In Model 1 a polymer ensemble is generated from 3D-GPC and rheometer data. 3D-GPC data is used to build an initial polymer ensemble, consisting of a MWD-BPD distribution and a cayley tree/comb structure distribution. This ensemble is used as an input for the “Branch-on-Branch” (BoB) model to predict rheology. In a fitting procedure the ensemble is iteratively refined to acquire the best fit between the predicted (BoB) rheology and the experimentally determined rheology. This polymer ensemble will be used in Model 2 to predict the structural changes of the LDPE samples during high temperature processing.
While we are able to find similar trends between Model 1 and experimental data, it is important to keep in mind certain assumptions that influence the final result. Structure factor 𝜀 accounts for the flow modification of the solvent caused by the polymer shape (section 1.2.2.1). A small linear polymer alters the flow differently than a large highly branched cayley tree polymer and therefore it would be ideal to define a 𝜀 for every polymer shape (and mass). In literature however it is difficult to find exact values for different polymer shapes or polymer mixtures and often a value is taken between 0.5 and 2.0 for the whole molecular weight range. In this project we use 𝜀 as a free variable that, in combination with structure distribution 𝜇, is used to find a more accurately polymer shape distribution then when assuming a fixed value for 𝜀.
To differentiate between branch-on-core and branch-on-branch structures, regular comb and cayley tree structures are used, respectively. This assumption makes the polymer ensemble much more comprehensible, but it could limit the accuracy of the predicted outcome. Most reports in literature where the properties of virgin LDPE are predicted, full-scale polymerization kinetic simulations are used to find the structural and viscoelastic properties. These models typically have much higher calculation times and are more complex than the models in this project.37,38
In the procedure of finding the structure distribution, we derived relations from Kramers theorem to calculate the amount of branch points from the branching ratio 𝑔 for well-defined structures. These relations correspond with 3D random walk simulations for the same polymer structures. Compared with the Zimm-Stockmayer functions these relations are increase the accuracy of the structure selection procedure, because these relations account for the difference in structure.
In Model 2 the structural modification of the generated polymer ensemble were generated. By assuming ideal PFR flow, a batch-like reactor configuration was modelled that included three different reaction types: chain scission, cross-linking and chain transfer. The MWD and 𝑔′ from Model 2 follow the same trend compared with experimental data. Residence constant 𝑎 for instance scales well with the extruder screw rpm, which directly affects the residence time. For both model and experimental results the dynamic moduli shift towards lower moduli values, but do not fit well.
This indicates that either the sequence of reactions does not produce the right polymer structures, or the structures obtained from Model 1 affect the result of Model 2.
To conclude, we have built a model that is able to simulate the structural modification of LDPE during high-temperature processing. The model is able to predict the trends of the structural modification in a relatively short calculation time (in the order of minutes). In future work the models can be improved to give more accurate results. In the next part an outlook is given for future work.
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4.1 Outlook
Model 1:
1) Execute more 3D-GPC and rheometer measurements for different LDPE samples to validate the model.
2) Use alternative techniques to determine branching/structure preferably directly. Light scattering could be used to determine the radius of gyration from where the 𝑔′ → 𝑔 step omitted. 13C-NMR could be used to reveal more information about the core structure of the polymers.
3) Compare the results of Model 1 with full-scale polymerization models in terms of number of branches and structure. This could give more information regarding the structure
distribution and BPD shape.
4) Find a way to estimate the short chain branching by comparing with experimental results or polymerization models. Short chain branching is especially important in Model 2 to use the right kinetics.
5) Use other types of structure to increase the accuracy of the structure distributions.
6) Automate the fitting procedure to find accurate fit results.
Model 2:
1) Execute more experiments to validate the model and find average values for the transient extensional viscosity.
2) Find the effect of different reaction steps in detail. Possibly update the kinetic scheme to a more complete scheme. When information about short chain branching is given, more detailed reaction mechanisms can be used.
3) Model the extruder more in detail. Now a batch reactor is assumed, but in extruders there are differences in temperatures, stresses and back mixing occurs.
4) Execute extrusion experiments for different temperatures and use this to find more realistic reaction rate constants.
5) Integrate CSTR reactor configuration in the model and use them in series to find a more realistic PFR reactor type.
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