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Eindhoven University of Technology

MASTER

Structural modification of LDPE during high-temperature processing

Simons, Jérôme

Award date:

2018

Link to publication

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1

Structural Modification of LDPE during High-Temperature Processing

Final Report Jérôme Simons

Eindhoven University of Technology, Laboratory of Physical Chemistry (SPC), Master Chemical Engineering, Graduation Project, September 2017-June 2018,

Supervisor: prof. dr. ir. C.F.J. den Doelder PDEng.

Abstract – Processability of polymers relates to the rheology of the material, which is a function of the molar mass and branching topology. Classical approaches treat the material as fixed when predicting the processability. Low-density polyethylene (LDPE) is often extruded at high temperatures. Experimental data show that chemical modification changes the material during such high-temperature processing and directly affects the rheology. In this project a model is built that is able to describe the molecular structure modification. Triple detector GPC data is used to obtain the molecular weight distribution and to estimate the long chain branching. The so-called “branch-on- branch” model of Das et al. (2006) is used to predict the rheological properties of the polymer molecules.

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2

Contents

Introduction ... 4

1 Theory ... 5

1.1 Polymers ... 5

1.1.1 Polyethylene Production ... 5

1.1.2 Polymer Characteristics ... 8

1.2 Polymer Characterization ... 9

1.2.1 Characterization of Molar Mass ... 9

1.2.2 Characterization of Architecture ... 12

1.2.3 The “Branch-on-Branch” (BoB) Model ... 17

2 Model 1: From Experimental Data to Detailed Polymer Ensemble ... 19

2.1 Flow Sheet ... 19

2.2 Detailed Description Model 1 ... 20

2.2.1 Experimental Data (GPC)... 20

2.2.2 Relate IV with branching ratio 𝒈 ... 20

2.2.3 Structure Selection and Distribution ... 21

2.2.4 Number of branch points from 𝒈 ... 22

2.2.5 Combing Polymer Information... 24

2.2.6 Reduce MWD-BPD ... 25

2.2.7 BoB model ... 25

2.3 Model 1 Results ... 26

2.3.1 LDPE A ... 27

2.3.2 LDPE B ... 28

2.4 Final Result LDPE A and LDPE B... 29

3 Model 2: Structural Modification Polymer Ensemble... 30

3.1 Flow Sheet ... 30

3.2 Model 2 Description ... 31

3.2.1 Modelling Method ... 31

3.2.2 Reaction Mechanism ... 32

3.2.3 Bookkeeping Method ... 33

3.3 Model 2 Results ... 36

3.3.1 LDPE A – 5 rpm ... 37

3.3.2 LDPE A – 25 rpm ... 38

3.3.3 LDPE A – 150 rpm ... 39

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3

3.3.3 LDPE A – 250 rpm ... 39

4 Conclusion ... 40

4.1 Outlook ... 42

References ... 43

Appendix 1 Kramer theorem: Branching Ratio Derivations ... 45

Kramers Theorem ... 45

A1.1 Star Shaped Polymers ... 46

A1.2 Comb Shaped Polymers ... 47

A1.3 Cayley Tree Structured Polymers ... 50

Appendix 2 3-D Random-Walk Simulations ... 53

Appendix 3 Model 1 Variables ... 55

A3.1 Structure Factor 𝜺 ... 56

A3.2 Structure Distribution Postion 𝝁 ... 57

A3.3 Structure Distribution Slope 𝒂 ... 58

A3.4 Branching Distribution ... 59

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4

Introduction

In the processes that convert newly synthesized polymers into their final products, for instance in film blowing, film/sheet casting and extrusion blow molding, extrusion is an important step operation. During extrusion conductive heat transfer and shear stresses, caused by heating and a rotating screw, produce a viscoelastic polymer melt that is subsequently deformed in a shaping device. Finally, the polymer melt is cooled to achieve a solid product.

Ideally, the polymer architecture obtained after synthesis should not be affected by heat and stresses in the extrusion process. As described by Koopmans1 an essential characteristic of the polymers should be “their ability to sustain a repetitive process of heating and cooling without, in principle, changing their chemical composition”. During LDPE processing, virgin LDPE polymers are exposed to various temperatures and stresses in order to obtain the final product. Furthermore this final product also undergoes ‘aging’ when it is used. It is shown that polymer processing under high temperature and stress can cause the polymer chains to be modified by reactions such as cross- linking and chain scission, depending on temperature and residence time.2,3 This chemical modification affects rheological properties and therefore the processability of the polymers.

Understanding this process is valuable for the design of polymer products and plastic recycling.

In this project, the structural modification of low-density polyethylene (LDPE) during extrusion as a function of temperature and residence time is modeled. The changes in molecular weight distribution, intrinsic viscosity and rheological properties for the linear and non-linear viscoelastic regions are used as indicator for structural changes. To connect polymer structures with rheological properties, the so-called “branch-on-branch” (BoB) model of Das et al.4 is used. In this model the extended tube model is applied to calculate the rheological properties of polymer melts with various molecular weights and architectures. By comparing experimentally obtained rheological properties with the rheological response from the BoB model, a polymer ensemble is created that is able to represent the experimentally tested LDPE samples in terms of MWD, number of branch points and structure. From here the LDPE samples are exposed to high temperature processing. Experimental data obtained from these samples are compared with data from a second, kinetic, model where the found representative polymer architectures are chemically changed and evaluated by the BoB model. With these two models, the structural change in LDPE as a function of temperature and residence time during extrusion can be predicted.

Background information regarding LDPE, used analytical instruments and models will be provided in Chapter 1. In chapter 2 the model that is used to translate experimental data to a polymer ensemble (‘Model 1’) is discussed. ‘Model 2’ is described in chapter 3, where the polymer ensemble is modified to match the structural modification of LDPE samples. The conclusion and outlook of the project are given in chapter 4.

