Simulation of Tie-Chains and Entanglements in Semi-Crystalline

Polyethylene

X I A O Y U L A N

Master of Science Thesis Stockholm, Sweden 2012

Simulation of Tie-Chains and Entanglements in Semi-Crystalline

Polyethylene

X I A O Y U L A N

DN240X, Master’s Thesis in Numerical Analysis (30 ECTS credits) Master Programme in Scientific Computing 120 credits Royal Institute of Technology year 2012 Supervisor at CSC was Michael Hanke Examiner was Michael Hanke TRITA-CSC-E 2012:028 ISRN-KTH/CSC/E--12/028--SE ISSN-1653-5715 Royal Institute of Technology School of Computer Science and Communication KTH CSC SE-100 44 Stockholm, Sweden URL: www.kth.se/csc

Abstract

It is well-known that the fraction of chains linking the crystalline regions together has a prime

role in many mechanical properties of semi-crystalline polymers, notably the resistance to slow

crack growth. An advanced computationally efficient computer program for modeling crystalline

layers and for calculating the amount of tie-chains and entanglements in these model systems has

been developed and evaluated. The Monte-Carlo model was used to investigate how the tie-chain

and entanglement fraction was affected by relative amorphous density ( /a c ), crystal thickness

(Lc), amorphous thickness (La), temperature ( T ), chain length and branches. The simulation

result shows that the influence of entanglements are usually greater than the influence of tie-

chains, even though they are often neglected in other commonly used models, like the famous

Huang-Brown model. The numerical efficiency of our algorithm is ( *log )O n n where n is the

filling degree of occupied stems in computational domain. Moreover, we did some initial

research of small molecules diffusion in semicrystalline polymers. The simulation program is

both reliable and time-efficient.

Simulering av kristallöverbryggande polymerkedjor

och av infrusna kedjeintrasslingar i semi-kristallin

polyeten.

Sammanfattning

Inom polymervetenskap är det välkänt att andelen kedjor som sammanlänkar olika kristallskikt i

semikristallina polymerer påverkar många mekaniska egenskaper hos materialet, i synnerhet

motståndskraft mot långsam spricktillväxt. Ett avancerat och beräkningseffektivt Monte-Carlo

datorprogram för att modellera växelvis kristallina och amorfa lager av polyeten och för att

beräkna andelen kristallöverbryggande kedjor och andelen infrusna kedjeintrasslingar i dessa

simulerade system har utvecklats och utvärderats. Modellen användes för att undersöka hur dessa

två egenskaper påverkas av relativ amorf densitet ( /a c ), kristalltjocklek (Lc), amorf tjocklek

(La), temperatur ( T ), molviktsfördelning (Mw) och förgreningsgrad. Simuleringsresultaten

indikerar att kedjeintrasslingarna verkar ha minst lika stort inflytande på materialets egenskaper

som de kristallöverbryggande kedjorna, detta trots att de försummas i de flesta andra modeller

inom detta fält. Detta gäller inte minst den berömda Huang-Brown modellen. Monte-Carlo

modellens numeriska komplexitet avseende antal fyllda positioner i kristallgittret är av ordning

( *log )O n n och implementeringen av programmet är både pålitlig och tidseffektiv. Utöver

utvecklandet av Monte-Carlo modellen gjordes även viss initial forskning rörande diffusion av

små molekyler i semikristallina polymerer.

Contents

1. Introduction .................................................................................................................................. 1

2. Background .................................................................................................................................. 4

2.1 Huang and Brown’s model ..................................................................................................... 4

3. Method ......................................................................................................................................... 9

3.1 Crystal layer building and tie-chain calculation ..................................................................... 9

3.2 The knot algorithm for entanglement ................................................................................... 14

3.3 The spline method ................................................................................................................ 22

3.4 Initial step for small molecules diffusion research ............................................................... 24

4. Result .......................................................................................................................................... 31

4.1 Result of tie-chain concentration .......................................................................................... 31

4.2 Result of entanglement concentration .................................................................................. 37

4.3 Numerical efficiency of tie-chain and entanglement algorithm ........................................... 49

4.4 Result of initial step for small molecules diffusion research ............................................... 51

5. Discussion and Future work ....................................................................................................... 55

5.1 Discussion ............................................................................................................................ 55

5.2 Future work .......................................................................................................................... 56

6. Summary .................................................................................................................................... 57

References ...................................................................................................................................... 58

Introduction

1

1. Introduction

In material science, fracture toughness describes the ability of a material containing a crack to

resist fracture. It is one of the most important properties of materials for design applications. For

instance, the polyethylene resins used for pipe manufacture must withstand rapid crack

propagation (RCP), which means a craze is propagating with speed greater than tens of meters

per second along the whole length of a pipe. Furthermore, polyethylene must also resist slow

crack growth (SCG), which is when a craze is growing slowly from a point of stress

concentration and finally causes fracture of the material. The slow crack growth behavior of

polyethylene depends primarily on the molecular structure. The important factors are molecular

weight distribution, type of short branches, density of the branches, the distribution of the branch

density relative to the molecular weight distribution and probably the distribution of the branches

within the individual molecule.

Polyethylene (PE) is the polymer of choice when modeling semicrystalline polymer. Results data

and ideas obtained from PE studies are valuable also in the study of other semicrystalline

polymers. Developments of polymerisation methods have made it possible to tailor-make

polyethylenes of different crystallinities, morphologies, molar distributions and branching

distributions. The availability of polyethylene samples of different molar masses and different

degrees of branching is one of the reasons why polyethylene morphology has been the subject of

so many investigations. Another reason is that polyethylene is used in large quantities and in

many different applications. The properties of polyethylene are controlled by the morphology.

When polyethylene and similar semi-crystalline polymers are cooled from melt, typically small

crystal nuclei eventually form throughout the material. From these nucleation positions, stacks of

thin crystal lamellae starts growing radially outwards, often forming superstructures like

spherulites and axialites. The orientation of each crystal lamellae is depending on both the

chemical and physical properties of the material and on the processing conditions. However, they

usually become more parallel with increasing crystallinity. Between the layers a portion of

amorphous (i.e. non-crystalline) polymer always remains. The existence of crystal lamellae in

melt-crystallized polyethylene was independently shown by Fischer[1]

and Kobayashi[2]

. They

Introduction

2

observed stacks of almost parallel crystal lamellae with amorphous material sandwiched between

adjacent crystals.

A polymer chain leaving a section of a crystal lamellae thus has the following options: (a)

immediately return into the same crystal from where it came, (b) enter the amorphous region for a

while but eventually return into the initial crystal, (c) enter the amorphous region and end there,

and (d) enter the amorphous region and propagate to an adjacent crystal lamellae (see figure 1-1).

The fourth alternative means that a tie-chain is formed. The number of tie-chains leaving a unit

area of fold surface largely controls the fracture toughness of semi-crystalline polymers like

polyethylene. From a functional point of view, the best definition of a tie molecule is a molecule

that supports the applied stress and joins the crystal lamella together. The existence of tie

molecules was beautifully demonstrated in the microscopy study by Keith et a1[3]

. The central

role of tie molecules in determining the fracture strength of crystalline polymers has been

proposed by Peterlin[4]

, Backman and DeVries[5]

, Gibson et al.[6]

, Brown and Ward[7]

, Lustiger

and Markham[8]

, and Lu et al[9]

. It is suggested that the sliding of the tie molecules through the

crystal and through the entanglements in the amorphous region is the fundamental process of

failure. It is the number of tie molecules and whether they are pinned that control the rate of crack

growth which involves the disentanglement of the fibrils in the craze.

Figure 1-1: Schematic description of the possible shapes of polymer chains entering the amorphous

region between two crystalline layers: (a) tight fold, (b) statistical loop, (c) loose chain end, (d) tie chain

Introduction

3

Several theoretical models for estimating the tie-chain concentration for linear polyethylene have

previously been suggested; among others the well-known Huang and Brown[10]

model. This

method captures the probability of a molecule with a particular molecular weight and hence chain

length, to form a tie molecule by traversing a critical distance between lamellae. The problem of

Huang and Brown’s method resides in the omission of the SCB distribution influence over the tie

molecules estimation and does not consider chain entanglements or branch type.

In many fields of science, old analytical tools are replaced with simulations. Modern computers

are fast and it is possible to carry out simulations and obtain results within reasonable short time

periods. Simulations allow more freedom in the design of the conditions for building a system

and the results obtained can be represented in many different ways in order to fit the requirements

set by the user.

This project used a Monte-Carlo method to build a stack of crystal lamellae. We systematically

study how the tie-chain concentration is affected by relative amorphous density ( /a c ), crystal

thickness (Lc), amorphous thickness (La), temperature ( T ), chain length and branches. The

trapped entanglements have also been assessed by simulation and we also investigate the

efficiency of the whole tie-chain and knot algorithm. What is more, we are also interested in

small molecule diffusion in semicrystalline polymers. Furthermore, some initial research work is

done in order to prepare for future molecular dynamics simulations. All these features will be

presented in this report.

