.M.; Fernandes, C. Verification and Validation of OpenInjMoldSim, an Open-Source Solver to Model the Filling Stage of Thermoplastic Injection Molding. Fluids
**2020**, 5, 84. [Google Scholar] [CrossRef] - Fernandes, C.; Fakhari, A.; Tukovic, Ž. Non-Isothermal Free-Surface Viscous Flow of Polymer Melts in Pipe Extrusion Using an Open-Source Interface Tracking Finite Volume Method. Polymers
**2021**, 13, 4454. [Google Scholar] [CrossRef] - Viana, J.C.; Cunha, A.M.; Billon, N. The Thermomechanical Environment and the Microstructure of an Injection Moulded Polypropylene Copolymer. Polymers
**2002**, 43, 4185–4196. [Google Scholar] [CrossRef] - Edwards, B.J.; Sefiddashti, M.H.N.; Khomami, B. Atomistic Simulation of Shear Flow of Linear Alkane and Polyethylene Liquids: A 50-Year Retrospective. J. Rheol.
**2022**, 66, 415. [Google Scholar] [CrossRef] - Katsarou, A.F.; Tsamopoulos, A.J.; Tsalikis, D.G.; Mavrantzas, V.G. Dynamic Heterogeneity in Ring-Linear Polymer Blends. Polymers
**2020**, 12, 752. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tsalikis, D.G.; Mavrantzas, V.G.; Vlassopoulos, D. Analysis of Slow Modes in Ring Polymers: Threading of Rings Controls Long-Time Relaxation. ACS Macro Lett.
**2016**, 5, 755–760. [Google Scholar] [CrossRef] - Tsalikis, D.G.; Mavrantzas, V.G. Size and Diffusivity of Polymer Rings in Linear Polymer Matrices: The Key Role of Threading Events. Macromolecules
**2020**, 53, 803–820. [Google Scholar] [CrossRef] - Papadopoulos, G.D.; Tsalikis, D.G.; Mavrantzas, V.G. Microscopic Dynamics and Topology of Polymer Rings Immersed in a Host Matrix of Longer Linear Polymers: Results from a Detailed Molecular Dynamics Simulation Study and Comparison with Experimental Data. Polymers
**2016**, 8, 283. [Google Scholar] [CrossRef] - Tsolou, G.; Stratikis, N.; Baig, C.; Stephanou, P.S.; Mavrantzas, V.G. Melt Structure and Dynamics of Unentangled Polyethylene Rings: Rouse Theory, Atomistic Molecular Dynamics Simulation, and Comparison with the Linear Analogues. Macromolecules
**2010**, 43, 10692–10713. [Google Scholar] [CrossRef] - Stephanou, P.S.; Mavrantzas, V.G. Accurate Prediction of the Linear Viscoelastic Properties of Highly Entangled Mono and Bidisperse Polymer Melts. J. Chem. Phys.
**2014**, 140, 214903. [Google Scholar] [CrossRef] - Smith, D.E.; Babcock, H.P.; Chu, S. Single-Polymer Dynamics in Steady Shear Flow. Science
**1999**, 283, 1724–1727. [Google Scholar] [CrossRef] [PubMed] [Green Version] - LeDuc, P.; Haber, C.; Bao, G.; Wirtz, D. Dynamics of Individual Flexible Polymers in a Shear Flow. Nature
**1999**, 399, 564–566. [Google Scholar] [CrossRef] - Sefiddashti, M.H.N.; Edwards, B.J.; Khomami, B. Individual Chain Dynamics of a Polyethylene Melt Undergoing Steady Shear Flow. J. Rheol.
**2015**, 59, 119–153. [Google Scholar] [CrossRef] [Green Version] - Edwards, C.N.; Sefiddashti, M.H.N.; Edwards, B.J.; Khomami, B. In-Plane and out-of-Plane Rotational Motion of Individual Chain Molecules in Steady Shear Flow of Polymer Melts and Solutions. J. Mol. Graph. Model.
**2018**, 81, 184–196. [Google Scholar] [CrossRef] [Green Version] - Batchelor, G.K.; Green, J.T. The Determination of the Bulk Stress in a Suspension of Spherical Particles to Order C2. J. Fluid Mech.
**1972**, 56, 401–427. [Google Scholar] [CrossRef] - Sefiddashti, M.H.N.; Edwards, B.J.; Khomami, B. Steady Shearing Flow of a Moderately Entangled Polyethylene Liquid. J. Rheol.
**2016**, 60, 1227–1244. [Google Scholar] [CrossRef] - Kim, J.M.; Baig, C. Precise Analysis of Polymer Rotational Dynamics. Sci. Rep.
**2016**, 6, 19127. [Google Scholar] [CrossRef] [Green Version] - Tsamopoulos, A.J.; Katsarou, A.F.; Tsalikis, D.G.; Mavrantzas, V.G. Shear Rheology of Unentangled and Marginally Entangled Ring Polymer Melts from Large-Scale Nonequilibrium Molecular Dynamics Simulations. Polymers
**2019**, 11, 1194. [Google Scholar] [CrossRef] [Green Version] - Schroeder, C.M.; Teixeira, R.E.; Shaqfeh, E.S.G.; Chu, S. Dynamics of DNA in the Flow-Gradient Plane of Steady Shear Flow: Observations and Simulations. Macromolecules
**2005**, 38, 1967–1978. [Google Scholar] [CrossRef] - Schroeder, C.M.; Teixeira, R.E.; Shaqfeh, E.S.G.; Chu, S. Characteristic Periodic Motion of Polymers in Shear Flow. Phys. Rev. Lett.
**2005**, 95, 018301. [Google Scholar] [CrossRef] - Huang, C.C.; Sutmann, G.; Gompper, G.; Winkler, R.G. Tumbling of Polymers in Semidilute Solution under Shear Flow. EPL
**2011**, 93, 54004. [Google Scholar] [CrossRef] [Green Version] - Xu, X.; Chen, J.; An, L. Shear Thinning Behavior of Linear Polymer Melts under Shear Flow via Nonequilibrium Molecular Dynamics. J. Chem. Phys.