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5

1 Theory 1.1 Polymers

The first key element in this project is establishing a polymer ensemble: a (preferably simple) set of polymers that accurately represents molecular properties (mass, branching, structure) and rheological properties (linear and non-linear viscoelastic behavior). Polymer, derived from Greek, consists of two elements with poly (πολλή) standing for many and mer (μέρος) for part. Polymers are large molecules (macromolecules), built up from many small molecules (monomers) that are held together by covalent bonds. Typically a polymer consists of over more than 1000 monomers. A molecule consisting of a few monomers is named an oligomer.

Polymers have been around for a very long time in nature, with rubber, cellulose and DNA being well known examples. In the 19th century the foundation of the polymer industry were established by discoveries concerning modified natural polymers. Around the beginning of the 20th century the first synthetic polymers were produced. Polyethylene for example was first discovered in 1898 by Hans von Pechmann and during the Second World War polyethylene was used as an insulator in radar installations.5,6 Nowadays countless of different synthetic polymers are produced in large quantities, reaching a global production of 335 million metric tons in 2016.7

Synthetic polymers are usually separated into different types of polymers: thermoplastics and thermosets. Thermoplastics, also known as plastics, are linear or branched polymers that are able to reversibly transition from solid to liquid and liquid to solid by heating or cooling. This is why these polymers are easy to form (and reform) into various shapes and include the most used polymers like polyethylene (PE), polypropylene (PP) and polyvinylchloride (PVC). In solid state, thermoplastics typically do not form a fully crystalline material because the entangled polymer chains, especially highly branched chains, are unable to order. Therefore most thermoplastics are simply ‘frozen’ in position, resulting in an amorphous material. The more linear or slightly branched polymer chains are able to order a fraction of their chains, resulting in a semi-crystalline material. Here both crystalline and amorphous regions are present in the material, giving the material a higher stiffness due to the present crystalline phases.

Thermosets are a type of polymers where a crosslinked three dimensional network is formed when irreversibly cured by heat or radiation and decompose before melting when heated. Thermosets with a low degree of crosslinking show elastic behavior, retaining their original shape after deformation. Highly crosslinked thermosets are more brittle and show permanent or plastic deformation under load. Well known examples of thermosets are Bakelite, epoxy resins and vulcanized rubber.

1.1.1 Polyethylene Production

In this project the focus lies on the thermoplastic low density polyethylene (LDPE). Polyethylene polymer chains are produced by addition polymerization of ethylene. Ethylene is obtained from cracking ethane/propane, naphtha or gas oil. LDPE is typically manufactured during high pressure operations.

The first steps in the production of polyethylene under high pressure took place in the ’30, when the British chemists Fawcett and Gibson synthesized the first grams of polyethylene in 1933, using high pressure and small amounts of oxygen. However, further productions were cancelled because of

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6 extreme hazardous risks: Fawcett and Gibson worked in open laboratories while the systems were highly instable and explosive.8 Perrin and Manning took over the program in 1935, constructed equipment that handled the required high pressures.

The polymerization of ethylene takes place at high pressure (1000-3000 bar) and temperature (420- 570 K). Conversion rates of 15-30% are obtained with short residence times around 20-50 seconds.1 Oxygen and/or organic peroxides are used as an initiator to start the radical polymerization (Eq. 1.1- 1.2).

Initiator decomposition

𝐼 ⟶ 𝑅+ 𝑅 (1.1)

Initiation

𝑅 + 𝑀 ⟶ 𝐿1 (1.2)

Here, 𝐼 is the initiator, 𝑅 the decomposed initiator, 𝑀 the monomer and 𝐿𝑛 the living chain with 𝑛 number of monomers. The chain growth propagates by adding monomers to the radical species formed in Eq. 1.2. These radicals react with other monomer species to form larger polymers (Eq.

1.3).

Propagation

𝐿𝑛 + 𝑀 ⟶ 𝐿𝑛+1 (1.3)

Termination of the radical species occurs by combination, coupling of two living polymers to form a single polymer chain (Eq. 1.4), or disproportioning, where a hydrogen atom is removed to form a double bound and to terminate a chain radical on another living polymer (Eq. 1.5). 𝐷𝑛 is the dead chain with 𝑛 number of monomers. Note that the dead polymer chain with the double bound can be activated; leading to possible branch formation.

Combination

𝐿𝑛 + 𝐿𝑚 ⟶ 𝐷𝑛+𝑚 (1.4)

Disproportioning

𝐿𝑛 + 𝐿𝑚 ⟶ 𝐷𝑛= + 𝐷𝑚 (1.5)

Because of the elevated pressure and temperature during polymerization, several additional reactions take place. Eq. 1.6 displays the radical chain transfer to species 𝐴, being initiator, monomer, solvent or transfer agents. Intramolecular chain transfer (back-biting) of the polymer chain results in short-chain branching (Eq. 1.7) and intermolecular chain transfer leads to long-chain branching (Eq. 1.8). The secondary radicals formed during inter- and intramolecular chain transfer are also able to divide the polymer chain in two parts by chain scission (Eq. 1.9).

Chain transfer to X

𝐿𝑛 + 𝐻 − 𝑋 ⟶ 𝐷𝑛 + 𝑋 (1.6)

Intramolecular chain transfer

𝐿𝑚 ⟶ 𝐿𝑚 (1.7)

Intermolecular chain transfer

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7

𝐿𝑛 + 𝐷𝑚 ⟶ 𝐷𝑛 + 𝐿𝑚 (1.8)

Chain scission

𝐿𝑚 ⟶ 𝐿𝑥+ 𝐷𝑚−𝑥 (1.9)

The polymerization of LDPE is typically conducted in autoclave or tubular reactors. Autoclave reactors are comparable with continuous stirred tank reactors (CSTR) and are operated around 1800 bar and 530 K. The final LDPE weight distribution is broad and the chains are highly branched.