Background

4

2. Background

The slow crack growth (SCG) behavior of polyethylene is determined mainly by the molecular

structure. Many important features could influent it, such as molecular weight distribution

(MWD), type of short chain branches(SCB), density of the branches, the distribution of the

branch density relative to the molecular weight distribution and possibly the distribution of the

branches within the individual molecule. In the past several analytical models have been

developed aiming at obtaining the qualitative and quantitative relation between SCG and

different factors of molecular structure. In the following, the well-known Huang and Brown’s

model will be introduced.

2.1 Huang and Brown’s model

2.1.1 The effect of molecular weight on slow crack growth

In Huang and Brown’s work[11]

, they inspected the rate of slow crack growth in polyethylene in

terms of molecular weight of a series of homopolymers. Their results can be summarized as:

5 110,000/

0( 18,000)

RT

w

Ae

(1)

where 0 is a measure of the slow crack growth rate in terms of the rate of the crack-opening

displacement; A is a material constant; is the stress on the single-edge notched tensile

specimen and R ( /J molK ) is the gas constant. Equation (1) indicates that the temperature and

stress reliance on the slow crack growth rate is independent of w which is the weight average

molar mass for these homopolymers. Moreover, the material is extremely brittle and slow crack

growth is impossible for w that are less than 18,000. Based on Huang and Brown’s analysis, the

effect of molecular weight is based on the increase in tie molecules with molecular weight.

2.1.2 The effect of branch density on slow crack growth

Then Huang and Brown investigated the effect of branch density on slow crack growth[12]

. They

used a series of ethylene-hexene copolymers with about the same molecular weight to measure

Background

5

the effect of the density of butyl branches. In experiment, slow crack growth was observed in

notched tensile specimens under plane-strain conditions.

They deduce the following equation to evaluate the effects of both molecular weight and branch

density:

0

0 0

1 1exp

( 18,000)

n

w

D Q

R T T

(2)

Here, 0 is a measure of the slow crack growth rate in terms of the rate of the crack opening

displacement. is the applied stress and 0 is a reference stress. Q is the activation energy, n

is a material parameter, and 0T is a reference temperature. Q is 115 kJ/mol for the

homopolymers and is 125 kJ/mol for the copolymers. n is equal to 5 and independent of

molecular weight and branch density except for its value is 3.3 for the material with 4.6 butyl

branches/ 1000 C. Thus, the main influence of branch density is in the material parameter D .

They proposed that the effect of branching is to increase the probability of forming a tie molecule.

The process for how the branching increases the tie molecule is as follows: as the branching

density increases, the lamella thickness decreases and the long period decreases. The probability

of forming a tie molecule is related to the size of the random coil in the melt with respect to the

long period. If a random coil is larger than the long period, then it is possible to form a tie

molecule.

Moreover, Huang and Brown proposed there are a number of ethylene-hexene copolymers whose

values of w and average branch density are almost the same, but the slow crack growth rates

varies by as much as 310 . This means that the details of the molecular weight distribution and of

the branch distribution relative to the molecular weight distribution are also very important

factors to influence slow crack growth.

2.1.3 Optimal value of branch density in resin manufacturing

Huang and Brown also found that if the density of butyl branches varies from 0 to 4.6

branches/l000C, the rate of slow crack growth decreases by a factor of 410 [12]

(see figure 2-1).

This effect is mainly caused by the difference in the fraction of tie molecules. This is a very

useful detection. A lot of research has been made by resin manufacturers for the purpose of

producing better polyethylenes with long life-time for critical applications such as water and gas

Background

6

pipes and as containers for toxic waste. One wildly used type of resin is the ethylene-hexene

copolymer with about 4.5 butyl chains/1000C. However, it seems the value 4.5 butyl chains per

1000 carbon atoms was simply based on trial and error and is not explained in any published

papers. In this case, it is useful to obtain scientific data on the effects of branch density on slow

crack growth to help us understanding the phenomenon and make the resin design more reliable

in the future.

Figure 2-1 : Initial rate of craze growth vs. branch density at 42 C ,3MPa[12]

.

Based on Huang and Brown’s research, it seems that 4.6 butyl branches per 1000 C is the optimal

value for minimizing slow crack growth. The likely explanation is as follows: when the branch

density increases, the number of tie molecules increases while the lamella crystal thickness

decreases. Then the distance that a tie molecule needs to go through before it is released from the

crystal decreases with lamella thickness. Hence, tie molecules increases and the lamella thickness

decreases with branch density increases cause a minimum in the function of 0 , versus d . Based

on figure 2-1, when branch density is 4.6 butyl branches per 1000 C, 0 is minimized.

2.1.4 The model of tie chain and slow crack growth

From the previous study, Huang and Brown reached the conclusion that the number of tie chains

leaving a unit area of fold surface largely controls the rate of slow crack growth of polyethylene.

They propose a model for tie chain calculation and a model for slow crack growth[10]

.

Experimentally, the lamellae thickness and the long period were evaluated as functions of the

branch density. The tie molecules calculation was based on the values of the molecular weight

and the long period. The model for slow crack growth was based on the rate of disentanglement

Background

7

of the tie molecules. The rate of disentanglement changes inversely with the number of tie

molecules and directly with the number of tie molecules which are not pinned by the branches.

The basic assumption to calculate the probability of a tie molecule forming is as follows: If the

end-to-end distance of a molecule in the melt is larger or equal to the distance between adjoining

lamellar crystals, then it is possible for a tie molecule forming. If the end to end distance is

smaller than the amorphous region thickness between the crystals, then a tie molecule can’t form.

The critical distance between the lamella crystals must be set to make a specific calculation. It is

assumed that if the end-to-end distance of the random coil in the melt is larger than (2 )c aL L , a

tie molecule will be formed, where cL is crystal thickness and aL is the amorphous layer thickness.

Note, the choice of the critical distance could influence the absolute value of the calculated tie

molecules concentration, but the relative number of tie molecules concentration, which is

obtained by multiplying tie-chain probability with the volume fraction crystallinity of polymer, is

little influenced by the certain choice of the critical distance.

The probability for forming a tie molecule is as follows:

2 2 2

2

2 2 2

0

exp( )1

3 exp( )

c aL Lr b r dr

Pr b r dr

(3)

Here, r is the end-to-end distance of a random coil, 2 23/ 2b r , 2r is the root-mean-square value

of the end-to-end distance of a random coil. The factor 1/3 was introduced since two dimensions

of the lamella crystals are much greater than the long period, thus the probability of forming tie-

chains in these two dimensions can be neglected.

Huang and Brown also developed a theoretical model to describe the rate of slow crack growth in

an ethylene-hexene copolymer as a function of the basic morphological parameters. The

parameters are: the spacing of the butyl branches, the number of tie molecules and the thickness

of the lamellar crystal.

/

0

n Q RT

c

Ae

BL t

(4)

Here, 0 is a measure of the slow crack growth rate in terms of the rate of the crack opening

displacement. A is a constant. is the fraction of tie molecules. B is a constant that is related to

the strength of the bonds between the tie molecule and the crystal, cL [nm] is crystal thickness, t

Background

8

is the number of tie molecules per unit cross-section area of the fibril within the amorphous

region, is the applied stress, Q is the activation energy, n is a material parameter and the gas

constant R is in ( /J molK ). By comparing eqs (1) and (4), the factor D , which determines the

experimental quantity, is better understood.

The slow crack growth process consists of the disentanglement of molecules in the fibrils of a

craze. Huang and Brown also showed how the morphology of the fibrils changes with the branch

density. The fibrils become coarser with increasing branch density, which is the fact that suggests

the coarser fibrils cause a greater resistance to the disentanglement process.

Method

9

3. Method

In the past, several theoretical models have been developed aiming at obtaining different

measures of the tie-chain concentration for linear polyethylenes; among others the well-known

Huang and Brown model. Many molecular dynamics studies have also been focused on the

amorphous interlayer of polyethylenes. Anyhow, to our knowledge, none of the previous

simulations have systematically studied how the tie-chain concentration is affected by relative

amorphous density ( /a c ), crystal thickness (Lc), amorphous thickness (La), temperature (T ),

chain length and branches. Neither have the concentration of trapped entanglements been

assessed by simulation. All these features have been taken into account in the model presented

here. Moreover, we are also interested in diffusion of small molecule compounds in

semicrystalline polymers. Some initial steps of this research are included also.

3.1 Crystal layer building and tie-chain calculation

3.1.1 Overview of the model

The aim of the model is to get a sufficiently realistic tool for estimating the tie-chain

concentration between parallel crystal layers (see figure 3-1). The main variables are: polymer

branching density, crystal thickness (Lc), amorphous thickness (La), amorphous density (a),

crystal density (c) and crystallization temperature (T). In this model, parallel semi-crystalline

layers were simulated with Monte-Carlo technique. In the amorphous layers between the crystals,

atomistic Flory ghost chains simulations are performed, continuing until the chain touches a

crystal layer. The positions of the carbon atoms in the inter-crystalline layers could be stored for

further analysis.

