Relate IV with branching ratio - Detailed Description Model 1

2 Model 1: From Experimental Data to Detailed Polymer Ensemble

2.2 Detailed Description Model 1

2.2.2 Relate IV with branching ratio

The intrinsic viscosity data is used to determine the (average) amount of branch points per polymer weight fraction. First branching ratio 𝑔′ with respect to the relative difference in IV is used.

𝑔′ = ^[𝜂]_[𝜂]^{𝑏𝑟𝑎𝑛𝑐ℎ𝑒𝑑}

𝑙𝑖𝑛𝑒𝑎𝑟 (2.1)

For the intrinsic viscosity of the linear polymers the semi-empirical law [𝜂]𝑙𝑖𝑛𝑒𝑎𝑟= 𝐾𝑀^𝛼 is used and with this 𝑔′ can be calculated. Branching ratio 𝑔, the ratio of the average radii of gyration of branch and linear polymers with equal molar mass, is determined from 𝑔′ by the following equation:

𝑔^′= 𝑔^𝜀 (2.2)

In general a value for structure factor 𝜀 is typically chosen between 0.5 and 2.0. The obtained branching ratio 𝑔 is shown in figure 2.1B.

Figure 2.1 – (A) (Blue) Example of molecular weight distribution of LDPE sample ‘A’, determined with 3D-GPC.

(Red) Intrinsic viscosity of sample ‘A’, represented by the solid line. The dotted line represents the intrinsic viscosity if the polymers would only be linear. This linear reference is determined from the Mark-Houwink equation [𝜼] = 𝑲 ∗ 𝑴^𝜶, with 𝑲 = 𝟑. 𝟖 ∗ 𝟏𝟎^−𝟒 and 𝜶 = 𝟎. 𝟕𝟑𝟐. (B) Example of branching ratios 𝒈 and 𝒈′, by making use of a structure factor 𝜺 = 𝟎. 𝟗.

B A

21 2.2.3 Structure Selection and Distribution

For building the polymer ensemble information about the architecture of the polymers and its distribution per weight fraction is required. Pladis et al.²⁷ use a kinetic-molecular topology Monte Carlo algorithm for the synthesis of LDPE to predict polymer chain architectures and suggest that for low molecular weight fractions (< 10³ g/mol) mainly linear polymer chains, for medium molecular weight fractions (~10⁴ g/mol) comb chains and for higher molecular weight fractions (> 10⁵) polymer chains with “branch on branch” type of architecture are present (figure 2.3A). Rungswant et al.²⁸ use

13C-NMR, 3D-GPC and rheology to predict the polymer chain structure. Here they discuss a transition from primitive core to branch-on-core to branch-on-branch structures for increasing molar masses, where the primitive core is linear or star/T-shaped.

In general, branched polymer structures can be divided into two different types: branch-on-core and branch structures. In our model comb shaped polymers are used to represent branch-on-core structures and cayley trees to represent branch-on-branch structures. Figure 2.2 shows the two different types for increasing number of branch points. All segments are chosen to be of equal length, in order to reduce the amount of variables and limit the complexity of the polymer

ensemble. Now only the structure type and number of branch points determine the architecture of the polymer.

Figure 2.2 – Polymer chain structures for 4 to 6 number of branch points. (A) Comb structure, (B) Cayley tree structure. The red segments are the newly added segments.

Figure 2.3 – (A) Structure distribution result, retrieved from Pladis et al. [27] (B) Example of structure distribution shape for branched polymers when using a sigmoidal curve (Eq. 2.2.3).

B A

22 To account for the transition from branch-on-core to branch-on-branch structure types for

increasing molar mass a sigmoidal curve is used (Eq. 2.2.3). 𝑓_{𝑐𝑎𝑦𝑙𝑒𝑦} or 𝑓_{𝑐𝑜𝑚𝑏} is the fraction of cayley tree or comb polymer structure, 𝑎 is the slope of the curve and 𝜇 is the position where 𝑓 =¹₂. A structure distribution when using a sigmoidal curve is shown in figure 2.3B.