Tubular reactors consist of a very long tube wherein the reaction mixture is transported as a plug flow (PFR). These reactors are operated around 2700 bar and include various heat zones, with peak temperatures of around 570 K. Because less back mixing takes place in the tubular reactors, the resulting weight distribution is often narrower than for the autoclave reactor and less long chain branching takes place.9 An example of the different weight distributions is shown in figure 1.1.

In the last stage of the polymer production process the polymers are shaped into granules during an extrusion step. These granules are typically cylindrical or spherical to ensure the product can be transported and handled with ease. Hereafter the granules can be deformed for various applications by making use of processes like film blowing, film/sheet casting and extrusion blow molding. In all processes the granules need to be melted to reach desired flow properties. This is achieved by making use of a second extrusion step. In figure 1.2 a schematic cross section of a single-screw extruder is shown. Granules are added to the feed hopper and a viscoelastic polymer melt is produced by the heater and the rotating screw. The polymer melt is subsequently deformed in a shaping device (film blowing etc.) and cooled down to obtain the final product.

W eigh t F ra ct ion

Molar Mass

Tubular Autoclave

Figure 1.1 - Hypothetical molecular weight distribution of LDPE produced in a tubular (blue) or autoclave (red) reactor type.

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8 1.1.2 Polymer Characteristics

The configuration of the monomers in the polymer chain determines the polymer architecture or topology. To describe the architecture of a polymer chains different properties are defined. The property degree of polymerization (DP) is used to account for the number monomers present in the polymer chain. DP multiplied with the molar mass of the monomer results in the total molar mass of the polymer chain. For linear chains, DP also describes the size of the polymer chain, but for branched polymers additional definitions are needed.

Degree of branching is another property that refers to the number of branch points present in the polymer chain. Branching occurs for example in LDPE production by back biting or intermolecular chain transfer. Here a distinction is made between short chain branches and long chain branches.

Short chain branches are associated with oligomers (for example methyl, propyl or hexyl side groups) that are present on the chain backbone. Short chain branches typically decrease the crystallinity of the material, because these chains permit the chains to align next to each other. Long chain branches are much longer than short chain branches and influence the rheological properties of the material. The minimum length of a long chain branch is not defined, but the property entanglement length can be used to define a criterion for long chain branches. In polymer melts, polymer chains are intertwined and form loops or connection points that impact the flow behavior.

Here the mass between two connection points (entanglement molar mass) can be determined from rheological experiments by making use of an analogy with crosslinked rubbers.10 Polyethylene at 140°C is determined to have an entanglement molar mass of 1000 g/mol and when divided with the

Star Comb Cayley Tree Random Tree

Figure 1.3 – Different polymer structure types. From left to right: star, comb, cayley tree and random tree shape.

Figure 1.2 – Schematic cross section of a single screw extruder. Granules are added to the feed hopper and a viscoelastic polymer melt is produced by the heater and the rotating screw.

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9 molar mass of the monomer ethylene (28 g/mol), the entanglement length of PE is equal to ~36 ethylene units.10 For this case, long chain branches should be around or larger than 36 ethylene units.

The final property needed to define the polymer architecture is the monomer and branch distribution on the polymer chain. Different options exist when incorporating these distributions in polymer chains, resulting in different polymer architectures as shown in figure 1.3. The first structure shown is a star shaped chain, where all branches (or arms) are connected to one single branch point. A comb structure consists of a backbone chain with a number of distributed linear branches attached to it. Cayley trees are typically defined in terms of generations. For the 1st generation Cayley trees the structure is identical to a star with 3 arms. The 2nd generation contains the 3-armed star, but now two segments are added to all free chain ends of the star, which results in a branch-on-branch structure as shown in figure 1.3. To obtain the next generation, segments are added to the free chain ends of the previous generation. Random trees consist of randomly distributed branches that are connected to the backbone or to other branches. The length of the segments in the polymer chain can be irregular for all previously described structures.

After polymerization a mixture of polymer chains is obtained, where nonuniformities are typically found in mass, branching and structure. During polymerization, polymers are mixed and exposed to different reactions, resulting in a distribution in mass, branching and structure. Here the term polydispersity is used to indicate the nonuniformity of the polymer mixture. In the next section, different analytical techniques are explained to find the polydispersity of the polymer mixtures.

1.2 Polymer Characterization

In this project, triple detector GPC (3D-GPC) and rheometer measurements are conducted to obtain information about the mass, branching and structure of the polymer samples. 3D-GPC makes use of several analytical methods to find the polymer characteristics, including size exclusion chromatography (SEC), light scattering (LS) and viscosity measurements. In the following section, the principles that are used to find polymer characteristics are explained.

1.2.1 Characterization of Molar Mass 1.2.1.1 Size Exclusion Chromatography

In Size Exclusion Chromatography (SEC) the differences in size of polymer molecules in dilute solutions are used to separate large molecules from small molecules. Here a column is filled with porous particles (the stationary phase) and subsequently filled with a liquid (the mobile phase).

When the polymer solution is added to the column, the small polymers are able to enter the pores and penetrate deep, while the large polymers that are larger than the pore size are not able to enter the pores at all. Therefore large polymers exit the column first and the small polymers last. This mechanism is named steric exclusion, where certain pore volume fractions are accessible for certain polymer sizes. Eq. 1.10 shows the relation between elution volume 𝑉𝑒, solvent volume outside the pores 𝑉0, solvent volume inside the pores 𝑉𝑖 and distribution coefficient 𝐾𝑑.11

𝑉𝑒= 𝑉0+ 𝐾𝑑𝑉𝑖 (1.10)

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10 The distribution coefficient 𝐾𝑑 is dependent on the size of the polymer chains and the used porous particles. Polymers that are larger than the largest pore size leave the column at elution volume 𝑉0

and polymers that are able permeate into all pores leave the column at elution volume 𝑉𝑖. The relation between elution volume and polymer size is shown in figure 1.4.