Method

10

Figure 3-1: Parallel lamellae stacks model[13]

. A sandwich based on two crystal lamellae and an

amorphous phase in between.

3.1.2 Geometry settings As one of the first steps of the model, the settings of the initial geometry are defined. The user

determines how many parallel crystal layers (positioned in the x-y plane) that should be used.

The size of the computational domain in the x-direction (Dx) and in the y-direction (Dy), are also

user defined. In order to obtain correct values even for long chains, these values must be chosen

large enough. The crystal thickness (Lc) can either be stated explicitly or be calculated from the

Gibbs-Thomson equation (equation 5), which gives the crystal thickness in nanometers as

function of melting temperature (Tm) in Kelvin for a given equilibrium melting temperature (Tm0).

mm

mc

TT

TL

0

0624.0 (5)

The amorphous thickness can either be stated explicitly or be calculated from equation 6 (see

DesLauriers et al’s research [14]

):

ca

ccc

aw

wLL

)1( (6)

ac

ac

cw

(7)

)log(0241.00748.1 M (8)

Once the size of the computational domain is set, the program pre-allocates a decent number of

empty matrices and pre-calculates a structure based on an orthorhombic grid including how the

mesh-points are positioned and connected to each other. The domain is periodic in all three

directions.

Method

11

3.1.3 Polymer Settings There are a number of different possibilities for the user to change the structure and the

conditions for the simulations. First and foremost the crystallization temperature (T) is stated and

then the chain-length distribution of the main-chains is chosen. If all chains have the same length

this parameter is a scalar value, otherwise it should be submitted as a n*2 matrix histogram,

where the first column contains molecular weights and the second column contains the relative

number fraction of respective molecular weight.

The weight fraction of branches and the average length of each branch should also be stated.

These variables can either be scalars or vectors/matrices. The average chain lengths can be

submitted as a distribution just as for the main chains. The weight fraction of branches can be a

function of chain-length and is stored in matrix histogram where the first column is chain-length

and the second is weight fraction of branches.

3.1.4 Main structure of the code

Pseudo code for the chain simulation for repeated simulation times(100) do

while filling degree of stems < set value (0.5) do

A chain starts at a random position in amorphous layer. 0 1p .

while the chain doesn’t touches a crystal layer do

Flory ghost chain simulation is performed

end while

if the chain touches one of the crystal surfaces and the remaining chain-length > Lc then

if one of the stem positions which is close enough is empty then

The chain enters the crystal and proceed to the other side of the crystal

if the chain emerges on the other side of the crystal surface then

calculate 1 ( / ) ( / )i i A C i A C AIMp p , generate random number q : 0 1q

if 1iq p then

Immediate re-entry at a adjacent position

else continue with a new random walk in amorphous region

end if

end if

else The chain continue walking in the amorphous region

end if

end if

end while

end for

Method

12

Detailed description:

At the beginning of each simulation, the starting-point of one polymer chain is positioned at a

random amorphous position pointing in a random direction. The positions of the subsequent

atoms are determined with the temperature dependent Flory Monte-Carlo ghost chain simulation

and carbon-carbon bond length (0.154nm) until either the other chain end is reached or until one

of the crystal layers is hit. Positions of eventual branches are stored continuously. The molecules

behave like ghosts if segments of a molecule don’t sense the other segments of the same

molecule. The basic idea of a Flory chain is that the probability for the chain to reach one of the

three main states trans, gauche, anti-gauche (see figure 3-2) is dependent on the previous state[15]

.

For polyethylene the relation matrix can be written as Eqn9:

01

01

1

U (9)

)exp(RT

E , E=5.4 kJ/mol, 8.31 300RT J/mol.

1

1 2 1 2 1 2

10

1 1

10

1 1

P

(10)

Figure 3-2 : Conformational states of n-butane. Carbon – dark, hydrogen – white. The carbon atoms are

all in one plane for a butane molecule in trans state. The carbon skeleton of butane in the two gauche

states is non-planar and the two states are mirror images with respect to a plane containing three of the

carbon atoms[15]

.

The interpretation is the following: If the previous state is trans, look on row 1 in U, if it is

gauche then look on row 2 and if it is anti-gauche then look on row three. The probability for the

Method

13

first state (trans) equals the number in the first column in the right row divided by all values in

the row et cetera. So the probability for the next carbon in a polyethylene chain to enter a specific

state can be determined from Eqn10 where the rows correspond to the previous state while the

columns correspond to the next state. This means for instance that the probability for obtaining

trans after gauche is )1/(1 .

If the chain touches one of the crystal surfaces and the remaining chain-length is larger than the

crystal thickness and there are no branches on that chain sequence, then the chain will try to enter

the crystal surface. All stem positions inside a (user-defined) radius are fetched. If at least one is

empty, the closest empty position is chosen as starting node for the new entry, otherwise the

chain has to continue walking in the amorphous region. Note that there are periodic boundary

conditions.

Once the chain emerges on the other side of the crystal surface it gets the opportunity to either do

an immediate re-entry at one random of the six adjacent positions (see figure 3-3), assuming any

of these are empty, or to continue with a new random walk in the amorphous region. The

probability ( 0 1ip ) of adjacent reentry is controlled by the model as follows:

1 ( / ) ( / )i i A C i A C AIMp p

so the desired proportions between amorphous and crystalline densities are obtained.

Figure 3-3: The six adjacent stem positions of the yellow point are all in the red circle[15]

.

If a chain emerges from one crystal layer and performs a random walk that ends in another layer,

then a tie-chain has been created. The number of tie-chains is stored as function of filling degree.

If branches are present, then each branch take one “step”, i.e. either a random walk or an adjacent

reentry, each time the main chain has performed its step. This continues until both the main chain

and all its branches are traversed. Then a new chain is initiated and the procedure begins again.

Method

14

When finally a sufficiently large fraction of the stems are filled (user defined scalar, preferably

0.4-0.8), the key results from the simulations are stored.

The whole simulation is then repeated a number of times, preferably at least 100 times, in order

to obtain sufficiently reliable average values of tie-chain fraction.

3.1.5 Stored matrices The properties that can be obtained as out-data so far is (1) matrices storing all particle positions

in 3D for the last simulation, (2) matrices storing information of how the stems were filled, (3)

matrices storing averaged values of tie-chain fractions, amorphous density, adjacent reentry

probabilities and other properties as function of filling degree. These matrices are sufficient for

the task to plot the amorphous density distributions in 1D-3D, to plot schematic folding

visualizations (see figure 3-4), to plot tie-chain fractions etc as function of branching degree and

to use as in-data for the knot-algorithm.

Figure 3-4: Schematic visualisation of four polyethylene crystal layers. Filling degree 50% and

computational domain 20x20x30 nm.

3.2 The knot algorithm for entanglement

3.2.1 Overview of the model It has previously been noted that other kinds of entanglements than tie-chains can be at least as

important as tie-chains (see figure 3-5). In the extension of Huang and Brown’s model proposed

by Yeh and Runt[16]

, chain entanglements in the amorphous layers are taken into consideration in

parallel with tie-chains. The calculations are based on the probability for two entangled chains to

Method

15

crystallize into two different adjacent lamellae. It turns out that the probability of chain

entanglement is much greater than that of conventional tie-chains. Chain entanglements are

complications not directly addressed by the classical statistical mechanics theory. It adds more

junction points to the covalent network. In order to evaluate the importance of the chain

entanglements, we also developed a code for calculating entanglements of the kind that is formed

when one loop with both ends on one crystal layer is connected to another loop with both ends on

the other side. Input data to the model comes from the main model previously described.

Figure 3-5 : Sketch of tie chain (left) and entanglement (right) [13]

.

3.2.2 Algorithm Details

Knot Algorithm:

1. Tie-chains and the chains with loose ends are first removed. The chains starting and ending

on the lower face are placed in one group and those starting and ending at the upper face are

placed in another group.

2. Chains are projected in xy plane.

3. The self-intersection points should be calculated and stored. It should be determined which

part of the chain is above the other at all self-intersection points.

4. All chains in the upper group should be compared with all chains in the lower groups. The

comparisons are always done pair-wise. Find the positions of all the intersection points. It

should be noted which of the lines is positioned above the other at the intersection point.

Method

16

5. When all well-defined intersection points (nodes) are found, and the order of the intersections

are determined for both chains, and it is known which chain is above the other, then it is

possible to determine if the sequence of intersections results in an entanglement or not.

(1) If one chain is followed and each intersection point where this chain is above the other is

denoted (+) and otherwise (-), then a row of signs will result, for instance [- + -].

