# Assessment of the Tumbling-Snake Model against Linear and Nonlinear Rheological Data of Bidisperse Polymer Blends

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Stress Tensor

## 3. Small Shear Rate Expansion in the Stationary State

#### 3.1. Stationary Regime, Constant $\epsilon $

#### 3.2. Stationary Regime, Variable $\epsilon $

## 4. Linear Viscoelastic Regime

## 5. Brownian Dynamics Results

#### 5.1. Linear Viscoelastic Behavior

#### 5.2. Non-Linear Regime

#### 5.3. Comparison with Experimental Data

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Doi, M.; Edwards, S.F. Dynamics of concentrated polymer systems. 1. Brownian-motion in equilibrium state. J. Chem. Soc. Faraday Trans. 2
**1978**, 74, 1789–1801. [Google Scholar] [CrossRef] - Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Clarendon: Oxford, UK, 1986. [Google Scholar]
- De Gennes, P.G. Reptation of a polymer chain in presence of fixed obstacles. J. Chem. Phys.
**1971**, 55, 572–579. [Google Scholar] [CrossRef] - Watanabe, H. Viscoelasticity and dynamics of entangled polymers. Prog. Polym. Sci.
**1999**, 24, 1253. [Google Scholar] [CrossRef] - McLeish, T.C.B. Tube theory of entangled polymer dynamics. Adv. Phys.
**2002**, 51, 1379–1527. [Google Scholar] [CrossRef][Green Version] - Stephanou, P.S.; Mavrantzas, V.G. Quantitative predictions of the linear viscoelastic properties of entangled polyethylene and polybutadiene melts via modified versions of modern tube models on the basis of atomistic simulation data. J. Non-Newton. Fluid Mech.
**2013**, 200, 111–130. [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] [PubMed] - Marrucci, G.; Grizzuti, N. Fast flows of concentrated polymers—Predictions of the tube model on chain stretching. Gazz. Chim. Ital.
**1988**, 118, 179–185. [Google Scholar] - Ianniruberto, G.; Marrucci, G. A simple constitutive equation for entangled polymers with chain stretch. J. Rheol.
**2001**, 45, 1305–1318. [Google Scholar] [CrossRef] - Stephanou, P.S.; Tsimouri, I.C.; Mavrantzas, V.G. Flow-induced orientation and stretching of entangled polymers in the framework of nonequilibrium thermodynamics. Macromolecules
**2016**, 49, 3161–3173. [Google Scholar] [CrossRef] - Marrucci, G. Dynamics of entanglements: A nonlinear model consistent with the Cox-Merz rule. J. Non-Newton. Fluid Mech.
**1996**, 62, 279–289. [Google Scholar] [CrossRef] - Ianniruberto, G.; Marrucci, G. On compatibility of the Cox-Merz rule with the model of Doi and Edwards. J. Non-Newton. Fluid Mech.
**1996**, 65, 241–246. [Google Scholar] [CrossRef] - Ianniruberto, G.; Marrucci, G. Flow-induced orientation and stretching of entangled polymers. Philos. Trans. R. Soc. A
**2003**, 361, 677–687. [Google Scholar] - Pattamaprom, C.; Larson, R.G. Constraint Release Effects in Monodisperse and Bidisperse Polystyrenes in Fast Transient Shearing Flows. Macromolecules
**2001**, 34, 5229–5237. [Google Scholar] [CrossRef] - Pearson, D.S.; Kiss, A.D.; Fetters, L.J.; Doi, M. Flow-induced birefringence of concentrated polyisoprene solutions. J. Rheol.
