Model 2 Description

During LDPE processing, virgin LDPE polymers are exposed to various temperatures and stresses in order to obtain the final product. Also this product undergoes ‘aging’ when it is used and these processes combined change the properties of the material in its life-cycle. Understanding these processes is valuable for the design of polymer products and plastic recycling.^39-41 In literature numerous studies can be found that describe the structural modification of LDPE during processing/recycling, but are typically only limited to experimental measurements. In this project, the polymer ensemble obtained from Model 1 is used to describe the structural modification and in this section the polymer ensembles are exposed to different chemical reactions to simulate the processing steps.

3.2.1 Modelling Method

In order to model polymerization, the population balance method is often used to keep track of the change in concentration of different (polymer) species. Here differential equations are defined for all reactive species, including rates for initiation, propagation, chain scission, branching and termination. For simple cases, like steady state polymerization of linear polymers, these equations can be solved analytically. When more elaborate kinetic schemes are used, numerical methods are usually required to solve the population balance.

The population balance method provides information about the molecular weight distribution and number of branches, but lacks information about the exact structure of the simulated polymers. In order to elucidate the polymer architecture from the population balance results the Conditional Monte Carlo Sampling method for example can be used to predict polymer architectures.⁴² Here however assumptions need to be made regarding the sequence of chemical reactions that potentially decrease the accuracy of the architecture distribution predictions.⁴³

An alternative method to model polymerization is the Monte Carlo method, where random numbers are used to predict the sequence of reactions to generate a distribution in polymer architectures.

These Monte Carlo methods have proven to be predict the molecular topology accurately.^44,45 Different from the population balance method, the Monte Carlo method gives detailed information about the polymer architectures. This detailed information is needed to calculate the viscoelastic response with the BoB model. In this project the Monte Carlo method is used to expose the (virgin) polymer ensemble to a number of different reactions that result in a modified polymer ensemble.

At time step 𝑡 a reaction is chosen stochastically according to the Gillespie method.⁴⁶ The probability that a reaction 𝑖 is selected at step 𝑡 is:

𝑃(𝑖, 𝑡) =_{∑ 𝑟(𝑖,𝑡)}^{𝑟(𝑖,𝑡)}

𝑖 (3.1)

Time step 𝑡 is incremented by:

Δ𝑡 = ^ln(

1 𝑎)

∑ 𝑟(𝑖,𝑡)_𝑖 (3.2)

Here 𝑎 is a random number between 0 and 1. For every time step 𝑟(𝑖, 𝑡) is calculated by assuming zeroth order reaction rates for the dead or living polymers. The reaction rates are subsequently used to calculate the probability 𝑃(𝑖, 𝑡). In the next part the reactions are specified.

32 3.2.2 Reaction Mechanism

In part 1.1.1 the chemical reactions that occur during LDPE polymerization were discussed. Because we will be modeling chemical reactions during high-temperature processing, a limited set of reactions is used. When LDPE A is extruded at 5 rpm, 3D-GPC data show that scission takes place, because the molecular weight distribution shifts to lower molecular weights (figure 3.1). For extrusion at 250 rpm cross-linking also is assumed to take place, because the polymer fraction of high molecular weight increases. Furthermore the 𝑔′ factor is lower for high weights, indicating the creation of more densely branched polymers. Therefore chain transfer has to be included. The following three reactions will be used:

𝐷_𝑛→ 𝐿_𝑛−𝑥+ 𝐿_𝑛 (chain scission) (3.1)

𝐿_𝑛+ 𝐿_𝑚 → 𝐷_𝑛+𝑚 (cross-linking) (3.2)

𝐿_𝑛+ 𝐷_𝑚→ 𝐷_𝑛+ 𝐿_𝑚 (chain transfer) (3.3)

𝐿_𝑛 is a living polymer (i.e. polymer with radical) with 𝑛 number of monomers and 𝐷𝑚 is a dead (without radical) polymer of 𝑚 number of monomers. For chain scission we assume that the chance scission occurs is proportional to the amount of C-C bounds present in the polymer chain. Therefore large polymers and their large segments are most likely to break. Furthermore we assume that due to the high shear stresses the chains that lie deep in the polymer molecule are exposed to higher stresses than the branches that have a free end. The stress makes those segments more likely to break. Therefore chain scission is assumed to be proportional to the priority of the chain (from the pom-pom model, part 1.2.3).

Cross-linking is only dependent on the presence of a radical on the polymer chain. During chain transfer a radical is extracted from a (living) polymer and added to a (dead) polymer. The transfer of a radical to polymer chain 𝑥 is proportional to the amount of monomers present. This type of chain transfer causes long chain branching. Short chain branches are produced via another chain transfer mechanism termed back-biting.⁴⁶ This reaction type is omitted in our model because Model 1 does not describe the presence of short chain branches. This is also why we assume that location where scission or chain transfer occurs is random. The location of scission for example is affected by primary or secondary carbon molecules and the presence of these molecule types is unknown.

Figure 3.1 – (A) Molecular weight distribution of LDPE sample for different extrusion rpm values. (B) Branching ratio 𝒈′.

B A

33 3.2.3 Bookkeeping Method

For the bookkeeping of the polymer structures an adjusted polyconf (section 1.2.3.1) matrix is used.

A large matrix is built up from a number of vertically stacked polyconf matrices. In a polyconf matrix the first four columns describe the connectivity of the segments. The fifth column is used to keep track of the segment weights and the sixth column is used to indicate how many polymer fractions are described by this polyconf matrix. To keep track of the general characteristics of the individual polymer fractions vectors are used that contain information about their weight, number of branch points, being living/dead, radical location and their structure (referring to the polyconf matrix).

When the chain scission reaction is selected, a polymer is selected randomly where the chance of being selected is dependent on is molar mass. Next a segment is chosen to be split into two, based on its mass. Chain scission separates the polyconf matrix into two parts, splitting selected segment 𝑥 randomly into part 𝑀 − 𝑛 and 𝑛. In order to accurately split the polymer in two the polymer segments that are connected to the left and right of segment 𝑥 are determined. From here two new polyconf matrices are created and added to the large polyconf matrix. A generalized procedure is

Figure 3.2 – Polyconf matrix modification procedure for chain scission.

x

34 After updating the polyconf matrix the vectors are modified to account for the two new polymers and the locations of the radicals are included (being segment 𝑥 for polymer 2 and segment 0 for polymer 3 in figure 3.2).

The cross-linking mechanism is reverse of the scission mechanism. During cross-linking two living polymers are randomly chosen and their polyconf matrices are combined (figure 3.3). Hereafter the vectors are updated.

During chain transfer a radical is moved from one polymer to another. A living polymer is chosen randomly and a dead polymer is, based on its total and segment weight, selected randomly. The radical is moved to a segment of the dead polymer on a random position. This mechanism is shown in figure 3.4. After updating the matrix and vectors the next reaction is selected. In part 3.3 the results of the structural modification are discussed.

y -1 -1 ^y+1 y+2 𝑛 1

Figure 3.3 - Polyconf matrix modification procedure for chain scission.

35

Figure 3.4 - Polyconf matrix modification procedure for chain scission.

36