**2014**, 140, 174902. [Google Scholar] [CrossRef] [PubMed] - Stephanou, P.S.; Kröger, M. Non-Constant Link Tension Coefficient in the Tumbling-Snake Model Subjected to Simple Shear. J. Chem. Phys.
**2017**, 147, 174903. [Google Scholar] [CrossRef] - Costanzo, S.; Huang, Q.; Ianniruberto, G.; Marrucci, G.; Hassager, O.; Vlassopoulos, D. Shear and Extensional Rheology of Polystyrene Melts and Solutions with the Same Number of Entanglements. Macromolecules
**2016**, 49, 3925–3935. [Google Scholar] [CrossRef] [Green Version] - Gordon, R.J.; Schowalter, W.R. Anisotropic Fluid Theory: A Different Approach to the Dumbbell Theory of Dilute Polymer Solutions. Trans. Soc. Rheol.
**1972**, 16, 79–97. [Google Scholar] [CrossRef] - Johnson, M.W.; Segalman, D. A Model for Viscoelastic Fluid Behavior Which Allows Non-Affine Deformation. J. Nonnewton. Fluid Mech.
**1977**, 2, 255–270. [Google Scholar] [CrossRef] - Larson, R.G. Constitutive Equations for Polymer Melts and Solutions, 1st ed.; Butterworth-Heinemann: Oxford, UK, 1988; ISBN 978-0-409-90119-1. [Google Scholar]
- Öttinger, H.C. Beyond Equilibrium Thermodynamics; John Wiley and Sons: Hoboken, NJ, USA, 2005; ISBN 0471666580. [Google Scholar]
- Stephanou, P.S.; Baig, C.; Mavrantzas, V.G. A Generalized Differential Constitutive Equation for Polymer Melts Based on Principles of Nonequilibrium Thermodynamics. J. Rheol.
**2009**, 53, 309–337. [Google Scholar] [CrossRef] - Beris, A.N.; Edwards, B.J. Thermodynamics of Flowing Systems: With Internal Microstructure; Oxford University Press: New York, NY, USA, 1994; ISBN 019507694X. [Google Scholar]
- Stephanou, P.S. The Rheology of Drilling Fluids from a Non-Equilibrium Thermodynamics Perspective. J. Pet. Sci. Eng.
**2018**, 165, 1010–1020. [Google Scholar] [CrossRef] - Hinch, E.J. Mechanical Models of Dilute Polymer Solutions in Strong Flows. Phys. Fluids
**1977**, 20, S22–S30. [Google Scholar] [CrossRef] - Rallison, J.M.; Hinch, E.J. Do We Understand the Physics in the Constitutive Equation? J. Nonnewton. Fluid Mech.
**1988**, 29, 37–55. [Google Scholar] [CrossRef] - Beris, A.N.; Stiakakis, E.; Vlassopoulos, D. A Thermodynamically Consistent Model for the Thixotropic Behavior of Concentrated Star Polymer Suspensions. J. Nonnewton. Fluid Mech.
**2008**, 152, 76–85. [Google Scholar] [CrossRef] - Stephanou, P.S.; Georgiou, G.G. A Nonequilibrium Thermodynamics Perspective of Thixotropy. J. Chem. Phys.
**2018**, 149, 244902. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Housiadas, K.D.; Beris, A.N. Extensional Behavior Influence on Viscoelastic Turbulent Channel Flow. J. Nonnewton. Fluid Mech.
**2006**, 140, 41–56. [Google Scholar] [CrossRef] - Souvaliotis, A.; Beris, A.N. An Extended White–Metzner Viscoelastic Fluid Model Based on an Internal Structural Parameter. J. Rheol.
**1992**, 36, 241–271. [Google Scholar] [CrossRef] - Stephanou, P.S.; Tsimouri, I.C.; Mavrantzas, V.G. Simple, Accurate and User-Friendly Differential Constitutive Model for the Rheology of Entangled Polymer Melts and Solutions from Nonequilibrium Thermodynamics. Mater.
**2020**, 13, 2867. [Google Scholar] [CrossRef] - Siepmann, J.I.; Karaborni, S.; Smit, B. Simulating the Critical Behaviour of Complex Fluids. Nature
**1993**, 365, 330–332. [Google Scholar] [CrossRef] [Green Version] - Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] [Green Version] - Nosé, S. Constant Temperature Molecular Dynamics Methods. Prog. Theor. Phys. Suppl.
**1991**, 103, 1–46. [Google Scholar] [CrossRef] [Green Version] - Hoover, W.G. Canonical Dynamics: Equilibrium Phase-Space Distributions. Phys. Rev. A
**1985**, 31, 1695. [Google Scholar] [CrossRef] [Green Version] - Parrinello, M.; Rahman, A. Polymorphic Transitions in Single Crystals: A New Molecular Dynamics Method. J. Appl. Phys.
**1981**, 52, 7182–7190. [Google Scholar] [CrossRef] - Todd, B.D.; Daivis, P.J. Nonequilibrium Molecular Dynamics: Theory, Algorithms and Applications; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Tuckerman, M.; Berne, B.J.; Martyna, G.J. Reversible Multiple Time Scale Molecular Dynamics. J. Chem. Phys.
**1992**, 97, 1990–2001. [Google Scholar] [CrossRef] [Green Version] - Williams, G.; Watts, D.C. Non-Symmetrical Dielectric Relaxation Behaviour Arising from a Simple Empirical Decay Function. Trans. Faraday Soc.
**1970**, 66, 80–85. [Google Scholar] [CrossRef] - MathWorks, T. MATLAB (R2020b); MathWorks Inc.: Portola Valley, CA, USA, 2020. [Google Scholar]
- Grmela, M. Stress Tensor in Generalized Hydrodynamics. Phys. Lett. A
**1985**, 111, 41–44. [Google Scholar] [CrossRef]

