# Predicting High-Density Polyethylene Melt Rheology Using a Multimode Tube Model Derived Using Non-Equilibrium Thermodynamics

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## Abstract

**:**

## 1. Introduction

## 2. The Constitutive Model

#### 2.1. State Variables

**c**, following the method of Stephanou et al. [23,35], is considered to be made dimensionless through $\tilde{c}=Kc/{k}_{B}T$, where K denotes the spring constant of the Hookean dumbbells that represents the entanglement strands at equilibrium, k

_{B}the Boltzmann constant, and T denotes the absolute temperature [36]. The conformation tensor $\tilde{c}$ refers to one entanglement strand, and at equilibrium (zero flow field applied), it coincides with the unit tensor. To characterize the multiple modes of the polymer chains, N conformation tensors are considered, with one tensor being considered for each mode [36]. Finally, the momentum density

**M**, which is the hydrodynamic variable, is further considered, meaning that, overall, the vector of state variables is expressed as $x=\{M,{c}^{(1)},{c}^{(2)},\dots ,{c}^{(i)},\dots ,{c}^{(N)}\}$.

#### 2.2. System Hamiltonian

^{th}mode, and

**I**is the unit tensor. The partial derivative of the potential with respect to the trace of the conformation tensor defines the (dimensionless) effective spring constant [37,38] for the i

^{th}mode

^{th}mode. As shown by the study of Stephanou et al. [35], ${b}_{e}=3\left[{\left(0.82\right)}^{2}/{C}_{\infty}\right]\left({M}_{e}/{M}_{0}\right)$ (when all FENE parameters are considered equal), where ${C}_{\infty}$ is the polymer characteristic ratio at infinite chain length, M

_{e}is the entanglement molecular weight, and M

_{0}is the average molar mass of a monomer. For example, for PS melts, b

_{e}= 54 [35].

#### 2.3. The Poisson and Dissipation Brackets

#### 2.4. The Matrices L and Λ

^{th}mode, which is considered to be half of the corresponding reptation time, ${\tau}_{\mathrm{CR}}^{(i)}={\scriptscriptstyle \frac{1}{2}}{\tau}_{d}^{(i)}$ [39] [we note that this time coincides with the CCR relaxation time at equilibrium, as shown in Equation (10)], and ${\tau}_{R}^{(i)}\left(tr\tilde{c}\right)$ is the Rouse relaxation time of the i

^{th}mode,

^{th}mode, which is given as ${\tau}_{d}^{(i)}=3Z{\tau}_{R,\mathrm{eq}}^{(i)}$ [19], and ${k}^{(i)}$ is the Extended White–Metzner (EWM) exponent [36] for the i

^{th}mode. We note that for the Rouse time, a shear rate dependency through the use of the trace of the conformation tensor of each mode is considered. The functions ${f}_{\mathrm{rep}}^{(i)}\left(tr{\tilde{c}}^{(i)}\right)$ and ${f}_{\mathrm{Rouse}}^{(i)}\left(tr{\tilde{c}}^{(i)}\right)$ are scalar functions of the trace of the conformation tensor, as defined via the following equation [23]:

^{th}mode. For the (dimensionless) mobility tensor ${\tilde{\mathsf{\beta}}}^{(i)}$ of the i

^{th}mode, the Giesekus’ postulate ${\tilde{\mathsf{\beta}}}^{(i)}=I+{\alpha}^{(i)}{\tilde{\mathsf{\sigma}}}^{(i)}$ is used [37] with ${\tilde{\mathsf{\sigma}}}^{(i)}={\mathsf{\sigma}}^{(i)}/{G}_{i}$, and ${\alpha}^{(i)}$ is the anisotropic mobility (Giesekus) parameter of the i

^{th}mode. Then, with ${\Lambda}_{\alpha \beta \gamma \epsilon}^{(ii)}={\Lambda}_{\alpha \beta \gamma \epsilon}^{\mathrm{rep},(ii)}+{\Lambda}_{\alpha \beta \gamma \epsilon}^{\mathrm{Rouse},(ii)}$, we obtain

^{th}mode. This parameter is important, as it allows for the prediction of a transient stress undershoot (following the overshoot) at high shear rates [23].

#### 2.5. Thermodynamic Admissibility

^{th}mode. Obviously, since the conditions $0\le {\alpha}^{(i)}\left(1-{\xi}^{(i)}\right)<1,0\le {\xi}^{(i)}<1,\forall i$ and ${\beta}_{ccr}^{(i)}\ge 0,\forall i$ [23] guarantee that each term of the summation is positive, the sum as a whole is also positive, meaning that the multimode version of the Stephanou et al. model [23] presented in this work is thermodynamically admissible.

#### 2.6. Conformation Tensor Evolution Equation

^{th}mode is given in Equation (10), and the (dimensionless) effective spring constant is given in Equation (4c). Finally, the expression for the polymeric stress tensor is obtained by substituting Equations (4b) and (11) into Equation (8) as follows:

## 3. Asymptotic Behavior of the Model in Steady State Shear

## 4. Results and Discussion

_{0}= 14 g/mol, whereas C

_{∞}= 7.3 and M

_{e}= 828 g/mol (see Table 3.3, p. 151 of Ref. [40]). These values yield b

_{e}= 16.34. We will compare against the experimental data of Konaganti et al. [15] that have performed rheological measurements of the sample HDPE-1 (reported by the same group [41]), for which M

_{w}= 206 kg/mol; thus, the number of entanglements is equal to Z ≈ 249 >> 1. The relaxation spectrum is the same as the one used by Konaganti et al. [15] (see their paper’s Table 2 for T = 200 °C, though is also provided in Table 1), and it was obtained by fitting the expressions of the storage and loss moduli, which are shown in Equation (17), with the corresponding experimental data. The comparison against the experimental storage and loss moduli is shown in Figure 1.