(2) If two neighbouring (+) or (-) are found to be caused by neighbouring chains nodes, then

they can be removed.

(3) If a (+ -), (+ - +) or (+ - + -) sequence is found to be adjacent to a self-cross node where

the end that is firstly crossed by this chain is below, then the whole sequence can be replaced

by (-). (And vice versa for a (- + - +) sequence).

(4) If the remaining sequence contains a (+ -) or (- +), then an entanglement has formed

between the two chains.

Implementation details:

Step 1: Given is a nx3 matrix with x, y and z positions for all chains. The chains in the matrix are

separated by rows with [-1 -1 -1]. The chains should be sorted so that the tie-chains and the

chains with loose ends are first removed. The chains starting and ending on the lower face are

then placed in one group and those starting and ending at the upper face are placed in another

group.

Details:

(1) Since the crystal lamellas are parallel with the x-y plane, the z position of the starting and

ending atoms in each chain are compared with the z position of lamellas, based on which one

chain is removed or placed in group.

(2) Although the dimension of the matrix which stores the chain group could changes

dynamically, we calculate the number of atoms in each group first. After that, setting the matrix

with fixed dimension, which makes it more efficient.

Step 2: For each chain, the x and y values stored should be used to fill some of the positions in a

sparse matrix. If the domain in the xy plane is [0 Dx]*[0 Dy], then the grid spacing should be

chosen such that it is slightly larger than the distance r between the carbon atoms. As an simple

example, if Dx=Dy=1000, r=0.9 and dx=dy=1, then a carbon atom positioned at [2.3, 4.6] should

Method

17

result in that the [3, 5] position in the sparse matrix becomes filled with a “1”. If several atoms in

a row are positioned in the square, then they should still just result in a “1”. However, it they are

separated, indicating that the chain is overlapping itself, the ones should add, giving a “2” or

more.

Step 3: Some of the chain-steps will result in pure diagonal movements in the sparse matrix. In

this case one of the two missing corners must be filled such that the sparse chain becomes

continuous. This is done by drawing a line between the two points in the diagonal squares and

finding a new point in one of the missing corners. Note that the coordinates of the new point

should also be stored, not only the sparse matrix representation of it.

Details:

(1) If one atom is in square (x,y) and the following atom is in square(x+1, y+1), then there is a

diagonal movement.

Notice that the boundary condition is periodic B.C, which means, if the x position of atom a is in

the boundary square nx and the x position of the atom b is in the boundary square 1, then there is

also a diagonal movement between a and b. We need to find the missing corners for this kind of

diagonal movement occurs on the boundary.

(2) To find the new point on the line between the two points a and b in diagonal squares, we use

the “Bisection method”:

[1] Draw a line between a and b.

[2] Choose the middle point c of the line.

[3] Judge if c is in one of the miss corners.

[4] If so, the new point is founded. Otherwise, if c and b are in the same squares, set c as b, go

back to [1]. Do the similar thing if c and a are in the same squares.

Step 4: The self-intersection points should be calculated and stored. It should be determined

which part of the chain is above the other at all self-intersection points.

Details:

(1) The way that how to calculate the self-intersection points is shown as following:

[1] Based on Step 2, find all the squares whose value is larger or equal to “2”.

Method

18

[2] For one square, find all the points that are in it. In figure 3-6, they are b, c and f. b and c are

neighbour points.

[3] For all the inner points, find all their neighbour points. Here are a, d, e and g.

[4] Calculate the intersection points of all the lines, which are between ab & ef, ab & fg, bc & ef,

bc & fg, cd & ef, cd & fg. If one of the intersection points is on both lines then it is the self-

intersection point. Here, the self-intersection point is on bc and fg.

For points b 1 1( , )x y , c 2 2( , )x y , f 3 3( , )x y , g 4 4( , )x y , the position of the intersection point ( , )x y is:

1 2 1 2 3 4 1 2 3 4 3 4

1 2 3 4 1 2 3 4

1 2 1 2 3 4 1 2 3 4 3 4

1 2 3 4 1 2 3 4

( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )

x y y x x x x x x y y xx

x x y y y y x x

x y y x y y y y x y y xy

x x y y y y x x

(11)

d

c

ba

e

f

g

Figure 3-6

(2) Note that the boundary condition is periodic B.C. If the square is on boundary, the position of

at least one point of a, d, e, g need to be found.

Step 5: When all chains are given sparse matrix representations, all chains in the upper group

should be compared with all chains in the lower groups. The comparisons are always done pair-

wise. The first step in the comparison is simply to add the sparse matrix 2D representations of the

two chains to each other and find all positions where intersections occurred, i.e. the matrix sum is

2 or more.

Step 6: The next step is to find the more exact positions of the intersections. This is done with

standard line-line intersection algorithm. It should be noted which of the lines is positioned above

the other at the intersection point.

Method

19

Details:

(1) The way to calculate the intersection points is similar with the way to calculate self-

intersection points:

[1] Based on Step 5, find all the squares whose sum value is larger or equal to “2” and the values

of the square for both chains are not 0.

[2] For one square, find all the points of two chains that are in it. In figure 3-7, they are b, e.

[3] For all the inner points, find all their neighbour points. Here are a, c, d and f.

[4] Using Equation 11, calculate the intersection points of all the lines, which are between ab &

de, ab & ef, bc & de, bc & ef. If one of the intersection points is on both lines then it is the self-

intersection point. Here, the self-intersection point is on bc and ef.

a

b

c

d

e

f

Figure 3-7

(2) Be noticed that the boundary condition is periodic B.C. If the square is on boundary, the

position of at least one point of a, c, d, f need to be found.

Step 7: When all well-defined intersection points (nodes) are found, and the order of the

intersections are determined for both chains, and it is known which chain is above the other, then

it is possible to determine if the sequence of intersections results in an entanglement or not.

(1) If one chain is followed and each intersection point where this chain is above the other is

denoted (+) and otherwise (-), then a row of signs will result, for instance [- + -].

(2) If two neighbouring (+) or (-) are found to be caused by neighbouring chains nodes, then they

can be removed.

Method

20

(3) If a (+ -), (+ - +) or (+ - + -) sequence is found to be adjacent to a self-cross node where the

end that is firstly crossed by this chain is below, then the whole sequence can be replaced by (-).

(And vice versa for a (- + - +) sequence).

(4) If the remaining sequence contains a (+ -) or (- +), then an entanglement has formed between

the two chains.

Details:

The implement of this step is shown as following.

[1] Condition (3),

For (+ -) sequence, we use figure 3-8 to interpret this condition.

There is (+ -) sequence on chain m. We find the corresponding intersection points a and b on

chain n since there maybe self-cross node on chain n. c and d are the pair of points that we used

to get the self-cross point in Step4. As shown in figure 3-8 a, if there is self-cross point, then the

points sequence on chain n should be c a b d. As shown in figure 3-8 b, if there is self-cross point,

then the points sequence on chain n should be a c b d. So if we could find pair of points c and d

which satisfy c b d, then the whole sequence (+ -) can be replaced by (-).

Figure 3-8 a b

For (+ - +) sequence, we use figure 3-9 to interpret this condition.

There is (+ - +) sequence on chain m. We find the corresponding intersection points a, b and c on

chain n since there maybe self-cross node on chain n. d and e are the pair of points that we used

to get the self-cross point in Step4. As shown in figure 3-9 a, if there is self-cross point, then the

points sequence on chain n should be a d b c e. As shown in figure 3-9 b, if there is self-cross

point, then the points sequence on chain n should be a d b e c. So if we could find pair of points d

and e which satisfy a d b e, then the whole sequence (+ - +) can be replaced by (-).

Method

21

Figure 3-9 a b

For (+ - + -) sequence, we use figure 3-10 to interpret this condition.

There is (+ - + -) sequence on chain m. We find the corresponding intersection points a, b, c and

d on chain n since there maybe self-cross node on chain n. e and f are the pair of points that we

used to get the self-cross point in Step4. If there is self-cross point, then the points sequence on

chain n should be a e b c f d. So if we could find pair of points e and f which satisfy a e f d, then

the whole sequence (+ - + -) can be replaced by (-).

Figure 3-10

[2] Condition (2)

Go through the row of sequence, remove the neighbouring (+) or (-) if they are caused be

neighbouring chain nodes.

[3] Condition (4)

Go through the row of sequence, count the number of (+ -) or (- +) which are the number of

entanglements between two chains.

Method

22

Note: If there is just one intersection point for one chain, then we add the starting and ending

points of the chain to get the row of sign. For instance, the chain starting and ending on the lower

face, we add two “-” at the beginning and ending of the row of sign.

3.3 The spline method

We use two-chain model to test and verify the knot algorithm for entanglement. In two-chain

model, one chain starts and ends at the lower face and the other chain starts and ends at the upper

face. In order to simulate the position of these two chains, we give the positions of a few atoms

and then use Kochanek-Bartels spline[17]

to do interpolation.