**1989**, 33, 517–535. [Google Scholar] [CrossRef] - Mead, D.W.; Larson, R.G.; Doi, M. A Molecular Theory for Fast Flows of Entangled Polymers. Macromolecules
**1998**, 31, 7895–7914. [Google Scholar] [CrossRef] - Ye, X.; Larson, R.G.; Pattamaprom, C.; Sridhar, T. Extensional properties of monodisperse and bidisperse polystyrene solutions. J. Rheol.
**2003**, 47, 443–448. [Google Scholar] [CrossRef] - Leygue, A.; Bailly, C.; Keunings, R. A tube-based constitutive equation for polydisperse entangled linear polymers. J. Non-Newton. Fluid Mech.
**2006**, 136, 1–16. [Google Scholar] [CrossRef][Green Version] - Read, D.J.; Jagannathan, K.; Sukumaran, S.K.; Auhl, D. A full-chain constitutive model for bidisperse blends of linear polymers. J. Rheol.
**2012**, 56, 823–873. [Google Scholar] [CrossRef] - Graham, R.S.; Likhtman, A.E.; McLeish, T.C.B.; Milner, S.T. Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release. J. Rheol.
**2003**, 47, 1171–1200. [Google Scholar] [CrossRef] - Bird, R.B.; Armstrong, R.C.; Hassager, O. Dynamics of Polymeric Liquids: Vol. 2, Kinetic Theory; John Wiley & Sons: New York, NY, USA, 1987. [Google Scholar]
- Curtiss, C.F.; Bird, R.B. A kinetic-theory for polymer melts. 1. The equation for the single-link orientational distribution function. J. Chem. Phys.
**1981**, 74, 2016–2025. [Google Scholar] [CrossRef] - Curtiss, C.F.; Bird, R.B. A kinetic-theory for polymer melts. 2. The stress tensor and the rheological equation of state. J. Chem. Phys.
**1981**, 74, 2026–2033. [Google Scholar] [CrossRef] - Stephanou, P.S.; Schweizer, T.; Kröger, M. Communication: Appearance of undershoots in start-up shear: Experimental findings captured by tumbling-snake dynamics. J. Chem. Phys.
**2017**, 146, 161101. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bird, R.B.; Saab, H.H.; Curtiss, C.F. A kinetic-theory for polymer melts. 4. Rheological properties for shear flows. J. Chem. Phys
**1982**, 77, 4747–4757. [Google Scholar] [CrossRef] - Bird, R.B.; Saab, H.H.; Curtiss, C.F. A kinetic-theory for polymer melts. 3. Elongational flows. J. Phys. Chem.
**1982**, 86, 1102–1106. [Google Scholar] [CrossRef] - Stephanou, P.S.; Kröger, M. Solution of the complete Curtiss-Bird model for polymeric liquids subjected to simple shear flow. J. Chem. Phys.
**2016**, 144, 124905. [Google Scholar] [CrossRef] - 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] [PubMed] - Stephanou, P.S.; Kröger, M. From intermediate anisotropic to isotropic friction at large strain rates to account for viscosity thickening in polymer solutions. J. Chem. Phys.
**2018**, 148, 184903. [Google Scholar] [CrossRef] [PubMed] - Stephanou, P.S.; Kröger, M. Tumbling-Snake Model for Polymeric Liquids Subjected to Biaxial Elongational Flows with a Focus on Planar Elongation. Polymers
**2018**, 10, 329. [Google Scholar] [CrossRef] - Kröger, M. Models for Polymeric and Anisotropic Liquids; Springer: New York, NY, USA, 2005; Volume 675. [Google Scholar]
- Luap, C.; Müller, C.; Schweizer, T.; Venerus, D.C. Simultaneous stress and birefringence measurements during uniaxial elongation of polystyrene melts with narrow molecular weight distribution. Rheol. Acta