𝑓_{𝑐𝑎𝑦𝑙𝑒𝑦}(𝑥) =_1+𝑒_{−𝑎(𝑥−𝜇)}¹ = 1 − 𝑓_{𝑐𝑜𝑚𝑏}(𝑥) (2.3)

A sigmoidal curve is used because its shape is similar to the distribution shapes found in literature and is easily altered with only parameters 𝑎 and 𝜇. Note that figure 2.3B only shows the structure distribution of cayley trees or combs and not of linear polymers. If a cayley tree or comb has zero branch points, the polymer is automatically linear. Therefore it is not necessary to include linear polymers into this distribution at this point. In the next section the number of branch points and its distribution is determined.

2.2.4 Number of branch points from 𝒈

In part 2.2.2 branching ratio 𝑔 is obtained. From here different relations can be used to determine the (average) amount of branch points in a polymer fraction. Equations 2.2.4 and 2.2.5 are typically used and were initially derived by Zimm and Stockmayer.²⁰

𝑔 =_𝑛⁶

Both functions are frequently used in literature, but however they are limited in use. These functions are valid for randomly branched polymers, but in our case polymers with well-defined regular structures are used. Figure 2.4A includes the two Zimm-Stockmayer equations and simulated cayley tree and comb structures. 3D-random walk simulations for freely jointed polymers were conducted

Figure 2.4 – (A) Branching ratio 𝒈 versus (average) number of branch points. The two Zimm-Stockmayer functions, Zimm 1 and Zimm 2, are the yellow and purple lines, respectively. The blue circles represent the simulated Cayley Tree polymer structures and the red triangles the combs. 3D-random walk simulations for freely jointed polymers were conducted by assuming 𝜽-conditions (no inter- or intramolecular interactions). Every structure is simulated around 1000 times to calculate the average branching ratio 𝒈. The polymer segments consist of 10-100 Kuhn segments of length 1. (B) Cayley tree, comb and star shaped polymers when simulated (blue circle, red triangle, yellow diamond, respectively) or calculated with Kramers theorem relations (solid red, blue, yellow, respectively)

B A

23 by assuming 𝜃 -conditions (no inter- and intramolecular interactions). More details of these simulations can be found in Appendix 2.

As can be seen in figure 2.4A, the simulations do not correspond with the Zimm-Stockmayer equations, especially the comb structures. An alternative method that considers the specific structure of polymer chains is the Kramers theorem:¹⁰

< 𝑅_𝑔²>= ^𝑏²

𝑁_{𝑡𝑜𝑡𝑎𝑙}² ∑^𝑁_𝑘=1𝑁(𝑘)[𝑁_{𝑡𝑜𝑡𝑎𝑙}− 𝑁(𝑘)] (2.6)

Here the average radius of gyration is described in terms of the amount of freely jointed Kuhn segments 𝑁 and of Kuhn length 𝑏. By using the Kramers theorem, the average of the number of possibilities to divide the polymer into two parts can be calculated. From here the classic Debye result can be derived for linear polymers (Eq. 2.2.7):

< 𝑅_𝑔²>_{𝑙𝑖𝑛𝑒𝑎𝑟}= ^𝑏²

𝑁_{𝑡𝑜𝑡𝑎𝑙}² ∫ 𝑁(𝑁₀^𝑁 _{𝑡𝑜𝑡𝑎𝑙}− 𝑁)𝑑𝑁=^𝑏²^𝑁^{𝑡𝑜𝑡𝑎𝑙}

6 (2.7)

Eq. 2.2.6 divided by Eq. 2.2.7 results in the branching ratio 𝑔:

𝑔 = _𝑁⁶

𝑡𝑜𝑡𝑎𝑙3 ∑^𝑁_𝑘=1𝑁(𝑘)[𝑁_{𝑡𝑜𝑡𝑎𝑙}− 𝑁(𝑘)] (2.2.8)

From here we derived unique expressions for branching ratio 𝑔 for cayley tree and comb shaped polymers. The detailed derivations are included in Appendix 1. The final relations are the following:

𝑔_{𝑐𝑎𝑦𝑙𝑒𝑦}=^𝑏

2(24𝑙𝑛(𝑏+2)−𝑙𝑛(3)