3.3.1 Overview of spline

In numerical approximation, spline is sufficiently smooth piecewise-polynomial function. Spline

interpolation is preferred to polynomial interpolation since it yields similar results, even when

using low-degree polynomials, while avoiding Runge’s phenomenon for higher degrees. The

most commonly used splines are cubic spline, i.e., of order 3.

A cubic Hermite spline is a third-degree spline with each polynomial of the spline in Hermite

form. The Hermite form consists of two control points and two control tangents for each

polynomial. The smoothness of cubic Hermite spline is 1C . For interpolation on a grid with

points ip for 0,...,i n , interpolation is performed on one subinterval , 1( )i ip p at a time, tangent

values are predetermined.

3.3.2 Kochanek-Bartels spline

A Kochanek-Bartels spline or Kochanek-Bartels curve is a cubic Hermite spline with tension,

bias, and continuity parameters defined to change the behavior of the tangents.

Given n + 1 knots, 0 ,..., np p , to be interpolated with n cubic Hermite curve segments, for each

curve we have a starting point ip and an ending point 1ip with starting tangent iT and ending

tangent 1iT defined by Equation 12:

Method

23

1 1

1 2 1 1

(1 )(1 )(1 ) (1 )(1 )(1 )( ) ( )

2 2

(1 )(1 )(1 ) (1 )(1 )(1 )( ) ( )

2 2

i i i i i

i i i i i

T p p p p

T p p p p

(12)

Where is the tension, is the bias, and is the continuity parameter. The tension

parameter changes the length of the tangent vector. The bias parameter primarily changes the

direction of the tangent vector. The continuity parameter changes the sharpness in change

between tangents. The impact of each of these values on the drawn curve is shown as following:

Tension = +1-->Tight = −1--> Round

Bias = +1-->Post Shoot = −1--> Pre shoot

Continuity = +1-->Inverted corners = −1--> Box corners

Figure 3-11 The effect of various parameter settings used together[17]

.The three lines under the figures

represent the value axes for tension, bias and continuity. The middle points represent value 0, the left

points are -1 and right points are 1.

With Kochanek–Bartels spline, we could choose the tangents given the date points 1ip , ip and

1ip with these three possible parameters , and , which is suitable to simulate polymer

Method

24

chains. In our implement, we choose 1 , 0 , 0 to simulate more smooth chains. (see

figure 3-11).

Interpolating p in the interval , 1( )i ip p can be done with Equation 13:

00 10 01 1 11 1( ) ( ) ( ) ( ) ( )i i i ip t h t p h t T h t p h t T (13)

with 1( ), [0,1]i i ip p t p p t and 00 10 01 11, , ,h h h h are Hermite basis functions. In our

implementation, we have 00 10 01 11, , ,h h h h as following:

3 2

00

3 2

10

3 2

01

3 2

11

2 3 1

2

2 3

h t t

h t t t

h t t

h t t

(14)

3.4 Initial step for small molecules diffusion research

3.4.1 Overview of the problem

We are interested in the diffusion of small molecule compounds in semicrystalline polymers. The

first step is to find the initial positions of small molecules 2 2 2 4, , ,H O N O CH which are randomly

added in a polymer chains filled domain. A new molecule should be sufficient far away from

these polymer chains and new molecules we have added previously.

3.4.2 Algorithm details

We implement it in the following two steps:

Step1 First, we abstract all kinds of small molecules to be spherical. Then the problem is as

following: Some polymer chains are located in a cubic domain. We want to add some new

“spheres” in randomly. A new “sphere” should be sufficient far away from these polymers and

these new “sphere” we have added in already. The problem is to find the positions of the centers

of these “spheres”.

Method

25

Program implementation

(1) We create a lot of points located randomly instead of those polymer chains. Since it is a cubic

domain, the coordinate ( , , )x y z of one point shows its position. Then the problem turns to be for a

randomly created new item whose position is 0 ( , , )p x y z , the distance between it and other items

( , , )p x y z should satisfy 0| |p p r , here r is pre-defined. For the boundary, periodic boundary

condition is implemented.

We use the following two stop conditions:

[1] The maximum number of added items is a predefined scalar number.

[2] If the program has tried to add a single item more than a certain times (typically around 100),

then break the loop even if condition [1] is not yet satisfied.

(2) In order to improve efficiency, we divide the domain into many boxes regularly. So those

original points are divided into different boxes also. To do this, we sort the indexes of all the

points. This progress is called “sorting” here and will be used later. For a new item that we create

randomly, first we find which box it belongs to. Then we just need to check if it is far away

enough with all the points that are in the 27 boxes around the one contains this new item. Note: if

the box that contains the new item is on the boundary, the periodic boundary condition is

implemented.

(3) Besides those original points, we need to check if a new item is far away enough with other

new items that we have added in already. It is memory wasted if we use a big matrix, whose rows

represent all the boxes and the number of columns are the max number of new items, to store all

the new atoms. Moreover, changing the dimensions of a matrix frequently is inefficient. It is also

not efficient to do “sorting” for all the new items which are added in already every time when we

are trying to add a new item. We implement two methods to store new items in order to make the

program more efficient.

*The first method (index method)

We store all the coordinates of new items in a vector in the sequence of adding in. And we use

three matrices to store the indexes of them in this coordinate vector. For the first matrix, each row

represents one box. The columns mean the items in this box. However, the number of columns is

smaller than the max number of new items, which means, we can’t guarantee the indexes of all

the new items could be stored in this matrix. So if it is not enough, we use the second one. And if

Method

26

the second one is full also, we use the third one. For this one, we could change the dimension of it

if it’s needed, while for the first two, we can’t change the dimensions of them.

So if we want to check the distance between a new item and other new items in a certain box, we

need to go through the certain row in the first matrix. And if the second or third matrices contain

indexes of items in this box also, we should go through the certain line in these two matrices.

*The second method (sorting method)

In this method, we store all the coordinates of new items in 3 different vectors. The length of the

first vector is the smallest and it is much less than the other two. The length of the third vector is

the max number of new items by 3. If we would like to store a new item, we check if we could

add it in the first vector. If it is not full, we just add it in. Otherwise, we check if the second one is

full. If not, we need to move all the items in the first one to the second one and do “sorting” to the

second one, then add the new item in the first vector. If the second one is full also, we should

move all the items in the second vector to the third one and do “sorting” to the third one first,

then move all the items in the first vector to the second one and do “sorting” to the second one,

after this, add the new item in the first vector.

When we want to check the distance between a new item and other new items, we go through the

first unsorted vector first. Then we could check the distance of the new item with other new items

in second and third vectors for a certain box, since these two vectors are sorted.

*Implementation details

We tried our best to improve the efficiency in some details.

[1] We use vectors to store the coordinates of old points and new items. Since we need to read the

value of the coordinate many times, it is always more efficient to read them from a vector than

from a matrix.

[2] For the index of the box, instead of a coordinate ( , , )i j k , we translate it to a number using

( 1) ( 1)x x yi j n z n n , where xn and yn are the number of boxes in x and y labels.

[3] To check if the distance is large enough, we judge it every time after we calculate 2

0( )x x

or 2

0( )y y . If the value of them is bigger than 2r , we don’t need to go on calculating 2

0( )z z ,

just continue to check the next. If we find the distance between the new item and another original

one is smaller than r , we don’t need to continue checking other points but create a new item.

Method

27

[4] Since periodic boundary condition is implemented, we judge if the new item belongs to a box

that on the boundary of the domain first. If so, there are some special treatments. To go through

27 boxes around it, we need to find the certain box in the domain for the one outside. And we

should choose the min value between 2

0( )x x , 2

0( )xx x L and 2

0( )xx x L instead

of 2

0( )x x , where xL is the domain length on the x label.

Step2 Then we replace these spheres as small molecules 2 2 2 4, , ,H O N O CH . We need to calculate

the position of all the atoms for these molecules.

We implement it by the following 3 steps:

(1) Find the position of the atoms when the center of the molecules is on the origin of coordinate.

* 2H O

We know the structure of 2H O is a triangle which is shown in figure 3-12. (pm = 1210 m)

If the centroid of the triangle is on origin of coordinate, then we get the position of H and O :

(Note: Centroid of a triangle is the point of intersection of its medians which are the lines joining

each vertex with the midpoint of the opposite side.)

Figure 3-12: Structure of 2H O and Position of H ,O and centroid

* 2N

We know the structure of 2N is a line which is shown in figure 3-13. (pm = 1210 m)

If the middle point of the line is on origin of coordinate, then we get the position of N :

Method

28

Figure 3-13: Structure of 2N and Position of N and middle point

* 2O

We know the structure of 2O is a line and the distance between two atoms is 120.8pm. (pm =

1210 m)

If the middle point of the line is on origin of coordinate, then we get the position of O as showed

in figure 3-14:

Figure 3-14: Position of O and middle point

* 4CH

We know the structure of 4CH is a tetrahedral which is shown in figure 3-15. (pm = 1210 m)

If the position of the C atom is on origin of coordinate, then we get the position of H :

Method

29

Figure 3-15: Structure of 4CH and Position of C and H

(2) Do coordinate rotation for the whole molecule including all atoms.