Ideally the distribution coefficient 𝐾𝑑 is directly related with the size and even molar mass of the polymers. This, however, has proven to be difficult because there are multiple separation mechanisms occurring during size exclusion chromatography.11 To obtain the absolute mass a calibration curve is sometimes used where the masses of known polymers are used. The calibration curve provides an easy way to obtain molar masses, but this method has a major pitfall. SEC only separates particles based on their hydrodynamic size and this size is, besides being dependent on the molar mass, dependent on the degree of branching and structure. Branching for example decreases the size of the polymers, resulting in a possible underestimated molar mass when using calibration curve. This is why additional analytical equipment in combination with SEC is needed to accurately determine the absolute mass of polymers.

1.2.1.2 Light Scattering

One of the few methods to for obtaining the absolute mass of polymers is light scattering. In light scattering, the oscillating field of the electric part of light creates dipoles from neutral molecules.

These polarized molecules are now able to affect the trajectory of incident radiation, resulting in a scattered beam. This event can take place multiple times, which impacts the intensity of the radiation beam. The decrease of intensity can be related with the mass, the Zimm formula is typically used:12

𝐾𝑐

𝑅𝜃=𝑀𝑃(𝜃)1 + 2𝐴𝑐𝑐 + ⋯ (1.11)

𝐾 is the optical constant, 𝑐 the concentration of the particles, 𝑅𝜃 the excess Rayleigh ratio, 𝑀 the molar mass, 𝑃(𝜃) the particle scattering function and 𝐴2 the second viral coefficient. The excess Rayleigh ratio 𝑅𝜃 is dependent on the intensity of the scattered light beam (Eq. 1.12). The particle scattering function 𝑃(𝜃) is dependent on the shape of the particle, but 𝑃(𝜃) can also be

Polym er Siz e

Elution Volume 𝑉

𝑒

𝑉 𝑖 𝑉 0

Figure 1.4 – Hypothetical elution curve: elution volume 𝑽𝒆 versus polymer size. 𝑽𝟎 is the solvent volume outside the pores, 𝑽𝒊 the solvent volume inside the pores.

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11 approximated with Eq. 1.13, for 𝑞2𝑅𝑔2 < 0.3.13 𝑅𝑔 is the radius of gyration and the scattering vector 𝑞 is dependent on the angle and wavelength of the incident light (Eq. 1.13).11

𝑅𝜃 =𝑟2

𝑉

𝐼𝜃−𝐼𝜃,𝑠𝑜𝑙𝑣𝑒𝑛𝑡

𝐼0 (1.12)

𝑃(𝜃) = 1 +𝑞23𝑅𝑔2 (1.13)

𝑞 =4𝜋

𝜆 sin (𝜃

2) (1.14)

𝐼𝜃, 𝐼𝜃,𝑠𝑜𝑙𝑣𝑒𝑛𝑡 and 𝐼0 are the intensities of the solution, solvent and the used light, 𝑟 is the distance between the solution and detector and 𝑉 is the solution volume. 𝜆 and 𝜃 are the wavelength in a solvent and angle of the used light beam.

With equation 1.11 the molar mass of polymer particles can be determined by static light scattering.

During static light scattering, the light intensity fluctuations are averaged over time. The averages are determined for various angles and concentrations and subsequently can be used to construct a Zimm-plot where the molar mass is obtained by doing extrapolations for 𝑐 → 0 and 𝑞 → 0. A hypothetical Zimm-plot is shown in figure 1.5. The spacing parameter 𝑘 is chosen arbitrarily to simplify the extrapolation process.

The Zimm-plot however is only useful in ‘batch mode’, where concentrations can be varied. In 3D- GPC experiments the elution volumes from the SEC equipment are analyzed by light scattering online, where the concentration is fixed and typically low. For these measurements high-intensity laser lights are applied to give reliable results. Low angle laser light scattering (LALLS), used to measure scattering at a very low angle (at ~7 degrees), or multiple angle laser light scattering (MALLS), where multiple angles are used (~15-160 degrees), measure the light scattering and facilitate the determination of the absolute molar mass.

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12 1.2.2 Characterization of Architecture

The absolute molar mass of polymers is found by combing SEC and LS. However, obtaining detailed information about the branching or structure is more complicated, because there is no direct method available to determine this. In this project intrinsic viscosity and rheological measurements are used to estimate the degree of branching and structure.

1.2.2.1 Dilute Solution Viscosity

When adding polymers to a solution the viscosity of solution increases. Here the hydrodynamic volume that is occupied by the polymer chain increases the viscosity. Specific viscosity 𝜂𝑠𝑝 is used to define the difference between the viscosity with and without the polymer (Eq. 1.15).

𝜂𝑠𝑝=𝜂−𝜂𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

𝜂𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (1.15)

The specific viscosity can be measured by determining the flow rate in glass capillaries or by a differential viscometer.14 If the concentration of these polymers in solution is low enough, polymers are assumed to be isolated and therefore intermolecular polymer-polymer interactions are typically neglected. From here the limit 𝑐 → 0 can be taken to calculate the intrinsic viscosity [𝜂] of the polymer chain (Eq. 1.16).