𝑙𝑛(2) −28)+𝑏(96𝑙𝑛(𝑏+2)−𝑙𝑛(3)

𝑙𝑛(2) −8)+(96𝑙𝑛(𝑏+2)−𝑙𝑛(3) 𝑙𝑛(2) +57)

(2𝑏+1)³ (2.9)

𝑔_{𝑐𝑜𝑚𝑏}=^4𝑏³^+12𝑏_(2𝑏+1)²^+4𝑏+1₃ (2.10)

Figure 2.5 – (A) Branching ratio 𝒈 versus number of branch points for different 𝝁 (sigmoid function), with 𝒂 = 𝟏. (B) Number of branch points per molar mass. Used variables: 𝜺 = 𝟎. 𝟗, 𝝁𝒔𝒊𝒈𝒎= 𝟏𝟎 and 𝒂 = 𝟏.

B A

24 These functions are plotted in figure 2.4B, where the results correspond with the random walk simulations. Because these 𝑔 ratios require an a priori description of the used structure in a polymer weight fraction, the structure distribution needs to be set first. With the sigmoidal curve the total 𝑔 ratio can be determined (Eq. 2.2.11).

𝑔 = 𝑔_{𝑐𝑎𝑦𝑙𝑒𝑦}𝑓_{𝑐𝑎𝑦𝑙𝑒𝑦}+ 𝑔_{𝑐𝑜𝑚𝑏}𝑓_{𝑐𝑜𝑚𝑏} (2.11)

In figure 2.5A 𝑔 versus number of branch points 𝑏 is shown for different structure distributions.

From this function the number of branch points can be determined (figure 2.5B).

2.2.5 Combing Polymer Information

In section 2.2.4 the average number of branch points is calculated. To estimate the full branch point distribution we add a folded normal distribution to the average number of branch points. A folded normal distribution is a normal distribution where the part of the curve that lies at the left side of 𝑥 = 0 is folded over by taking the absolute value and adding it to the remainder of the curve. The probability function is given by:

𝑃 = ¹

√2𝜋𝜎²𝑒⁻^{(𝑥−𝑏)2}^2𝜎2 + ¹

√2𝜋𝜎²𝑒⁻^(𝑥+𝑏)2^2𝜎2 (2.12)

Here 𝜎 affects the width of the distribution. This function is applied to the result from part 2.2.4 to acquire the branch point distribution (BPD) (figure 2.6A). BPD combined with MWD gives the full molecular weight distribution-branch point distribution of the polymer ensemble (figure 2.6B).

Figure 2.6 – (A) Example of branch point distribution (BPD) for LDPE ‘A’, ε=1.268. The solid blue line is the average number of branch points, while the dotted lines represent the spread in branch points per molecular weight. The spread is based on a folded normal distribution, with σ=0.2. (B) Example of 2-D joint (BPD (figure 6A) – MWD (figure 1A)) distribution for LDPE ‘A’. (C) Reduced BPD-MWD distribution, where the total (weight) number of polymers is 4000. In other words, 4000 polymers are selected (based on their weight fraction) that represent the MWD-BPD distribution.

B A

25 2.2.6 Reduce MWD-BPD

In the next part a model will be used to estimate the rheological response of the polymer ensemble.

The ensemble generated in 2.2.5 is very large and detailed. To reduce the calculation time, but still obtain a reliable result, the ensemble is filtered to a more simple set (figure 2.6C). A number of polymer fractions 𝑁𝑝𝑜𝑙𝑦 are selected based on their weight fractions.

2.2.7 BoB model

With the reduced MWD-BPD and structure distributions a polyconf file can be generated, containing linear, comb and cayley tree structures. Linear polymers are the simplest to define where only two segments are needed:

Comb polymers with 𝑏 = 1 is a 3-armed star. For increasing number of branch points two segments are added to the last segment. The polyconf matrix is modified by adding two rows at the bottom:

Cayley tree polymers start with the same 3-armed star at 𝑏 = 1. From here segments are added in a counterclockwise manner. The polyconf matrix will look like:

The segment mass and the weight fraction that are taken from the MWD-BPD distribution are

26 Where 𝑀 is the mass of the polymer, 𝑁𝑠 the number of segments, 𝑁𝑒 the entanglement length, 𝑀0

the monomer weight and 𝑏 the number of branches. For linear polymers, 𝑁_𝑠 is equal to 2.

