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

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## Abstract

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## 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

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**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 ${\epsilon}_{0}=0.18(1-2{\varphi}_{L}/3)$.

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**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