[𝜂] = lim𝑐→0𝜂𝑠𝑝 (1.16)

For linear polymers, the Mark-Houwink equation is used to relate the intrinsic viscosity with molar mass (Eq. 1.17). 𝑀𝑣 is the viscosity average molar mass and 𝐾 and 𝛼 are constants that depend on the polymer type.

𝐾𝑐

𝑞 2 + 𝑘𝑐

c=0

q=0 c1 c2

cn

q1 q2

qn 𝑹𝒈𝟐

𝟑𝑴

𝟐𝑨𝟐𝒌𝒄 𝟏

𝑴

Figure 1.5 – Hypothetical Zimm-plot. The molar mass, radius of gyration and second viral coefficient are determined for various angles and concentrations and their quantity is obtained by doing extrapolations for c→0 and q→0.

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13

[𝜂] = 𝐾𝑀𝑣𝛼 (1.17)

Both hydrodynamic volume and intrinsic viscosity are dependent on the degree of polymerization, degree of branching, structure and polymer-solvent interaction. The hydrodynamic volume is typically defined in terms of radius of gyration (Eq. 1.18). The mean square radius of gyration is the average square distance between monomer 𝑖 and all other monomers. The average distance is required if the polymer chain fluctuates in size over time (for instance because of Brownian motion).

< 𝑅𝑔2> = 1

2𝑁2𝑁𝑖=1𝑁𝑗=1< (𝑅𝑖− 𝑅𝑗)2> (1.18)

For polymers with the same molar mass, linear polymers have a larger hydrodynamic volume than highly branched and compact polymers, resulting in a larger contribution to the viscosity. Branching ratios 𝑔 and 𝑔′ are used to compare branched chains with its linear equivalent for both hydrodynamic volume and intrinsic viscosity:12

𝑔 =<𝑅<𝑅𝑔2>𝑏𝑟𝑎𝑛𝑐ℎ𝑒𝑑

𝑔2>𝑙𝑖𝑛𝑒𝑎𝑟 |

𝑀 (1.19)

𝑔′ =[𝜂][𝜂]𝑏𝑟𝑎𝑛𝑐ℎ𝑒𝑑

𝑙𝑖𝑛𝑒𝑎𝑟 |

𝑀 (1.20)

[𝜂]𝑏𝑟𝑎𝑛𝑐ℎ𝑒𝑑 is taken from viscosity measurements and [𝜂]𝑙𝑖𝑛𝑒𝑎𝑟 is obtained via the Mark-Houwink relationship. Now 𝑔′ can be calculated directly. When evaluating 𝑔 and 𝑔′ for polymers with specific architecture, the “shielding effect” has to be taken in account.

If we consider a polymer in dilute conditions, the chain segments at the edges of the polymer particle alter the flow of the solvent near the chain segments at the center in such a way that these are partially shielded from hydrodynamic interaction with the solvent flow outside the particle. For large polymers, the contribution of the center segments becomes negligibly small to the resistance to the flow of solvent outside the particle.15,16 Additionally, the amount of branching and the architecture of the polymers affect this contribution.

To include this effect in the evaluation of the two branching ratios, Zimm and Stockmayer defined the structure factor 𝜀:12

𝑔= 𝑔𝜀 (1.21)

This structure factor is derived to be 𝜀 = 0.5 for non-free drained star branched polymers and estimated to be 𝜀 = 1.5 for comb shaped polymers, where the branches are small relative to the backbone.17 For autoclave and tubular LDPE, the values 𝜀 = 1.2 and = 2.0 , respectively, have been found by Kuhn.18 Scholte et al. report values of 𝜀 = 0.8 − 1.0 for low-density polyethylene.19 Tackx et al. determined the 𝑔 factor with SEC-MALLS measurements and found 𝜀 to decrease with increasing molar weight. The structure factors varied for one size autoclave and tubular LDPE between 𝜀 = 1.0 − 1.5 and 𝜀 = 1.2 − 1.8. In general a value for 𝜀 is chosen between 0.5 and 2.0.

𝑔 can be calculated by using relationships derived by Zimm and Stockmayer.20 The first relation is valid for polydisperse trifunctional randomly branched polymers.

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14 𝑔 = 6

𝑛𝑤(1

2(2+𝑛𝑤

𝑛𝑤 )

1

2ln ((2+𝑛𝑤)

1 2+𝑛𝑤

1 2

(2+𝑛𝑤)12−𝑛𝑤12) − 1) (1.22)

Here the weight average 𝑔 is a function of 𝑛𝑤, the weight average number of branch points per molecule. For monodisperse trifunctional randomly branched polymers the next relation can be used:

𝑔 = (1 + (𝑚7)

1

2+49𝑚𝜋)

12

(1.23)

In this relation the average 𝑔 per molecule is described in the average branch points per molecule 𝑚. In addition, Zimm and Stockmayer give a relation to find the structure factor when using the relation mentioned above:

𝑔= 𝑔𝜀 = 𝑔2−𝛼 (1.24)

Here 𝛼 is the exponent in the semi-empirical law [𝜂] = 𝐾𝑀𝛼 for linear polymers. However, Zimm and Stockmayer note “it is still hazardous to draw inferences about branching from empirical viscosity-molecular weight relationships”.20

1.2.2.2 Rheology

An additional way to find more information about the structure of polymers is by conducting rheological measurements. Rheology is used to describe the flow behavior of materials under stress and it is shown that rheological measurement are able to reveal the structure and the degree of branching of polymers.21,22 Introducing even low amounts of long chain branches to linear polymers has a significant effect on the rheological properties.

To measure rheological behavior an oscillatory rheometer is typically used. Here a sinusoidal shear deformation is applied to the sample from which the stress response is measured. In figure 1.6, a rheometer is schematically shown, where the bottom plate oscillates with a certain frequency and the top plate measures the stress response. . Because of the rotating bottom plate, the strain is time dependent (Eq. 1.25). In this equation 𝜔 is the frequency of the strain and 𝑡 the time. The stress response can be proportional to the strain or, for most polymers, can lag behind. Here Eq. 1.26 is used, where 𝛿 is the phase lag.