The input parameters for the BoB model that can be found in section 1.2.3.1 (Table 1) are kept constant. With the polyconf file and the input parameters the viscoelastic response is calculated.

2.3 Model 1 Results

In short there are 4 variables that determine the result of Model 1 besides the selected polymer sample input:

1) Structure factor 𝜀

2) Structure distribution position 𝜇 3) Structure distribution slope 𝑎 4) Branching distribution width 𝜎

In this part 3D-GPC and rheometer data of polymer samples LDPE A and LDPE B will be used as an input of Model 1. The 3D-GPC data of LDPE A and LDPE B are shown in figure 2.7. With certain values for variables 𝜀, 𝜇, 𝑎 and 𝜎 the BoB model will give an estimation of the viscoelastic response of the polymer ensemble. The storage modulus 𝐺^′, loss modulus 𝐺^′′ and transient extensional viscosity 𝜂𝐸+

output from the BoB model will be used to fit on experimental data. Note that the extensional viscosity with 𝑟𝑎𝑡𝑒 = 1 𝑠⁻¹ will be used in the fitting procedure.

The values for the variables that give the best fit with experimental rheological data will be selected, from where the final polymer ensemble is obtained. Here the coefficient of determination 𝑅² will be used. 𝑅² is defined by:

𝑅²= 1 −^𝑆𝑆_𝑆𝑆^𝑟𝑒𝑠

𝑡𝑜𝑡 =^𝑆𝑆_𝑆𝑆^𝑟𝑒𝑔

𝑡𝑜𝑡 =^{∑ (𝑓}_{∑ (𝑦}^𝑖 ^𝑖^−𝑦)²

𝑖−𝑦)²

𝑖 (2.16)

In the right hand side of equation 2.16 𝑓𝑖 is the BoB data, 𝑦𝑖 the experimental data and 𝑦 the mean of the experimental data (𝑦 =¹_𝑛∑^𝑛_𝑖=1𝑦_𝑖). The average will be taken of the 𝑅² values of 𝐺^′, 𝐺^′′and 𝜂_𝐸⁺ fits to determine the final 𝑅² that is used to select the best fitting polymer ensemble.

Figure 2.7 – 3D-GPC data LDPE A (blue) and LDPE B (red). (A) Molecular weight distribution. (B) Intrinsic viscosity versus molecular weight. The linear reference is obtained from [𝜼] = 𝑲 ∗ 𝑴^𝜶, with 𝑲 = 𝟑. 𝟖𝟏𝟒𝟖 ∗ 𝟏𝟎^−𝟒and 𝜶 = 𝟎. 𝟕𝟑𝟐.

A B

27 2.3.1 LDPE A

Because of having 4 variables to fit the model output on the experimental data, the calculation time increases severely. In order to restrain the calculation time, the effect of the individual variables will be checked first. In figure 2.8 shifting the BPD up or down, while 𝜎 only adds some dispersity. 𝜇 affects the transition position of comb to cayley tree structures, while 𝑎 affects the smoothness of

this transition. This is why we fix 𝑎 at 𝑎 = 1 and 𝜎 = 0.25 during the fitting procedure. Afterwards the effect of varying 𝑎 and 𝜎 individually is checked to possible refine the best fit. As can be seen in figure 2.8C, the effect of 𝑎 on the overall fit is marginal. Because 𝜇 is more important here, we keep 𝑎 = 1. Figure 2.8D shows that also 𝜎 can be kept at 𝜎 = 0.25. Note that this reduction favors the calculation time, but reduces potentially the accuracy of the fitting procedure. In Appendix 3.1 the effect of varying the 4 variables to the dynamic moduli and extensional viscosity can be found.