_{ccr}= 4 × 10

^{−4}, and k = −3.5. For comparison, we also depict, in the following figures, the predictions of the multimode Giesekus model [which is a special case of our model in which β

_{ccr}= ξ = k = 0 and b

_{e}infinite or the function h = 1 in Equation (4c)] with α = 0.3 and the relaxation spectrum of Table 1.

#### 4.1. Comparison with Start-Up Shear Flow Data

_{e}and the anisotropic mobility (Giesekus) parameter α. As mentioned above, the former parameter is not a free parameter, as it is directly dictated by structural parameters. The latter parameter is a free parameter, and by increasing its value, the start-up shear viscosity overshoots noted at the two larger shear rates (0.5 1/s and 1 1/s) shift downwards and broaden (results not shown), thus more closely agreeing with the experimental data; however, the good comparison identified at the smaller shear rate (0.05 1/s) is reduced. This result might hint that the parameter α should not be a constant, but should increase with the applied strain rate. Similar arguments have been put forth and resulted in a variable non-affine/slip parameter proposed by Nikiforidis et al. [42] and a variable link tension coefficient proposed by Stephanou and Kröger [26]. We note that although a non-zero value of ξ is employed, the undershoots produced are too small to be noted via the scale used in Figure 2. Although no experimental data are provided, we provide the corresponding prediction of the growth of the first and second normal stress coefficients in Figure 3, as well as the steady-state values of all viscometric material functions in Figure 4.

#### 4.2. Comparison with Start-Up Uniaxial Elongational Flow Data

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Model prediction of the growth of the shear viscosity prediction (lines) upon the inception of the shear flow at three different dimensionless shear rates, along with comparison with the experimental data (symbols) considered in Ref. [15]. In panel (

**b**), we depict the same comparison for the multimode Giesekus model. The thick black line depicts the LVE envelope, which is shown in Equation (15a).

**Figure 3.**Model predictions (red, blue, and orange lines) of the growth of the first (

**a**) and second (

**b**) normal stress coefficients upon the inception of shear flow. The thick black lines depict the LVE envelope, which is shown in Equations (15b) and (15c). The s parameter values are the same as those used in in Figure 2. The multimode Giesekus model’s predictions are also depicted (red, blue, and orange dashed lines). The thick black dashed lines depict the LVE envelope, which are again shown in Equations (15b) and (15c).

**Figure 4.**Model predictions of the steady-state (

**a**) shear viscosity and the (

**b**) first and (

**c**) second normal stress coefficients of the HDPE-1 sample. The parameter values are the same as those used in Figure 2. The multimode Giesekus model prediction is also depicted (black dashed lines).

**Figure 5.**(

**a**) Comparison between the model predictions (lines) and the experimental measurements (symbols) of Konaganti et al. [15] for the growth of the elongational viscosity as a function of time for several stretch rates. The thick black line depicts the LVE envelope, as given in Equation (16). The sparameter values are the same as those used in Figure 2 except ξ = 0. In panel (

**b**), we depict the same comparison for the multimode Giesekus model.

**Figure 6.**Model prediction of the steady-state uniaxial elongational viscosity. The parameter values are the same as those used in Figure 2 except ξ = 0. The multimode Giesekus model prediction is also depicted (black dashed line).

**Table 1.**Relaxation spectrum [15].

Mode | ${\tilde{\mathit{G}}}_{\mathit{e}}^{(\mathit{i})}\text{}(\mathbf{Pa})$ | ${\mathit{\tau}}_{\mathbf{CR}}^{(\mathit{i})}\text{}(\mathbf{s})$ |
---|---|---|

1 | 387,808 | 0.00086 |

2 | 185,307 | 0.0075 |

3 | 93,338 | 0.0548 |

4 | 37,766 | 0.403 |

5 | 12,934 | 2.99 |

6 | 5025 | 30.78 |

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**MDPI and ACS Style**

Konstantinou, P.C.; Stephanou, P.S.
Predicting High-Density Polyethylene Melt Rheology Using a Multimode Tube Model Derived Using Non-Equilibrium Thermodynamics. *Polymers* **2023**, *15*, 3322.
https://doi.org/10.3390/polym15153322

**AMA Style**

Konstantinou PC, Stephanou PS.
Predicting High-Density Polyethylene Melt Rheology Using a Multimode Tube Model Derived Using Non-Equilibrium Thermodynamics. *Polymers*. 2023; 15(15):3322.
https://doi.org/10.3390/polym15153322

**Chicago/Turabian Style**

Konstantinou, Pavlina C., and Pavlos S. Stephanou.
2023. "Predicting High-Density Polyethylene Melt Rheology Using a Multimode Tube Model Derived Using Non-Equilibrium Thermodynamics" *Polymers* 15, no. 15: 3322.
https://doi.org/10.3390/polym15153322