𝜀(𝑡) = 𝜀0sin(𝜔𝑡) (1.25)

𝜎(𝑡) = 𝜎0sin(𝜔𝑡 + 𝛿) (1.26)

When a material is ideally elastic, the stress response is proportional to the applied strain (figure 1.6). This behavior is modeled by using a Hookean spring, where the applied stress 𝜎 is equal to the strain 𝜀 times the spring (or elastic) constant 𝐸. Equation 1.27, similar to Hooke’s law, is used to model the elastic behavior. The phase lag 𝛿 can be found by inserting Eq. 1.25 and Eq. 1.26 to 1.27.

For an ideally elastic material the phase lag is equal to zero, because the stress is proportional to the strain.

𝜎(𝑡) = 𝜀(𝑡)𝐸 (1.27)

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15 The stress response of an ideally viscous material is proportional to the applied strain rate (figure 1.6). A Newtonian dashpot model can be used to describe the viscous response. Eq. 1.28 is used here, where the applied force 𝜎 is equal to the strain rate 𝑑𝜀 𝑑𝑡⁄ times the viscosity. Here the phase lag is shifted one fourth of a period (𝛿 =𝜋2).

𝜎(𝑡) = 𝜂𝑑𝜀(𝑡)𝑑𝑡 (1.28)

Most polymers are viscoelastic, showing both viscous (dashpot) and elastic (spring) behavior. The phase lag has therefore a value between 0 and 𝜋2. An example of a simple model for viscoelastic behavior is the Maxwell model. In a Maxwell model, the spring and dashpot are connected in series.

The overall strain rate equation results in: 𝑑𝜀𝑑𝑡 =𝑑𝜀𝑑𝑡|

𝑆+𝑑𝜀𝑑𝑡|

𝐷=1𝐸𝑑𝜎𝑑𝑡+𝜎𝜂. Dynamic modulus 𝐺 can be derived from this equation, that is the ratio of stress 𝜎 over strain 𝜀. This modulus can be split into two elements, storage modulus 𝐺′ and loss modulus 𝐺′′, that represent the elastic and viscous behavior of the viscoelastic material, respectively.10 Rheometer measurements typically give the storage and loss moduli.

In these measurements, it is assumed that the applied strain scales linearly with the stress response.

Therefore the dynamic moduli are independent of the applied strain if this assumption holds true.

When the strain amplitude is increased however, deformations, structural changes and phase transitions can occur that give different rheological behavior.20 The strain does not scale linearly with the stress anymore, resulting in nonlinear rheological response. It is shown that nonlinear rheological measurements potentially can be used to provide information about the properties of

𝑅𝑜𝑡𝑎𝑡𝑒

𝑀𝑒𝑎𝑠𝑢𝑟𝑒 𝑅𝑒𝑠𝑝𝑜𝑛𝑠𝑒

𝑀𝑎𝑡𝑒𝑟𝑖𝑎𝑙

𝐸𝑙𝑎𝑠𝑡 𝑖𝑐 𝑉𝑖𝑠𝑐𝑜 𝑢𝑠 𝑉𝑖𝑠𝑐𝑜 𝑒𝑙𝑎𝑠𝑡 𝑖𝑐

𝛿 = 0

𝛿 = 𝜋/2

0 < 𝛿 < 𝜋/2 𝑆𝑡𝑟𝑒𝑠𝑠

𝑆𝑡𝑟𝑎𝑖𝑛

Figure 1.6 – (Left) Schematic representation of a rheometer. The bottom cylinder is oscillating at with a certain frequency.

The material responds to these deformations and this is measured at the top cylinder. (Right) Typical response curves of elastic, viscous and viscoelastic materials. For elastic materials the stress response is proportional to the strain, for viscous material the stress response is proportional to the strain rate and for viscoelastic material the stress response lies in between the elastic and viscous stress response.

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16 the materials, including branching and structure.24,25 Blood vessels for example stiffen (or harden) when a large amount of strain is applied.26 An example of this strain hardening for several biological gels is shown in figure 1.7.26 Here can be seen that for lower strain values the dynamic moduli are independent of the strain, while a dependency occurs for higher strain values. Large Amplitude Oscillation Shear (LAOS) test reveal the nonlinear behavior of materials, where for example the extensional viscosity as a function of time is used. The transient extensional viscosity is the ratio between the extensional stress and the strain rate:

𝜂𝐸+= 𝜎𝑒𝑥𝑡

𝑑𝜀/𝑑𝑡 (1.29)

In experiments, however, it might be difficult to measure the steady state extensional viscosity accurately. Therefore 𝜂𝑒+ is typically compared with linear viscoelastic behavior via the Boltzmann superposition principle:

𝜂𝐸+= 3 ∫ 𝐺(𝑡)𝑑𝑡0𝑡 = 3𝜂+ (1.30)

For a low 𝑡 the extensional viscosity 𝜂𝐸+ over steady state shear viscosity 𝜂+ is equal to 3 (also named the Trouton ratio). With the empirical Cox-Merz rule the steady state shear viscosity is linked to the complex viscosity, when shear rate 𝛾̇ is equal to frequency 𝜔:29,30

𝜂+(𝛾̇) = 𝜂(𝜔) (1.31)

The complex viscosity 𝜂 is obtained from dynamic moduli 𝐺 and 𝐺′′:

𝜂=√𝐺(𝜔)2𝜔+𝐺′′(𝜔)2 (1.32)

For higher 𝑡 values the extensional viscosity can deviate from the linear viscoelastic behavior and is an indicator for certain structural characteristics of the polymer sample. In our case highly branched LDPE is used where strain hardening occurs during extensional viscosity measurements. Here the applied strain strongly orientates the polymer chains that increases the extensional viscosity and stiffens the material. In film blowing for example strain hardening is crucial for stabilizing the resin.