The fit results of the final polymer ensemble from Model 1 for LDPE A are shown in figure 2.9. The best fits are found for:

Figure 2.9 – Viscoelastic behavior LDPE A. (A) Storage modulus 𝑮^′ and loss modulus 𝑮^′′ from the BoB model and experimental rheometer measurements. (B) Transient extensional viscosity 𝜼𝑬+ from the BoB model and experimental rheometer measurements. The dashed black line is the linear viscoelastic behavior as discussed in section. Model 1 variable values:

𝜺 = 𝟎. 𝟗𝟏, 𝝁 = 𝟗, 𝒂 = 𝟏, 𝝈 = 𝟎. 𝟐𝟓.

B A

28 2.3.2 LDPE B

The effect of varying the 4 variables individually for LDPE B is shown in figure 2.10. Similar to the procedure of LDPE A 𝑎 and 𝜎 are fixed at 𝑎 = 1 and 𝜎 = 0.25 during the fitting procedure. Afterwards the effect of varying 𝑎 and 𝜎 individually is checked to find the best fit. As can be seen in figure 2.10C, the effect of 𝑎 on the overall fit is marginal. Because 𝜇 is more important here, we keep 𝑎 = 1.

Figure 2.10D shows that also 𝜎 can be kept 𝜎 = 0.25. In Appendix 3.2 the effect of varying the 4 variables to the dynamic moduli and extensional viscosity can be found.

The fit results of the final polymer ensemble from Model 1 for LDPE A are shown in figure 2.11. The best fits are found for:

1) 𝜀 = 0.82 2) 𝜇 = 15 3) 𝑎 = 1 4) 𝜎 = 0.25

Figure 2.10– Fit results for LDPE B. Coefficient of determination 𝑹^𝟐 for varying (A) structure factor 𝜺, (B) structure distribution position 𝝁, (C) structure slope 𝒂, (D) BPD width 𝝈. Base constants: 𝜺 = 𝟎. 𝟖𝟐, 𝝁 = 𝟏𝟓, 𝒂 = 𝟏, 𝝈 = 𝟎. 𝟐𝟓.

A B

D C

Figure 2.11 – Viscoelastic behavior LDPE A. (A) Storage modulus 𝑮^′ and loss modulus 𝑮^′′ from the BoB model and experimental rheometer measurements. (B) Transient extensional viscosity 𝜼𝑬+

from the BoB model and experimental rheometer measurements. The dashed black line is the linear viscoelastic behavior as discussed in section.

Model 1 variable values: 𝜺 = 𝟎. 𝟗𝟏, 𝝁 = 𝟗, 𝒂 = 𝟏, 𝝈 = 𝟎. 𝟐𝟓.

A B

2.4 Final Result LDPE A and LDPE B

On the this page, the characteristics of the generated polymer ensembles of LDPE A and LDPE B are shown. LDPE B has a larger fraction of polymers with a higher molecular weight (figure 2.12A), while this fraction is predicted to be more highly branched than in LDPE A (figure 2.12D). The transition of branch-on-core to branch-on-branch structures is expected to occur for a higher mass for LDPE B (figure 2.12C).

A B

D C

F E

Figure 2.12 – Comparison polymer ensemble LDPE A (blue) and LDPE B (red). The variable values described in 2.3.1 and 2.3.2 are used to generate the polymer ensembles. (A) Molecular weight distribution. (B) Structure factor 𝒈. (C) Upper window: structure distribution in number of branches. Lower window: structure distribution in molecular weight (when the BPD is known (D) this plot can be obtained). (D) Branch point distribution. (E) Dynamic Moduli. (F) Transient extensional viscosity (𝒓𝒂𝒕𝒆 = 𝟏𝒔^−𝟏).

3 Model 2: Structural Modification Polymer Ensemble

The Model 2 is used to estimate the structural modification of (virgin) LDPE during high temperature processing. Here the polymer ensemble, obtained from Model 1, is exposed to several kinetic mechanisms. The resulting modified polymer ensemble is compared with 3D GPC and rheometer data for virgin LDPE A and the reaction constants are iteratively refined to find the best results..

Section 3.1 includes a flow sheet of Model 2. In section 3.2 a detailed description of the different steps taken is given and in section 3.3 the results are shown.

3.1 Flow Sheet

3.2 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).

In document Eindhoven University of Technology MASTER Structural modification of LDPE during high-temperature processing Simons, Jérôme (pagina 21-0)