To perform a rotation in Euclidean space, we use rotation matrix.

The following three basic rotation matrices rotate vectors about the , ,x y z axis in three

dimensions:

1 0 0

( ) 0 cos sin

0 sin cos

cos 0 sin

( ) 0 1 0

sin 0 cos

cos sin 0

( ) sin cos 0

0 0 1

x

y

z

R

R

R

(15)

To rotate the position of atom point ( , , )p x y z , we choose the rotation angles , , randomly.

After rotation, the new position of the point '( ', ', ')p x y z is as following:

'

' ( ) ( ) ( )

'

x y z

x x

y R R R y

z z

(16)

(3) Do coordinate translation for the whole molecules include all atoms. Move the center of

molecule to the center of “sphere” which we got in Step 1.

Method

30

After (2), we got the position of the point '( ', ', ')p x y z . If the position of center of the “sphere”

which we got in Step1 is ( , , )x y zv v v , then after translation the new position of the point

* * * *( , , )p x y z is as following:

*

*

*

'

'

'

x

y

z

x x v

y y v

z z v

(17)

* * * *( , , )p x y z is the final position for the atom.

Result

31

4. Result

4.1 Result of tie-chain concentration We investigate how the tie-chain fraction is affected by relative amorphous density ( /a c ), re-

entry probability, crystal thickness (Lc), amorphous thickness (La), temperature (T ), radius of

fetched stem position, number of C in each chain (chain length) and branches. Where nothing

else is said, 100x yD D nm , 10c aL L nm , 300T K , fragFill = 0.5, rAttach = 1nm,

/a c = 0.852 and 140wM kD . The algorithms are implemented in Matlab and run on Apple

MacBook Air with Intel Core i5/1.7GHz, 2 cores/hyperthreads, 3MB L3 cache, and 4G RAM.

4.1.1 Relative amorphous Density ( /a c )

First, we verify the correctness of algorithm and program. The relative amorphous density

( /a c ) and tie-chain degree (number of tie chain/filled points in crystal layer) should not be

influenced by filling degree (filled points/all points in crystal layer). The program results are as

follows:

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.2

0.4

0.6

0.8

1

1.2

Filling Degree

de

nsityA

mo

rph

/den

sityC

rysta

l

rhoA=0.65

rhoA=0.70

rhoA=0.75

rhoA=0.80

rhoA=0.85

rhoA=0.90

rhoA=0.95

rhoA=1.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Filling Degree

Tie

Cha

in D

eg

ree

rhoA=0.65

rhoA=0.70

rhoA=0.75

rhoA=0.80

rhoA=0.85

rhoA=0.90

rhoA=0.95

rhoA=1.00

Figure 4-1: The affect of filling degree to relative amorphous density and tie-chain degree

From figure 4-1, we could see the lines of relative amorphous density ( /a c ) and tie-chain

degree are almost straight. Just when filling degree is close to 0 or 1, there are oscillation parts.

This is what we expect.

Result

32

Then we investigate how the tie-chain fraction is affected by relative amorphous density ( /a c ).

The program result is as follows:

0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.016

0.018

0.02

0.022

0.024

0.026

0.028

0.03

densityAmorph/densityCrystal

Tie

Ch

ain

Fra

ctio

n

Figure 4-2: The affect of relative amorphous density to tie-chain degree

From figure 4-2, we could see that the tie-chain fraction linearly increases with relative

amorphous density ( /a c ). This is what we expect. If other parameters fixed, /a c increases

which means more parts of chains lay in amorphous layer, then the possibility of creating tie-

chain increases. So the tie-chain fraction increases.

4.1.2 Re-entry probability We investigate how the tie-chain fraction is affected by re-entry probability. The program result

is as follows:

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0.005

0.01

0.015

0.02

0.025

pTurn

Tie

Cha

in F

raction

Figure 4-3: The affect of re-entry probability to tie-chain degree

Result

33

From figure 4-3, we could see that the tie-chain fraction linearly decreases with re-entry

probability. That is what we expect. Re-entry probability increases which causes more parts of

chains lay in crystal layer, then the possibility of creating tie-chain decreases. So the tie-chain

fraction decreases.

4.1.3 Crystal thickness (Lc)

We investigate how the tie-chain fraction is affected by crystal thickness (Lc). The program result

is as follows:

2 4 6 8 10 12 14 16 18 200.0225

0.023

0.0235

0.024

0.0245

0.025

0.0255

0.026

0.0265

0.027

0.0275

Lc [nm]

Tie

Cha

in F

raction

Figure 4-4: The affect of crystal thickness to tie-chain degree

From figure 4-4, we could see that the tie-chain fraction decreases with crystal thickness (Lc) in

total. That is what we expect. If crystal thickness (Lc) increases, when a chain touches one of the

crystal surfaces, the probability of the remaining chain-length is larger than the crystal thickness

(Lc) decreases, then the probability of the chain enters the crystal surface decreases, which causes

the possibility of creating tie-chain decreases. So the tie-chain fraction decreases.

4.1.4 Amorphous thickness (La)

We investigate how the tie-chain fraction is affected by amorphous thickness (La). The program

result is as follows:

Result

34

2 4 6 8 10 12 14 16 18 20

0.01

0.02

0.03

0.04

0.05

0.06

La [nm]

Tie

Cha

in F

raction

Figure 4-5: The affect of amorphous thickness to tie-chain degree

From figure 4-5, we could see that the tie-chain fraction decreases with amorphous thickness (La).

That is what we expect. If other parameters fixed, amorphous thickness (La) increases, then it is

more difficult to create tie-chain. So the tie-chain fraction decreases.

4.1.5 Temperature (T ) We investigate how the tie-chain fraction is affected by temperature (T ). The program result is

as follows:

50 100 150 200 250 300 350 4000

0.05

0.1

0.15

0.2

0.25

Temperature [K]

Tie

Cha

in F

raction

Figure 4-6: The affect of temperature to tie-chain degree

From figure 4-6, we could see that the tie-chain fraction decreases with temperature (T ).That is

what we expect. Based on morphology of semicrystalline polymer, the chains are stiff when the

Result

35

temperature is low. When the temperature increases, the chains become more coiled, then it is

more difficult to create tie-chain. So the tie-chain fraction decreases.

4.1.6 Radius of fetched stem position

We investigate how the tie-chain fraction is affected by radius of fetched stem position. The

program result is as follows:

0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.023

0.0235

0.024

0.0245

0.025

0.0255

0.026

0.0265

0.027

rAttach

Tie

Cha

in F

raction

Figure 4-7: The affect of radius of fetched stem position to tie-chain degree

From figure 4-7, we could see that the tie-chain fraction oscillates with radius of fetched stem

position, which is a straight line in total. This is what we expect. Since the smallest number of

stem positions around one point is 6, and it is not so easy to fill all of them. The tie-chain fraction

should be independent with radius of fetched stem position.

4.1.7 Number of C in each chain (chain length)

We investigate how the tie-chain fraction is affected by number of C in each chain (chain length).

The program result is as follows:

Result

36

0 2 4 6 8 10 12 14

x 104

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

nC

Tie

Cha

in F

raction

DX=100

DX=50

DX=25

DX=12.5

Figure 4-8: The affect of chain length to tie-chain degree

From figure 4-8, we could see that the tie-chain fraction increases with number of C in each chain

(chain length). That is what we expect. If other parameters fixed, number of C in each chain

(chain length) increases, then it is easier to create tie-chain. So the tie-chain fraction increases.

Moreover, the tie-chain fraction doesn’t change much with Dx increases. This is what we expect.

4.1.8 Branches We investigate how the tie-chain fraction is affected by branches. The program result is as

follows:

0 200 400 600 800 1000 1200 14000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

nC in each branch

Tie

Cha

in F

raction

branchFrac=0.05

branchFrac=0.10

branchFrac=0.15

branchFrac=0.20

branchFrac=0.25

Figure 4-9: The affect of branches to tie-chain degree

Result

37

From figure 4-9, we could see that the tie-chain fraction increases with branch fraction increases.

That is what we expect. If other parameters fixed, the branch fraction increases, then it is easier to

create tie-chain. So the tie-chain fraction increases.

4.2 Result of entanglement concentration

4.2.1 Result of two chain model

We use two-chain model to test and verify the knot algorithm for entanglement. In two-chain

model, one chain starts and ends at the lower face and the other chain starts and ends at the upper

face. The programming result is shown as following. Figure b is always the xy-plane projection

view.

*Test1

This is the simplest test case.