Figure 1.7B shows a hypothetical plot for strain hardening compared with the linear viscoelastic result.

Extensional Viscosity 𝜂𝐸+

Time t

Linear Viscoelastic Behavior Nonlinear Viscoelastic Behavior

Figure 1.7 – (Left) Nonlinear response of various biological materials. Figure obtained from [23]. (Right) Hypothetical transient extensional viscosity curve for a material showing nonlinear viscoelastic behavior (red). The dashed blue line represents the material if it would show linear viscoelastic behavior. The line is obtained from the dynamic moduli and by using equations 1.30-1.32.

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17 1.2.3 The “Branch-on-Branch” (BoB) Model

Rheological measurements give valuable information about polymer samples, but it is hard to extract detailed structure characteristics. In the models of this project the “Branch-on-Branch” (BoB) model of Das et al.4 is incorporated to link molecular characteristics to viscoelastic responses. In this BoB model theoretical understanding of polymer dynamics is used to predict the viscoelastic response of polymer melts. The model is specifically suited for estimating the response of highly branched polymers.

Branched polymers differ in motion with respect to linear polymers because of the presence of branches. A common example is the motion of a 3-armed star, shown in figure 1.8. In a polymer melt the star is entangled with other polymers that restrict the movement of the star. The remaining space the star is allowed to move in has a tube-like shape, as shown in figure 1.8. The tube model, developed by de Gennes31, is used to define the motion of polymers in the space the polymer is allowed to move in. Linear polymers in this tube model would be able to move through the tube with a snake like motion, termed reptation. Adding a branch to the polymer restricts the movement:

the side-arm has to be dragged into to tube section where the polymer is moving into, which entropically unfavorable. This slows down the stress relaxation of the star polymer.

For polymers with more than one branch the relaxation becomes more difficult. Short polymer segments that have a free end relax faster than large segments that are on both sides connected to other segments. To describe the relaxation order of branched polymer hierarchical relaxation is used. First all segments with a free end relax until all segments are relaxed. Next the segments that are connected to these segments are allowed to relax but with added drag that is caused by the relaxed segments. This relaxation procedure is repeated until the whole polymer is relaxed. Here the quantity seniority is used here to describe the hierarchy of relaxation. This quantity is determined by finding the number of segments to the furthers free end on both sides of segment 𝑥. The smallest value of these two is the seniority. Segments with a free end have a seniority of 1 and relax first, the segments connected to these have a seniority of 2 and relax second, et cetera. With this value the

‘deepness’ of the segment in the polymer is specified. Besides hierarchical relaxation additional effects like primitive path fluctuations and constraint release are added to this theoretical frame from which ultimately the linear viscosity behavior is predicted.32

Figure 1.8 – In a polymer melt the star is entangled with other polymers that restrict the movement of the star.

The blue dots represent the topological constrains set by the polymer melt. The star is allowed to move inside the tube represented by the dashed black lines. If the star polymer leaves the tube, the side-arm has to be dragged into to tube section where the polymer is moving into.

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18 Nonlinear viscoelastic behavior is modelled by using the Pom-Pom model.33 A pom-pom polymer consists of a backbone that is connected on both sides with 𝑦 arms that all have a free end. With the pom-pom model the orientation and stretch of the backbone segment is described from which the strain hardening can be estimated. The value priority is used to account for the maximum stretch of segment 𝑥. The number of free ends that lie at the outer ends of the polymer are counted for both sides of segment 𝑥, from which the smallest value of the two is taken as the priority. The priority and seniority values can be used as an indicator for the structure of the polymer.

1.2.3.1 BoB Model Input

In order to use the BoB model, a couple of parameters have to be known a priori: the dynamic dilation exponent 𝛼, the number of polymer units in a entanglement 𝑁𝑒, equilibrium time 𝜏𝑒, the molar mass of a polymer unit 𝑀0, the temperature 𝑇 and the polymer melt density 𝜌. The dynamic dilation exponent 𝛼 is typically varied between 1 and 4/3. 𝑁𝑒 is obtained from the entanglement molar mass 𝑀𝑒 which can be determined from 𝑀𝑒=𝜌𝑅𝑇𝐺

𝑁0 or 𝑀𝑒=45𝜌𝑅𝑇𝐺

𝑁0, where 𝐺𝑁0 is the plateau modulus.34,35 For the equilibrium time 𝜏𝑒= 𝜁𝑎2𝑀𝑒

3𝜋2𝑘𝑇𝑀0 can be used, where 𝜁 is the monomeric friction coefficient and 𝑎 the tube diameter.32 𝑀0 is 28 g/mol for LDPE. In our model the values used by Read et al.36 will be used (Table 1).

The polymers that are used in the BoB model are represented by a matrix that is included in the polyconf file. This polyconf notation is the following. The first line is used for segment 0 that has two possible segments or free ends on each side. If segment 𝑥 is connected to segment 0 the value 𝑥 is added to the first line. If segment contains a free end, the value −1 is added to the line. For a polymer with 2 branches, the polyconf file will look like:

The first four columns are used for the connectivity specification. The fifth and sixth columns are used for the segment mass 𝑀𝑠 (in terms of 𝑀𝑒) and weight fraction 𝑓𝑠 (relative to the total mass of the polymer ensemble).