**2005**, 45, 83–91. [Google Scholar] [CrossRef] - Schweizer, T.; Hostettler, J.; Mettler, F. A shear rheometer for measuring shear stress and both normal stress differences in polymer melts simultaneously: The MTR 25. Rheol. Acta
**2008**, 47, 943–957. [Google Scholar] [CrossRef] - Auhl, D.; Ramirez, J.; Likhtman, A.E.; Chambon, P.; Fernyhough, C. Linear and nonlinear shear flow behavior of monodisperse polyisoprene melts with a large range of molecular weights. J. Rheol.
**2008**, 52, 801–835. [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] - Sefiddashti, M.H.N.; Edwards, B.J.; Khomami, B. Individual chain dynamics of a polyethylene melt undergoing steady shear flow. J. Rheol.
**2015**, 59, 1–35. [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 analyis of polymer rotational dynamics. Sci. Rep.
**2016**, 6, 19127. [Google Scholar] [CrossRef] [PubMed] - Huang, Q.; Alvarez, N.J.; Matsumiya, Y.; Rasmussen, H.K.; Watanabe, H.; Hassager, O. Extensional rheology of entangled polystyrene solutions suggests importance of nematic interactions. ACS Macro Lett.
**2013**, 2, 741–744. [Google Scholar] [CrossRef] - Huang, Q.; Mednova, O.; Rasmussen, H.K.; Alvarez, N.J.; Skov, A.L.; Almdal, K.; Hassager, O. Concentrated polymer solutions are different from melts: Role of entanglement molecular weight. Macromolecules
**2013**, 46, 5026–5035. [Google Scholar] [CrossRef] - Huang, Q.; Hengeller, L.; Alvarez, N.J.; Hassager, O. Bridging the gap between polymer melts and solutions in extensional rheology. Macromolecules
**2015**, 48, 4158–4163. [Google Scholar] [CrossRef] - Schieber, J.D.; Curtiss, C.F.; Bird, R.B. Kinetic Theory of Polymer melts. 7. Polydisprese Effects. Ind. Chem. Fundam.
**1986**, 25, 471–475. [Google Scholar] [CrossRef] - Schieber, J.D. Kinetic theory of polymer melts. VIII. Rheological properties of polydisperse mixtures. J. Chem. Phys.
**1987**, 87, 4917–4927. [Google Scholar] [CrossRef] - Schieber, J.D. Kinetic theory of polymer melts. IX. Comparisons with experimental data. J. Chem. Phys.
**1987**, 87, 4928–4936. [Google Scholar] [CrossRef] - Rubinstein, M.; Colby, R.H. Polymer Physics; Oxford University Press: Oxford, UK, 2003. [Google Scholar]
- Stephanou, P.S.; Baig, C.; Mavrantzas, V.G. Toward an improved description of constraint release and contour length fluctuations in tube models for entangled polymer melts guided by atomistic simulations. Macromol. Theory Simul.
**2011**, 20, 752–768. [Google Scholar] [CrossRef] - Stephanou, P.S.; Baig, C.; Mavrantzas, V.G. Projection of atomistic simulation data for the dynamics of entangled polymers onto the tube theory: Calculation of the segment survival probability function and comparison with modern tube models. Soft Matter
**2011**, 7, 380–395. [Google Scholar] [CrossRef] - Kröger, M.; Hess, S. Viscoelasticity of polymeric melts and concentrated solutions. The effect of flow-induced alignment of chain ends. Physica A
**1993**, 195, 336–353. [Google Scholar] [CrossRef] - Fang, J.; Kröger, M.; Öttinger, H.C. A thermodynamically admissible reptation model for fast flows of entangled polymers. II. Model predictions for shear and extensional flows. J. Rheol.
**2000**, 44, 1293–1317. [Google Scholar] [CrossRef] - Öttinger, H.C. Thermodynamically admissible reptation models with anisotropic tube cross sections and convective constraint release. J. Non-Newton. Fluid Mech.
**2000**, 89, 165–185. [Google Scholar] [CrossRef]