We got the result from program:

self-intersection points: non

intersection points: (25.0000 25.0000)

sequence of sign before Step7(for the lower chain) : +

final sequence of sign(for the lower chain): - + -

number of entanglement : 1

From figure 4-10, we could get the same result.

Note: The beginning and ending signs in sequence are added by the beginning and ending points

of the chain. The reason is illustrated in Step7.

Figure 4-10 a b

Result

38

*Test 2 3 4

This is a group test as we could know that these 3 test cases have the same xy-plane projection

view from figure 4-11b.

From the result we got from the program, they have the same self-intersection points and

intersection points:

self-intersection points: non

intersection points: (17.8 21.8) (22.2 24.2)

But the result of the sequence of sign for lower chain and number of entanglement are different:

Table 4-1

Test case number 2 3 4

Sequence of sign before Step7 + - non non

Sequence of sign + - non non

number of entanglement 1 0 0

We could get the same result from figure 4-11.

The different sequence of sign causes the different number of entanglement in these 3 test cases.

2

Result

39

3

4

Figure 4-11 a b

*Test5

From figure 4-12b, we could see there is self-intersection point in this test case.

We got the result from program:

self-intersection points: (20.0951 21.2922)

intersection points: (23 21) (22.1 23.6) (16.4 26.6)

sequence of sign before Step7(for the lower chain): - - +

sequence of sign (for the lower chain): - +

number of entanglement : 1

From figure 4-12, we could get the same result.

Note: the last two sign - + in the sequence of sign before Step7 is the condition (3) in Step7,

which is replaced by +.

Result

40

Figure 4-12 a b

*Test 6 7

This is a group test as we could know that these 2 test cases have similar xy-plane projection

view from figure 4-13b.

For test case 6, the result is as following:

self-intersection points: (19.5870 20.4716)

intersection points: (14.1 13) (16.7 16.1) (18.1 18.1) (17.8 24.3) (17.1 25)

sequence of sign before Step7 (for the lower chain): - - + + -

sequence of sign (for the lower chain): -

number of entanglement : 0

From figure 4-13, we could get the same result.

Note: the sequence of sign satisfy condition (2) in Step7, so - - and + + are removed.

For test case 7, the result is as following:

self-intersection points: (19.0611 22.0785)

intersection points: (17.8096 20.7812) (20.1812 20.9103)

(17.9580 23.1445) (19.9938 23.9810)

sequence of sign before step7(for the lower chain): - + - +

sequence of sign(for the lower chain): +

number of entanglement : 0

From figure 29, we could get the same result.

Result

41

Note: The sequence of sign (- + - +) before Step7 is the condition (3) in Step7, which is replaced

by +.

6

7

Figure 4-13 a b

*Test 8 9 10 11

This is a group test as we could know that these 4 test cases have the same xy-plane projection

view from figure 4-14.

From the result we got from the program, they have the same self-intersection points and

intersection points:

self-intersection points: non

intersection points: (24.2481 27.1777) (11.7571 30.0000) (14.0000 30.0000)

(16.1008 30.0000) (19.8992 30.0000) (21.9799 30.5988)

But the result of the sequence of sign for the lower chain and number of entanglement are

different:

Result

42

Table 4-2

Test case number 8 9 10 11

sequence of sign before

Step7

+ - + - + - - - + - + -

- - - - - + - - - - + +

sequence of sign + - + - + - + - + - - + non

number of entanglement 3 2 1 0

We could get the same result from figure 4-14. The different sequence of sign causes the different

number of entanglement in these 4 test cases.

Note:

Test 8: The sequence of sign (+ - + - + -) is the condition (4) in Step7, so the number of

entanglement is 3.

Test 9: The sequence of sign (- - + - + -) before Step 7 is the condition (2) in Step7, so the

first “- - ” is removed. The number of entanglement is 2.

Test 10: The sequence of sign (- - - - -) before Step 7 is the condition (2) in Step7, so the

first “- - - -”is removed. The number of entanglement is 1.

Test 11: The sequence of sign (- - - - + +) before Step 7 is the condition (2) in Step7, so

the whole sequence is removed. The number of entanglement is 0.

8

Result

43

9

10

11

Figure 4-14 a b

4.2.2 Result of the knot algorithm for entanglement concentraion

We investigate how the entanglement fraction is affected by relative amorphous density ( /a c ),

crystal thickness (Lc), amorphous thickness (La), temperature (T ), radius of fetched stem position,

number of C in each chain (chain length). Then we compared the results of both tie-chain fraction

Result

44

and entanglement fraction. We plot error bars alone curve by Matlab command “errorbar (X, Y,

E)” to show the result more reliable. The “E” in “errorbar (X, Y, E)” is the standard deviation of

100 repeating running results. Error bars could show the confidence level of data or the deviation

along a curve.

* Relative amorphous Density ( /a c )

We investigate how the entanglement fraction is affected by relative amorphous density ( /a c ).

The program result is as follows:

Figure 4-15: The affect of relative amorphous density to entanglement fraction

From figure 4-15, we could see that the entanglement fraction linearly increases with relative

amorphous ( /a c ). This is what we expect. If other parameters fixed, /a c increases which

means more parts of chains lay in amorphous layer, then the possibility of creating entanglement

increases. So the entanglement fraction increases.

Compare the results of both tie-chain fraction and entanglement fraction, the entanglement

fraction is greater than tie-chain fraction corresponding of a fixed /a c . When /a c increases,

entanglement fraction increases faster than tie chain fraction increases.

* Crystal thickness (Lc)

We investigate how the entanglement fraction is affected by crystal thickness (Lc). The program

result is as follows:

Result

45

Figure 4-16: The affect of crystal thickness to entanglement fraction

From figure 4-16, we could see that the entanglement fraction decreases with crystal thickness

(Lc) in total. That is what we expect. If crystal thickness (Lc) increases, when a chain touches one

of the crystal surfaces, the probability of the remaining chain-length is larger than the crystal

thickness (Lc) decreases, then the probability of the chain enters the crystal surface decreases,

which causes the possibility of creating entanglement decreases. So the entanglement fraction

decreases.

Compare the results of both tie-chain fraction and entanglement fraction, the entanglement

fraction is greater than tie-chain fraction corresponding of a fixed crystal thickness (Lc). When

crystal thickness (Lc) increases, entanglement fraction decreases faster than tie chain decreases.

*Amorphous thickness (La)

We investigate how the entanglement fraction is affected by amorphous thickness (La). The

program result is as follows:

Result

46

Figure 4-17: The affect of amorphous thickness to entanglement fraction

From figure 4-17, we could see that the entanglement fraction first increases then decreases with

amorphous thickness (La). That is what we expect. If the amorphous thickness (La) is too small, it

is easy to create tight folds by immediate reentering the crystal (figure 34(a)) and difficult to

create entanglement. That is why when amorphous thickness (La) increases from 0.65nm to

0.75nm, the entanglement fraction increases. However, when amorphous thickness (La) increases

from 0.75nm to 1nm, the amorphous thickness (La) is too large to create entanglement fractions.

So the entanglement fraction decreases. Figure 4-18 illustrates this.

(a) (b) (c)

Figure 4-18: The figure illustrates the relation between entanglement fraction and amorphous thickness

(La) .The red lines represent the crystal layers.

Compare the results of both tie-chain fraction and entanglement fraction, the entanglement

fraction is greater than tie-chain fraction corresponding of a fixed amorphous thickness (La) when

amorphous thickness (La) is larger than 0.75nm.

Result

47

* Temperature (T )

We investigate how the entanglement fraction is affected by temperature (T ). The program result

is as follows:

Figure 4-19: The affect of temperature to entanglement fraction

From figure 4-19, we could see that the entanglement fraction decreases with temperature (T ) in

total. That is what we expect. Based on morphology of semicrystalline polymer, the chains are

stiff when the temperature is low. When the temperature increases, the chains become more

coiled, then it is more difficult to create entanglement. So the entanglement fraction decreases.

Compare the results of both tie-chain fraction and entanglement fraction, the entanglement

fraction is greater than tie-chain fraction corresponding of a fixed temperature ( T ) when

temperature ( T ) is larger than 100K. When temperature ( T ) increases from 100K to 400K,

entanglement fraction decreases faster than tie-chain decreases. This could indicate that it might

be worth examining if the resistance of slow crack growth could potentially be improved by

lowering T during the manufacturing process.

* Radius of fetched stem position

We investigate how the entanglement fraction is affected by radius of fetched stem position. The

program result is as follows:

Result

48

Figure 4-20: The affect of radius of fetched stem position to entanglement fraction

From figure 4-20, we could see that the entanglement fraction oscillates with radius of fetched

stem position which is a straight line in total. This is what we expect. Since the smallest number

of stem positions around one point is 6, and it is not so easy to fill all of them. The entanglement

fraction should be independent with radius of fetched stem position.

