A One-Dimensional Steady-State Model for Polymer Decomposition into Vapor-Phase Monomer

Schematic (shown for polyoxymethylene as example)

Nomenclature

Symbol	Units	Description
$T$	K	temperature
$x$	m	depth
$k$	W/(m$\cdot$K)	liquid-phase (molten-layer) thermal conductivity
$\rho$	kg/m3	liquid-phase density
${\rm MW}_0$	kg/mol	monomer molecular weight
${\rm MW}$	kg/mol	polymer molecular weight
$A_\beta$	s-1	intrinsic pre-exponential factor for $\beta$-scission
$E_a$	J/mol	intrinsic activation energy of beta scission
$\gamma$		correction factor to the pre-exponential factor
$\Delta_rH$	J/kg	heat of decomposition reaction
$\Delta h_{\rm LH}$	J/kg	latent heat of polymer fusion
$R_u$	J/(mol$\cdot$K)	universal gas constant
$T_S$	K	temperature at liquid-gas interface
$T_{\rm melt}$	K	polymer melting point (temperature at solid-liquid interface)
$\dot{Q}_0''$	J/(s$\cdot$m2)	heat flux at liquid-gas interface
$r_b$	m/s	regression rate of the molten polymer surface

Version 0.9

Assumptions

• Uniform in $y$, $z$ directions. 1D steady state treatment.
• Decomposition occurs in liquid phase only.
• Regression is due to beta-scission to release monomer.
• Radical initiation reaction is neglected.
• Constant $k$, MW, $\rho$, $\gamma$.

Model

An analytical steady-state model for calculating polymer surface regression rate at a given heat flux or surface temperature.

1. Regression rate for given heat flux:

$$r_b=\frac{\dot{Q}_0''}{\rho(\Delta h_{\rm LH}+\frac{\Delta_rH}{{\rm MW}_0})}.$$

2. Regression rate for given surface temperature:

$$r_b=\frac{kE_a}{\Delta h_{\rm LH}\rho R_u}\sqrt{\frac{2\tilde{A}[g(\tilde{\theta}_S)-g(\tilde{\theta}_{\rm melt})]}{(1+\tilde{h})^2-1}},$$

where

$$\begin{gathered} \tilde{A}=\frac{2\Delta_rH\rho A_\beta R_u\gamma}{{\rm MW}kE_a}, \\\ \quad \tilde{\theta}_S=\frac{R_uT_S}{E_a},\\\ \quad \tilde{\theta}_{\rm melt}=\frac{R_uT_{\rm melt}}{E_a},\\\ \quad \tilde{h}=\frac{\Delta_rH}{{\rm MW}_0\Delta h_{\rm LH}}, \\\ g(u)={\rm Ei}\left(-\frac{1}{u}\right)+u\exp\left(-\frac{1}{u}\right),\\\ \quad {\rm Ei}(u)=\int_{-\infty}^u\frac{\exp(t)}{t}{\rm d}t. \end{gathered}$$

Sample parameter values for Polyoxymethylene (POM)

• Decomposition product: formaldehyde (CH₂O)
• $k$ = 0.14 W/m$\cdot$K ¹
• $\rho$ = 1.2 g/cm^{3 2}
• ${\rm MW}_0$ = 30 g/mol
• ${\rm MW}$ = 1$\times 10^5$ g/mol ³
• $A_\beta$ = 1.8$\times 10^{13}$ s^{-1 4}
• $E_a$ = 30 kcal/mol ⁵
• $\Delta_rH$ = 56 kJ/mol ⁶
• $\Delta h_{\rm LH}$ = 150 J/g ⁷
• $T_{\rm melt}$ = 438 K ⁸⁹
• $\gamma$ = 1
• $R_u$ = 8.314 J/mol$\cdot$K

Sample regression rate values

Input: Liquid-gas interface temperature $T_S$:

$T_S$ (K)	$r_b$ (m/s)
800	5.34$\times 10^{-4}$
700	1.22$\times 10^{-4}$
600	1.75$\times 10^{-5}$

Input: Liquid-gas interface heat flux $\dot{Q}_0''$:

$\dot{Q}_0''$ (J/(s$\cdot$m2))	$r_b$ (m/s)
3$\times 10^5$	4.13$\times 10^{-5}$
2$\times 10^5$	8.26$\times 10^{-5}$
1$\times 10^5$	1.24$\times 10^{-4}$

Gas phase kinetics model

In "mechanism" folder: 13 species, 47 reactions. FFCM-2 ¹⁰ formaldehyde sub-model.

• *.inp, *.dat: Chemkin format files
• *.cti, *.yaml: Cantera format files

Contributors

Wendi Dong, Nikolaos Kateris, Nicholas J. Montes, Amitesh S. Jayaraman, Hai Wang

Mechanical Engineering Department, Stanford University, Stanford, California 94305, United States

References

Footnotes

1. S. Luftl, P. Visakh, S. Chandran, Polyoxymethylene handbook: structure, properties, applications and their nanocomposites, John Wiley & Sons, 2014. ↩

2. H. W. Starkweather Jr., G. A. Jones, P. Zoller, The pressure-volume-temperature relationship and the heat of fusion of polyoxymethylene, Journal of Polymer Science Part B: Polymer Physics 26 (2) (1988) 257–266. ↩

3. Estimated, ranging from 1x10⁴ to 2x10⁵ g/mol. ↩

4. L.-S. Tran, J. Pieper, H.-H. Carstensen, H. Zhao, I. Graf, Y. Ju, F. Qi, K. Kohse-Hoinghaus, Experimental and kinetic modeling study of diethylether flames, Proceedings of the Combustion Institute 36 (1) (2017) 1165–1173. ↩

5. G. Berkowicz, T. M. Majka, W. ̇Zukowski, The pyrolysis and combustion of polyoxymethylene in a fluidised bed with the possibility of incorporating CO₂, Energy Conversion and Management 214 (2020) 112888. ↩

6. Group additivity calculation using NIST data. ↩

7. D. Czarnecka-Komorowska, T. Sterzynski, Effect of polyhedral oligomeric silsesquioxane on the melting, structure, and mechanical behavior of polyoxymethylene, Polymers 10 (2) (2018) 203. ↩

8. DURACON® POM report. https://www.polyplastics.com/en/support/mold/duracon/pom04c.html#:~:text=The%20melting%20point%20of%20DURACON,cylinder%20temperature%20(front%20section). ↩

9. K. Pielichowska, The influence of molecular weight on the properties of polyacetal/hydroxyapatite nanocomposites. Part 1. Microstructural analysis and phase transition studies, Journal of Polymer Research 19 (2) (2012) 9775. ↩

10. Y. Zhang, W. Dong, L. Vandewalle, R. Xu, G.P. Smith and H. Wang, Foundational Fuel Chemistry Model Version 2.0 (FFCM-2), https://web.stanford.edu/group/haiwanglab/FFCM2, 2023. ↩