# Analysis of extrusion foaming of polyolefins using isobutane and CO2

Carlos Sandino, Edith Peuvrel-Disdier, Jean-François Agassant, Séverine A.E. Boyer, Patrice Laure, Geoffrey Hibert, Yves Trolez

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Carlos Sandino, Edith Peuvrel-Disdier, Jean-François Agassant, Séverine A.E. Boyer, Patrice Laure, et al.. Analysis of extrusion foaming of polyolefins using isobutane and CO2. 55ème Congrès du Groupe Français de Rhéologie, Oct 2021, Bordeaux, France. . [hal-03410720](https://hal.science/hal-03410720)

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Submitted on 17 Nov 2022

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Distributed under a Creative Commons Attribution 4.0 International License
# Analysis of extrusion foaming of polyolefins using isobutane and CO₂

C. Sandino, E. Peuvrel-Disdier, J.F. Agassant, S.A.E. Boyer, P. Laure

MINES ParisTech, PSL Research University, CEMEF, UMR CNRS 7635, CS 10207, 06904 Sophia Antipolis cedex (France)

G. Hibert, Y. Trolez

TotalEnergies One Tech Belgium, Zone Industrielle Feluy C, B 7181 Feluy (Belgium)

## Objective

### Physical extrusion foaming

<figure><img src="image_2.png" /></figure>

Better understanding of extrusion foaming

<figure><img src="image_3.png" /></figure>

* Analysis of foaming conditions

* Bubble growth model predictions

* Analysis of pressure conditions along the die

## Materials, foaming conditions

**Two sets of foaming experiments:**

### LDPE grades:

CD0230 and LD0304 from TotalEnergies

Foaming with 5 and 8 wt% isobutane

Range of $T_{max}$ during foaming: 120 to 100 °C at the die exit

CD0230 and LD0304: $T_t = 97°C$ (start 100°C)

### PP experimental grades:

PP A (linear chains) and PP B (branched chains) from TotalEnergies

Foaming with 1 and 2 wt% CO₂

Range of $T_{max}$: 170 to 130°C at the die exit

PP A: $T_t = 97°C$ (start 105°C), PP B: $T_t = 104°C$ (start 110°C)

**Rheological behaviour**

<figure><img src="image_4.png" /></figure>

**Foam characterization:**

* SEM → Cell density (cells/m³)

* Mean bubble size

* Foam density (kg/m³)

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

## Bubble growth modeling

**Model equations and hypotheses [1]**

Growth of a gas bubble in a polymer shell containing polymer and dissolved gas

<figure><img src="image_7.png" /></figure>

$$ \text{Constant cell density: } n_{b,0} \rightarrow S_0 = \sqrt{\frac{3}{4 \pi n_{b,0}}} \quad R_0 = 1 \mu\text{m} \\ c(r, 0) = c_0 $$

<figure><img src="image_8.png" /></figure>

Diffusion equation

Bubble growth

Momentum equation

$$ \frac{\partial c}{\partial t} + \frac{\dot{R} R^2}{r^2} \frac{\partial c}{\partial r} = \frac{D}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial c}{\partial r} \right) $$

$$ \frac{d}{dt}(P_{\text{B}} R^3) = 3 \frac{D}{R} T R^2 \frac{\partial c}{\partial r} \Big|_{r=R} \\ P_{\text{B}} - P_{\text{a}} = \frac{2Y}{R} - \tau_{rr}(S) - \frac{2}{3} \int_{0}^{S_2} \frac{r^3}{r^3} \tau_{rr} - \tau_{\theta\theta} dy $$

Viscoelastic matrix: Multi-Maxwell or Phan-Thien-Tanner model

**Bubble growth prediction**

CD0230

<table><tr><td>%wt isobutane</td><td>8.23</td></tr><tr><td>T<sub>max</sub> (°C)</td><td>104</td></tr><tr><td>n<sub>b,a</sub> (cell/cm³)</td><td>9500</td></tr><tr><td>D (m²/s)</td><td>1.23E-09</td></tr><tr><td>H<sub>a</sub> (mol/(m²s))</td><td>1.818E-03</td></tr></table>