**Figure 1.**Model predictions for the slip parameter as a function of Wi and dependence on: (

**a**) the parameters ${\xi}_{0},\gamma $, and $\epsilon $ for $\alpha =0.4,b=50$, and (

**b**) the parameters $\alpha ,\epsilon ,$ and b for ${\xi}_{0}=0.05$ and $\gamma =1$. The dark yellow dotted lines depict the analytical expression Equation (4e) for each case.

**Figure 2.**Variation of the conformation tensor elements (

**a**) c

_{xx}, (

**b**) c

_{xy}, (

**c**) c

_{yy}, and (

**d**) c

_{zz}with Wi and dependence on the parameters ${\xi}_{0},\gamma $, and $\epsilon $ for $\alpha =0.4,b=50$. The dark yellow dashed line depicts the asymptotic behavior at small shear rates given by Equations (4).

**Figure 3.**Variation of the conformation tensor elements (

**a**) c

_{xx}, (

**b**) c

_{xy}, (

**c**) c

_{yy}, and (

**d**) c

_{zz}with Wi and dependence on the parameters $\alpha ,\epsilon ,$ and b for ${\xi}_{0}=0.05$ and $\gamma =1$.

**Figure 4.**Variation of the (

**a**) shear viscosity, (

**b**) the first normal stress coefficient, and (

**c**) the negative second normal stress coefficient as a function of Wi and dependence on the parameters ${\xi}_{0},\gamma $, and $\epsilon $ for $\alpha =0.4,b=50$.

**Figure 5.**Variation of the (

**a**) shear viscosity, (

**b**) the first normal stress coefficient, and (

**c**) the negative second normal stress coefficient as a function of Wi and dependence on the parameters $\alpha ,\epsilon $, and b for of ${\xi}_{0}=0.05$ and $\gamma =1$.

