Modelling random scission of linear polymers

Abstract

A mathematical model for the random scission of linear polymers is presented. The model takes the form of a set of ordinary differential equations which describe the evolution of the MW distribution as a function of the fraction of bonds broken. An exact solution, valid for large initial MW, is derived from the equations and compared with results from a Monte-Carlo type simulation. Isothermal thermogravimetric experiments using polyethylene are used to suggest a relationship between the rate of bond breaking and temperature and this is then used to compare model predictions for the rate of degradation with constant heating rate thermogravimetric experiments. Excellent agreement is found between theoretical predictions and experimental results for the case of a standard sample of polyethylene with number-average MW 2015.

Introduction

At the heart of modelling ignition and combustion behaviour of solids (as well as other important processes) is a general description of the thermal degradation of the material. For materials such as polymers, four thermal degradation mechanisms are thought to exist, namely random scission, end-chain scission (or unzipping), chain stripping and cross linking [1]. Traditional descriptions of these mechanisms have relied on applying one or more kinetic rate equations of the form

$$\text{d}\text{μ}\text{d}\text{t}\text{=−μ}{}^{\mathrm{v}}\text{κ}\text{T}\text{,}$$

where μ is the mass fraction of a particular component, ν is the reaction order and κ is usually assumed to be of Arrhenius form $\text{κ}\text{T}\text{=A}\text{exp}\text{−T}{}_{\mathrm{<mtext>A</mtext>}}\text{/T}$ [2], [3]. Note that T_A is also sometimes written in terms of an activation energy E_A, as T_A=E_A/R, where R is the gas constant. This approach essentially borrows ideas from statistical mechanics and the description of reactions between gas particles [4] and attempts to apply them to the degradation of solids with little or even no theoretical justification. Having stated this, it is proper to redress the balance a little and record the fact that these descriptions often do reproduce experimental data such as thermogravimetric analyses well. For example, the work of Bockhorn and collaborators [5], [6], [7] has produced good results for a number of polymers.

For the case of random scission, alternative models, which are sounder in theory, exist. For example, Emsley and Heywood [8] used a numerical Monte-Carlo type simulation to investigate degradation of linear polymers where a number of bond-breaking regimes were investigated. Platowski and Reichert [9], again using Monte Carlo methods, investigated modelling the kinetics of polymerisation reactions (the reverse process of the present subject admittedly, but still relevant in concept). Other more involved treatments include molecular dynamics simulations for depolymerisation of simple polymers [10], [11], [12] and the recent contribution of Doruker et al., [13] for the random scission of C–C bonds in polyethylene.

The present work extends the approach of Emsley and Heywood [8]. A mathematical model is developed in the form of a set of ordinary differential equations describing the evolution of the MW distribution for a linear polymer undergoing scission by a random process. An exact solution is derived and then used to compare predictions with Monte Carlo type simulations and thermogravimetric experiments using polyethylene.

For the purposes of the mathematical model, the polymer molecule will be viewed as consisting of a linear chain of repeat units linked by bonds which may break during the degradation process. Consider a large number N(r) of these idealised molecules initially consisting of n repeat units and n−1 bonds between repeat units, where r is the fraction of bonds broken at any stage. Let N_m(r) denote the number of molecules in the distribution with m repeat units and let f_m(r) be the corresponding relative frequency, i.e. $\text{f}{}_{\mathrm{m}}\text{r}\text{=N}{}_{\mathrm{m}}\text{r}\text{/N}\text{r}$. Thus the total number of unbroken bonds B(r) in the distribution will be $\text{B}\text{r}\text{=}\text{∑}\text{m=1}\text{n}\text{m−1}\text{N}{}_{\mathrm{m}}\text{r}$ and r is given implicitly by $\text{r=1−B}\text{r}\text{/B}\text{0}$. Now, defining $\text{m}\text{̄}\text{r}$ in an analogous fashion to the number-average molecular weight, i.e. $\text{m}\text{̄}\text{r}\text{=}\text{∑}\text{m=1}\text{n}\text{mf}{}_{\mathrm{m}}\text{r}$, it follows that $\text{B}\text{r}\text{=N}\text{r}$ $\text{m}\text{̄}\text{r}\text{−1}$ and so we can write

$$\text{1−r=}\text{N}\text{r}\text{m}\text{̄}\text{r}\text{−1}\text{N}\text{0}\text{m}\text{̄}\text{0}\text{−1}\text{.}$$

Now since each time a bond is broken the total number of molecules in the distribution increases by 1, so it follows that N increases linearly with r according to $\text{N}\text{r}\text{=N}\text{0}\text{+rB}\text{0}$, and so

$$\text{N}\text{r}\text{N}\text{0}\text{=1+r}\text{m}\text{̄}\text{0}\text{−1}\text{.}$$

Using this last relation with (2) gives

$$\text{m}\text{̄}\text{r}\text{=}\text{m}\text{̄}\text{0}\text{1+r}\text{m}\text{̄}\text{0}\text{−1}\text{,}$$

and we see that, as expected, $\text{N}\text{r}\text{m}\text{̄}\text{r}\text{=}\text{constant}\text{=}$ $\text{N}\text{0}\text{m}\text{̄}\text{0}$.

The degree of polymerisation χ(r) may be defined as the ratio of the initial number of repeat units in the distribution to the number of molecules in the distribution, i.e. $\text{χ}\text{r}\text{=N(0)}\text{m}\text{̄}\text{0}\text{/N}\text{r}$. Using Eq. (3) for the ratio $\text{N}\text{r}\text{/N}\text{0}$, it follows that $\text{χ}\text{r}\text{=}\text{m}\text{̄}\text{r}$ and so

$$\text{1}\text{χ}\text{r}\text{−}\text{1}\text{χ}\text{0}\text{=}\text{m}\text{̄}\text{0}\text{−1}\text{r}\text{m}\text{̄}\text{0}\text{,}$$

i.e. that the reciprocal of the degree of polymerisation is proportional to r. Hence for degradation processes in which r∝t, Ekenstam's relation [8] is recovered. Furthermore, we see that for distributions where $\text{m}\text{̄}\text{0}$ is large, a plot of 1/χ(r) against r should have gradient very close to 1.

Lastly, for the purposes of comparing theoretical predictions with experimental thermogravimetric data, we shall assume that species with fewer than m_v repeat units will be volatile gases and so the ratio of mass remaining in the distribution to initial mass μ(r) will be given by

$$\text{μ}\text{r}\text{=}\text{∑}\text{m=m}{}_{\mathrm{<mtext>v</mtext>}}\text{n}\text{mN}{}_{\mathrm{m}}\text{r}\text{∑}\text{m=1}\text{n}\text{mN}{}_{\mathrm{m}}\text{0}\text{=}\text{1}\text{m}\text{̄}\text{r}\text{∑}\text{m=m}{}_{\mathrm{<mtext>v</mtext>}}\text{n}\text{mf}{}_{\mathrm{m}}\text{r}\text{.}$$

The final form on the RHS follows from the fact that $\text{N}{}_{\mathrm{m}}\text{r}\text{=N}\text{r}\text{f}{}_{\mathrm{m}}\text{r}$ and using , to eliminate $\text{N}\text{r}\text{/N}\text{0}\text{.}$