<figure><img src="image_9.png" /></figure>

<table><tr><td>P<sub>foam</sub> (kg/m³)</td><td>922</td></tr><tr><td>P<sub>foam model</sub> (kg/m³)</td><td><b>31.6</b></td></tr><tr><td>P<sub>foam measured</sub> (kg/m³)</td><td>35.4</td></tr><tr><td>Relative error (%)</td><td>10.7</td></tr></table>

<figure><img src="image_10.png" /></figure>

**Foam density → comparison between model predictions and experiments**

<figure><img src="image_11.png" /></figure>

Low density foams → Well predicted by the bubble growth model

PP B + 1% CO₂

<figure><img src="image_12.png" /></figure>

Different densities of PP foams: → Analysis of pressure drop in the die

## Analysis of pressure drop in the die

<figure><img src="image_13.png" /></figure>

Why analysing ΔP in the die ?

Pressure drop in the die = responsible for the foam formation (nucleation + growth)

<figure><img src="image_14.png" /></figure>

$$ \Delta P_{\text{cls}} = \Delta P_{\text{conv}} + \Delta P_{\text{cap}} $$

$$ \begin{align} \Delta P_{\text{conv}} &= \frac{2K L_c}{3m(R_0 - R_1)} \left( \frac{3m + 1}{m} \frac{Q}{\pi} \right)^m \left[ \frac{1}{R_1} - \frac{1}{R_0} \right] \nonumber \\ \Delta P_{\text{cap}} &= 2K L_f \left( \frac{3m + 1}{m} \frac{Q}{\pi} \right)^m \frac{1}{R_1^{(1+3m)}} \tag{2} \end{align} $$

$$ \text{Power law: } K, m \nonumber \\ \text{Eq. 1} $$

$$ \text{If } \Delta P_B > \Delta P_{\text{cls}} \text{ ⇒ Additional contribution: } \Delta P_{\text{el}} = \Delta P_B - \Delta P_{\text{cls}} \tag{Eq. 1} $$

### Steps:

1. Calculation of $\Delta P_{\text{cls}}$ ✓

2. Comparison with $\Delta P_B$ ✓

$$ \text{If } \Delta P_B > \Delta P_{\text{cls}} \text{ ⇒ calculation } \Delta P_{\text{el}} = \Delta P_B - \Delta P_{\text{cls}} \text{ (Eq. 1)} $$

3. If viscoelastic effect, ⇒ $\Delta P_{\text{el}}$ variation with T follow an Arrhenius law

4. If $\Delta P_{\text{el}}$ (T) > Arrhenius law

⇒ Crystallization at the origin of $\Delta P_{\text{el}}$ excess

→ Calculation of $\eta_{\text{el}}$ (Eq. 2)

→ Estimation of a crystallized fraction $\Phi$ (Eq. 3)

$$ \eta_{\text{el}} = \frac{\pi \Delta P_{\text{el}} (1 - \cos \alpha)}{Q \sin^3 \alpha} \frac{R_0^2 R_1^2}{(R_1 + R_0) (\ln \frac{R_1}{R_0})} \tag{Eq. 2} $$

$$ \eta_{\text{susp}} = \eta_{\text{matrix}} \left[ 1 - \frac{\Phi}{B} \right]^{-2} \tag{Eq. 3} $$

B = 0,54 for spherulites [4]

<figure><img src="image_15.png" /></figure>

* Existence of an additional contribution ($\Delta P_{\text{el}}$) in the die

* Low $\Delta P_{\text{el}}$ =f(T) (higher T) follow an Arrhenius dependence

* The presence of a crystalline state in the die could explain the pressure drop excess and help in some cases the foaming process

### References:

[1] Reglero R. et al. (2016). Int. Polym. Proc. 31: 26-36

[2] Agassant J.F. et al., "Mise en forme des polymeres-approche thermomecanique de la plasturie" Lavoisier, (2015)

[3] Cogswell F.N., Polym. Eng. Sci. (1972) 12, 64-73, Y. Demay et al., Int. Polym. Proc. (2018) 3, 363-370

[4] Tanner R.I., J. Non-Newton. Fluid Mech. (2002) 102, 397-408; (2003) 112, 251-268