BoB input parameters 𝛼 (−) 1.0 𝑀0 (𝑚𝑜𝑙𝑔 ) 28 𝑁𝑒 (−) 57 𝜌 (𝑘𝑔

𝑚3) 785 𝜏𝑒 (𝑠) 5.8 ∗ 10−8

𝑇 (𝐾) 423

Table 1 – Input parameter values for BoB algorithm.

-1 -1 1 2 𝑀𝑠,0 𝑓𝑠,0

0 2 -1 -1 𝑀𝑠,1 𝑓𝑠,1

0 1 3 4 𝑀𝑠,2 𝑓𝑠,2 -1 -1 2 4 𝑀𝑠,3 𝑓𝑠,3

2 3 -1 -1 𝑀𝑠,4 𝑓𝑠,4

0 1

2 3

4

Figure 1.9 – Polyconf matrix structure for a regular comb polymer with two branch points.

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19

2 Model 1: From Experimental Data to Detailed Polymer Ensemble

The first model is used to generate a polymer ensemble from experimental data. Here triple detector GPC data is used, specifically molar mass and intrinsic viscosity data, to define an initial polymer ensemble. This ensemble is used as an input for the ‘Branch-on-branch’ (BoB) model of Das et al. (2006).4 The BoB model calculates the linear and nonlinear viscoelastic response of the ensemble and this is compared with experimental rheometer results. The variables of Model 1 are refined iteratively to find the best fit, giving a final polymer ensemble that will be used in part 3 as input for Model 2. Section 2.1 includes a flow sheet of Model 1. In section 2.2 a detailed description of the different steps taken is given and in section 2.3 results are shown for different LDPE samples.

2.1 Flow Sheet

Experimental Data (GPC)

Relate IV with Branching

Ratio 𝒈

Number of branch points

from 𝒈 Structure Selection and

Distribution

Molecular Weight

Intrinsic Viscosity

Number of Branch Points

Generate MWD-BPD

Reduce MWD-BPD

Polym er Ensem

ble

BoB Model

Predicted Rheological

Response

Experimental Data (Rheometer)

Polymer Ensemble

Acceptable

Y

Fit?

N

2.2.1.

Set Variables

2.2.2.

2.2.3.

2.2.4.

2.2.5.

2.2.6.

2.2.7.

2.3.

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20

2.2 Detailed Description Model 1

2.2.1 Experimental Data (GPC)

Experimental data from triple detector GPC are used as input for the model. These include molecular weight distributions (MWD) and the intrinsic viscosity (IV) per molecular weight fraction for LDPE samples. For calculating branching ratio 𝑔, information of linear polymers is required. Here we use Mark-Houwink equation [𝜂] = 𝐾 ∗ 𝑀𝛼, with 𝐾 = 3.8 ∗ 10−4 and 𝛼 = 0.732 In figure 2.1A the MWD and IV of LDPE sample ‘A’ is shown. In this part of the report LDPE A will be used to show the different steps taken in the model.

2.2.2 Relate IV with branching ratio 𝒈

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= 𝐾𝑀𝛼 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= 𝑔𝜀 (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 ∗ 𝑴𝜶, 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

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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-on-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.

A

B

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

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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). 𝑓𝑐𝑎𝑦𝑙𝑒𝑦 or 𝑓𝑐𝑜𝑚𝑏 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.

𝑓𝑐𝑎𝑦𝑙𝑒𝑦(𝑥) =1+𝑒−𝑎(𝑥−𝜇)1 = 1 − 𝑓𝑐𝑜𝑚𝑏(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

𝑤(12(2+𝑛𝑛 𝑤

𝑤 )

1

2ln ((2+𝑛𝑤)

1 2+𝑛𝑤12 (2+𝑛𝑤)12−𝑛𝑤

1

2) − 1) (2.4)

𝑔 = (1 + (𝑚7)

1 2+4

9 𝑚 𝜋)

12

(2.5)

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

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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

< 𝑅𝑔2>= 𝑏2

𝑁𝑡𝑜𝑡𝑎𝑙2𝑁𝑘=1𝑁(𝑘)[𝑁𝑡𝑜𝑡𝑎𝑙− 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):

< 𝑅𝑔2>𝑙𝑖𝑛𝑒𝑎𝑟= 𝑏2

𝑁𝑡𝑜𝑡𝑎𝑙2 ∫ 𝑁(𝑁0𝑁 𝑡𝑜𝑡𝑎𝑙− 𝑁)𝑑𝑁=𝑏2𝑁𝑡𝑜𝑡𝑎𝑙

6 (2.7)

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

𝑔 = 𝑁6

𝑡𝑜𝑡𝑎𝑙3𝑁𝑘=1𝑁(𝑘)[𝑁𝑡𝑜𝑡𝑎𝑙− 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:

𝑔𝑐𝑎𝑦𝑙𝑒𝑦=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)

𝑔𝑐𝑜𝑚𝑏=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

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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 = 𝑔𝑐𝑎𝑦𝑙𝑒𝑦𝑓𝑐𝑎𝑦𝑙𝑒𝑦+ 𝑔𝑐𝑜𝑚𝑏𝑓𝑐𝑜𝑚𝑏 (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)22𝜎2 + 1

√2𝜋𝜎2𝑒(𝑥+𝑏)22𝜎2 (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

C

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Based on our observations in the Rh‐catalyzed hydroformylation of styrene in  combination  with  the  electronic  properties  as  determined  in  section  6.3.2 

We introduce an efficient, scalable Monte Carlo algorithm to simulate cross-linked architectures of freely jointed and discrete wormlike chains.. Bond movement is based on the

(That is, denatured proteins are excluded from plastics.) For the most part plastics have moderate to low chain persistence (chain rigidity), so display sufficient flexibility that

Synthesized by accident while heating diazomethane (Called Poly-“methylene” due to repeating –CH 2 group).