**Figure 1.**Model predictions for the zero-rate shear viscosity (

**a**) and the first normal stress coefficient (

**b**), scaled with their corresponding values for the pure long component, for various model parameters as a function of ${\varphi}_{L}$. ${N}_{S}$ and ${N}_{L}$ denote the polymerization degree of the short and long component, respectively, and $\beta $ is the chain constraint exponent.

**Figure 2.**Model predictions for $|{\eta}^{*}|$, scaled with the zero-rate viscosity of the pure long component, as a function of the dimensionless frequency, $\tilde{\omega}=\omega {\lambda}_{L,p}$, when (

**a**) ${\epsilon}_{0}^{\prime}=0$, ${N}_{S}/{N}_{L}=0.25$, and $\beta =0$, (

**b**) ${\epsilon}_{0}^{\prime}=0.5$, ${N}_{S}/{N}_{L}=0.25$, and $\beta =0$, (

**c**), ${\epsilon}_{0}^{\prime}=0.5{N}_{S}/{N}_{L}=\beta =0$, and (

**d**) ${\epsilon}_{0}^{\prime}={N}_{S}/{N}_{L}=\beta =0.5$, for various values of ${\varphi}_{L}$.

**Figure 3.**Model predictions for the storage modulus, scaled with G, as a function of the dimensionless frequency, $\tilde{\omega}=\omega {\lambda}_{L,p}$, when (

**a**) ${\epsilon}_{0}^{\prime}=0$, ${N}_{S}/{N}_{L}=0.25$, and $\beta =0$, (

**b**) ${\epsilon}_{0}^{\prime}=0.5$, ${N}_{S}/{N}_{L}=0.25$, and $\beta =0$, (

**c**), ${\epsilon}_{0}^{\prime}=0.5{N}_{S}/{N}_{L}=\beta =0$, and (

**d**) ${\epsilon}_{0}^{\prime}={N}_{S}/{N}_{L}=\beta =0.5$, for various values of ${\varphi}_{L}$.

**Figure 4.**Model predictions for the loss modulus, scaled with G, as a function of the dimensionless frequency, $\tilde{\omega}=\omega {\lambda}_{L,p}$, when (

**a**) ${\epsilon}_{0}^{\prime}=0$, ${N}_{S}/{N}_{L}=0.25$, and $\beta =0$, (

**b**) ${\epsilon}_{0}^{\prime}=0.5$, ${N}_{S}/{N}_{L}=0.25$, and $\beta =0$, (

**c**), ${\epsilon}_{0}^{\prime}=0.5{N}_{S}/{N}_{L}=\beta =0$, and (

**d**) ${\epsilon}_{0}^{\prime}={N}_{S}/{N}_{L}=\beta =0.5$, for various values of ${\varphi}_{L}$.

**Figure 5.**Predictions for the reduced shear viscosity, using the zero-rate viscosity of the pure long component, as a function of dimensionless shear rate and for various values of volume fraction ${\varphi}_{L}$ from our analytical result Equation (8), shown by solid lines up to about $\mathrm{Wi}$ = 10, and from the BD simulations (symbols) for (

**a**) the DE model (${\epsilon}_{0}^{\prime}={\epsilon}_{0}=0$), (

**b**) the analytically solvable Curtiss-Bird model (${\epsilon}_{0}^{\prime}=0,{\epsilon}_{0}=0.1$), and the tumbling snake model, when ${\epsilon}_{0}^{\prime}=0.5$, and (

**c**), ${\epsilon}_{0}=0$, and (

**d**), ${\epsilon}_{0}=0.1$; in all cases ${N}_{S}/{N}_{L}=0.5$ and $\beta =0$. Note that different colors were employed for the BD simulations (symbols) and for the analytical results at small shear rates (lines) for better visibility.

**Figure 6.**Predictions for the reduced shear viscosity as a function of dimensionless shear rate and for various values of volume fraction ${\varphi}_{L}$ from our analytical result Equation (8) (lines) and from the BD simulations (symbols) for (

**a**) $\beta =0$ and (

**b**) $\beta =0.5$, keeping ${N}_{S}/{N}_{L}=0.5$, ${\epsilon}_{0}^{\prime}=0.5$, and ${\epsilon}_{0}=0.1$ constant.

**Figure 7.**Predictions for the reduced first normal stress coefficient, using the zero-rate first normal stress coefficient of the pure long component, as a function of dimensionless shear rate and for various values of volume fraction ${\varphi}_{L}$ from our analytical result Equation (8) (lines) and from the BD simulations (symbols) for (

**a**) ${\epsilon}_{0}^{\prime}={\epsilon}_{0}=0$, (

**b**) ${\epsilon}_{0}^{\prime}=0,{\epsilon}_{0}=0.1$, (

**c**) ${\epsilon}_{0}^{\prime}=0.5$, ${\epsilon}_{0}=0$, and (

**d**) ${\epsilon}_{0}^{\prime}=0.5$, ${\epsilon}_{0}=0.1$; in all cases ${N}_{S}/{N}_{L}=0.5$ and $\beta =0$. Note that different colors were employed for the BD simulations (symbols) and for the analytical results at small shear rates (lines) for better visibility.