**Figure 6.**Variation of the slip parameter upon the inception of shear flow as a function of $t/{\tau}_{\mathrm{R},\mathrm{eq}}$ at two different values of the dimensionless shear rate $\mathrm{Wi}=\dot{\gamma}{\tau}_{R,\mathrm{eq}}$, and dependence on: (

**a**) the parameters ${\xi}_{0}$ and $\epsilon $ for $\alpha =0.4,b=50$; and (

**b**) the parameters $\alpha ,\epsilon ,$ and b for ${\xi}_{0}=0.05$ and $\gamma =1$. In all cases, $\gamma =1$.

**Figure 7.**Variation of the growth of the conformation tensor elements (

**a**) c

_{xx}, (

**b**) c

_{xy}, (

**c**) c

_{yy}, and (

**d**) c

_{zz}upon the inception of shear flow as a function of $t/{\tau}_{\mathrm{R},\mathrm{eq}}$ at two different values of the dimensionless shear rate $\mathrm{Wi}=\dot{\gamma}{\tau}_{R,\mathrm{eq}}$, and dependence on the parameters ${\xi}_{0},\gamma $, and $\epsilon $ for $\alpha =0.4,b=50$.

**Figure 8.**Variation of the growth of the conformation tensor elements (

**a**) c

_{xx}, (

**b**) c

_{xy}, (

**c**) c

_{yy}, and (

**d**) c

_{zz}upon the inception of shear flow as a function of $t/{\tau}_{\mathrm{R},\mathrm{eq}}$ at two different values of the dimensionless shear rate $\mathrm{Wi}=\dot{\gamma}{\tau}_{R,\mathrm{eq}}$, and dependence on the parameters $\alpha ,\epsilon $, and b for of ${\xi}_{0}=0.01$ and $\gamma =1$.

**Figure 9.**Variation of the growth of the (

**a**) shear viscosity, (

**b**) the first normal stress coefficient, and (

**c**) the negative second normal stress coefficient upon the inception of shear flow as a function of $t/{\tau}_{\mathrm{R},\mathrm{eq}}$ at two different values of the dimensionless shear rate $\mathrm{Wi}=\dot{\gamma}{\tau}_{R,\mathrm{eq}}$, and dependence on the parameters ${\xi}_{0},\gamma $, and $\epsilon $ for $\alpha =0.4,b=50$. The dotted dark yellow line in each panel depicts the LVE envelope given by Equation (6).

**Figure 10.**Variation of the growth of the (

**a**) shear viscosity, (

**b**) the first normal stress coefficient, and (

**c**) the negative second normal stress coefficient upon the inception of shear flow as a function of $t/{\tau}_{\mathrm{R},\mathrm{eq}}$ at two different values of the dimensionless shear rate $\mathrm{Wi}=\dot{\gamma}{\tau}_{R,\mathrm{eq}}$, and dependence on the parameters $\alpha ,\epsilon $, and b for of ${\xi}_{0}=0.01$ and $\gamma =1$. Note that two LVE envelopes are provided for $-{\mathsf{\Psi}}_{2}^{+}\left(t\right)$, one for each $\alpha $ value.

**Figure 11.**Model predictions (blue line) for the conformation tensor elements (

**a**) c

_{xx}, (

**b**) c

_{xy}, (

**c**) c

_{yy}, and (

**d**) c

_{zz}in steady shear for the C

_{48}PE melt along with comparison with the NEMD simulation results (red squares).

**Figure 12.**Model predictions (blue line) for the (

**a**) shear viscosity, (

**b**) first normal stress coefficient, and (

**c**) negative second normal stress coefficient in steady shear for the C

_{48}PE melt along with comparison with the NEMD simulation results (red squares).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nikiforidis, V.-M.; Tsalikis, D.G.; Stephanou, P.S.
On the Use of a Non-Constant Non-Affine or Slip Parameter in Polymer Rheology Constitutive Modeling. *Dynamics* **2022**, *2*, 380-398.
https://doi.org/10.3390/dynamics2040022

**AMA Style**

Nikiforidis V-M, Tsalikis DG, Stephanou PS.
On the Use of a Non-Constant Non-Affine or Slip Parameter in Polymer Rheology Constitutive Modeling. *Dynamics*. 2022; 2(4):380-398.
https://doi.org/10.3390/dynamics2040022

**Chicago/Turabian Style**

Nikiforidis, Vasileios-Martin, Dimitrios G. Tsalikis, and Pavlos S. Stephanou.
2022. "On the Use of a Non-Constant Non-Affine or Slip Parameter in Polymer Rheology Constitutive Modeling" *Dynamics* 2, no. 4: 380-398.
https://doi.org/10.3390/dynamics2040022