**Figure 8.**Predictions for the reduced first normal stress coefficient as a function of dimensionless shear rate and for various values of volume fraction ${\varphi}_{L}$ from our analytical result Equation (8) (lines) and from the BD simulations (symbols) for (

**a**) $\beta =0$ and (

**b**) $\beta =0.5$, keeping ${N}_{S}/{N}_{L}=0.5$, ${\epsilon}_{0}^{\prime}=0.5$, and ${\epsilon}_{0}=0.1$ constant.

**Figure 9.**Predictions for the reduced second normal stress coefficient, using the zero-rate second normal stress coefficient of the pure long component, as a function of dimensionless shear rate and for various values of volume fraction ${\varphi}_{L}$ from our analytical result Equation (8) (lines) and from the BD simulations (symbols) for (

**a**) ${\epsilon}_{0}^{\prime}={\epsilon}_{0}=0$, (

**b**) ${\epsilon}_{0}^{\prime}=0,{\epsilon}_{0}=0.1$, (

**c**) ${\epsilon}_{0}^{\prime}=0.5$, ${\epsilon}_{0}=0$, and (

**d**) ${\epsilon}_{0}^{\prime}=0.5$, ${\epsilon}_{0}=0.1$; in all cases ${N}_{S}/{N}_{L}=0.5$ and $\beta =0$. Note that different colors were employed for the BD simulations (symbols) and for the analytical results at small shear rates (lines) for better visibility.

**Figure 10.**Predictions for the reduced second normal stress coefficient as a function of dimensionless shear rate and for various values of volume fraction ${\varphi}_{L}$ from our analytical result Equation (8) (lines) and from the BD simulations (symbols) for (

**a**) $\beta =0$ and (

**b**) $\beta =0.5$, keeping ${N}_{S}/{N}_{L}=0.5$, ${\epsilon}_{0}^{\prime}=0.5$, and ${\epsilon}_{0}=0.1$ constant.

**Figure 11.**Comparison of the predictions of the tumbling-snake model (lines) against experimental data (symbols) [14] for (

**a**) the zero-shear viscosity (where also the prediction for $\beta =0$ is shown), and (

**b**) the $|{\eta}^{*}|$; the parameter values are ${N}_{L}=42$, ${N}_{S}=14.7$, $\beta =0.626$, ${\eta}_{0,L}=1.8\times {10}^{4}$ Pa·s, ${\epsilon}_{0}^{\prime}={10}^{-3}$ and ${\lambda}_{L,p}=450$ s (the latter two needed only for $|{\eta}^{*}|$).

**Figure 12.**Comparison of the predictions of the tumbling-snake model (lines) against experimental data (symbols) [14] for (

**a**) the shear viscosity, and (

**b**) the first normal stress difference, as a function of shear rate; in addition to the parameter values mentioned in Figure 11, a volume-fraction dependent ${\epsilon}_{0}$ is employed of the form ${\epsilon}_{0}=0.18(1-2{\varphi}_{L}/3)$.

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stephanou, P.S.; Kröger, M. Assessment of the Tumbling-Snake Model against Linear and Nonlinear Rheological Data of Bidisperse Polymer Blends. *Polymers* **2019**, *11*, 376.
https://doi.org/10.3390/polym11020376

**AMA Style**

Stephanou PS, Kröger M. Assessment of the Tumbling-Snake Model against Linear and Nonlinear Rheological Data of Bidisperse Polymer Blends. *Polymers*. 2019; 11(2):376.
https://doi.org/10.3390/polym11020376

**Chicago/Turabian Style**

Stephanou, Pavlos S., and Martin Kröger. 2019. "Assessment of the Tumbling-Snake Model against Linear and Nonlinear Rheological Data of Bidisperse Polymer Blends" *Polymers* 11, no. 2: 376.
https://doi.org/10.3390/polym11020376