Compare the results of both tie-chain fraction and entanglement fraction, the entanglement

fraction is greater than tie-chain fraction corresponding of a fixed radius of fetched stem position.

* Number of C in each chain (chain length)

We investigate how the entanglement fraction is affected by number of C in each chain (chain

length). The program result is as follows:

Result

49

Figure 4-21: The affect of chain length to entanglement fraction

From figure 4-21, we could see that the entanglement fraction increases with number of C in each

chain (chain length). That is what we expect. If other parameters fixed, number of C in each

chain (chain length) increases, then it is easier to create entanglement. So the entanglement

fraction increases.

Compare the results of both tie-chain fraction and entanglement fraction, the entanglement

fraction is greater than tie-chain fraction corresponding of a fixed number of C in each chain

(chain length). When number of C in each chain (chain length) increases, entanglement fraction

increases faster than tie chain increases.

4.3 Numerical efficiency of tie-chain and

entanglement algorithm

We investigate the efficiency of the whole tie-chain and knot algorithm. First, we test the running

time of the entire model with the area of computational domain increasing.

The result is shown as following:

Table 4-3

,x yD D (nm) 10 2 20 20 2 40 40 2 80 80 2

Area of domain 200 400 800 1600 3200 6400 12800

Running time(s) 420.82 855.41 1772.99 3346.47 6846.78 13942.20 29460.52

From figure 4-22, we could see that the efficiency of the entire algorithm is ( )O n where n is the

area of computational domain. It is in prefect agreement with what we expected. Since the filling

degree is 0.5 which is fixed and is not quite large, the amount of tie-chain and entanglement

increases linearly with the domain area. So the running time also increases linearly and the

efficiency of the entire algorithm is ( )O n .

Result

50

102

103

104

105

100

101

102

103

104

105

Efficiency Analysis

Area of the domain Dx*Dy

Tim

e /

Tim

e0

Ordo(n)

Ordo(n*log(n))

Ordo(n2)

Model

Figure 4-22 Efficiency analysis of the entire model

Then, we test the running time (s) of the entire model and the knot algorithm with filling degree

increasing, respectively.

The result is shown as following:

Table 4-4

Filldegree 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time(model) 6463.30 10726.74 16260.39 22576.61 29227.48 36976.91 44868.33

Time( knot) 4676.3 8187.7 12695 18101 23799 30527 37343

From figure 4-23, we could see that the efficiency of the entire algorithm is ( *log )O n n and the

efficiency of the knot algorithm is between ( *log )O n n and2( )O n where n is the filling degree of

occupied stems in computational domain. It matches what we expected. Since the amount of

entanglements increases with the square of filling degree while amount of tie-chains increases

linearly. So the efficiency of the entire algorithm should between ( )O n and2( )O n . And the

efficiency of the knot algorithm is even better than what we expect.

Result

51

0.2 0.3 0.4 0.5 0.6 0.7 0.810

0

101

102

Efficiency Analysis Of Entire Model

Filling Degree

Tim

e /

Tim

e0

Ordo(n)

Ordo(n*log(n))

Ordo(n2)

Model

0.2 0.3 0.4 0.5 0.6 0.7 0.810

0

101

102

Efficiency Analysis Of Knot Algorithm

Filling Degree

Tim

e /

Tim

e0

Ordo(n)

Ordo(n*log(n))

Ordo(n2)

Model

Figure 4-23 Efficiency analysis of the entire model (left) and knot algorithm (right)

4.4 Result of initial step for small molecules diffusion

research

To test the program, we predefine the constant numbers and the result is shown as following:

Number of points added randomly to simulate polymer chains (the green points in figure4-24): 16

Number of molecules added in randomly: 12

Figure 4-24 Simulation of positions of molecules

From figure 4-24, we could see the positions of the molecules are reasonable.

Then we test the number of atoms in each box.

Result

52

We predefine the constant numbers and the result is shown as following:

Number of points added randomly to simulate polymer chains: 1

Number of molecules added in randomly: 10000

Number of boxes in each direction: 10

From figure 4-25 and 4-26, we could see that the distribution of atoms is normal distribution in

all boxes which is what we expected.

Figure 4-25: distribution of atoms by method 1(Index method)

Figure 4-26: distribution of atoms by method 2(Sorting method)

Result

53

At last, we test the efficiency of the two methods. We predefined the constant numbers and the

result is shown as follows:

Number of points added randomly to simulate polymer chains: 1

Number of molecules that are added in (Num accepted) and running time (s) for the two methods:

Table 4-5

Num accepted 2500 5000 10000 20000 40000 80000 160000

Time method 1 0.3131 0.8413 2.3879 7.6341. 28.6086 115.6322 546.9271

Time method 2 0.3882 0.9979 2.6136 8.2403 29.2583 114.5613 519.8957

From figure 4-27, we could see that the two methods have almost the same efficiency. As the

computational domain is fixed, it will be more difficult to find a place to add in a new molecule

as more and more molecules have been added in already.

103

104

105

106

100

101

102

103

104

105

Efficiency Analysis

Number of points that be added in

Tim

e /

Tim

e0

Ordo(n)

Ordo(n*log(n))

Ordo(n2)

Method1

Method2

Figure 4-27 Efficiency analysis of two methods with number of points that are added in

So we tested the running time (s) for the two methods (M1, M2) with number of molecules that

are tried to be added in (Num tried):

Table 4-6

Num tried 2500 5000 10000 20000 40000 80000 160000

Num accepted M1 2488 4947 9812 19188 36849 68642 120067

Num accepted M2 2487 4954 9788 19177 36873 68678 120240

Result

54

Time method 1 0.3798 0.8532 2.4113 7.1925. 24.2737 85.2093 256.4930

Time method 2 0.4345 0.9075 2.3240 7.0197 22.4000 77.1705 243.3891

From figure 4-28, we could see that the numerical efficiency of these two methods is between

( log )O n n and 2( )O n where n is the number of molecules that are tried to be added in.

103

104

105

106

100

101

102

103

104

105

Efficiency Analysis

Number of points that tried to be added in

Tim

e /

Tim

e0

Ordo(n)

Ordo(n*log(n))

Ordo(n2)

Method1

Method2

Figure 4-28 Efficiency analysis of two methods with number of points that are tried to be added in

Discussion and Future work

55

5. Discussion and Future work

5.1 Discussion

For tie-chains and entanglements we have compared the simulation results obtained with our

Monte-Carlo model with Huang and Brown’s statistical approach, which has proved to be one of

the most attractive models for predictive purposes.

Based on Huang and Brown’s model (equation 3), reduction in lamella thickness (La) or crystal

thickness (Lc) produces more tie molecules, which is consistent with the results of our model (see

section 4.1.3, 4.1.4). Moreover, we also got the result of how the tie-chain fraction is affected by

relative amorphous density ( /a c ), re-entry probability, temperature ( T ), radius of fetched

stem position, number of C in each chain (chain length) and branches. All of the simulation

results are realistic and in line with our expectations and with common sense.

One of the shortcomings of Huang and Brown’s model is that chain entanglements are neglected

as active elements for the stress transfer between crystallites. An entanglement can transfer stress

between neighboring crystallites just as two close tie-chains can. We investigated how the

entanglement fraction was affected by relative amorphous density ( /a c ), crystal thickness (Lc)

amorphous thickness (La), temperature (T ), radius of fetched stem position and number of C in

each chain (chain length). The knot algorithm simulation in our research (section 4.2.2) showed

that chain entanglements are often more numerous than tie-chain, which means that the influence

of entanglements is most probably larger than the influence of tie-chains. This result emphasizes

the role of chain entanglements, which are not taken into account in Huang and Brown’s model,

but which actually contribute to the experimental measurement of the strain hardening.

Huang and Brown’s model is a statistical model. Yeh and Runt proposed to improve Huang and

Brown’s model by taking trapped entanglement into consideration. Drawbacks of statistical

methods include the use of an arbitrary L , the assumption that L is equally influenced by a

change in aL and Lc *2 and the assumption that trapped entanglements are always formed if the

two chains are sufficiently close to each other. Our computational Monte-Carlo model can be

made more realistic than statistical methods. One advantage of simulations is that it allows more

Discussion and Future work

56

freedom in the design of the conditions for building a system and the results obtained can be

represented in many different ways in order to fit the requirements set by the user. The numerical

efficiency of our algorithm is ( )O n with respect to domain increasing, which is in prefect match

with what we expected and hoped for. It allowed molecular systems of 100nm to be analyzed

quickly on ordinary PC. The company Borealis has verified our Monte-Carlo model and knot

algorithm to be sufficiently realistic for existing polymers. Our model could be used to make the

searching for material with better mechanical properties more systematic. Borealis AB has also

decided to finance a follow-up study on this project. Finally, parts of this thesis have been used in

a paper[18]

which has been accepted for publication in the Scientific Journal Polymer.

5.2 Future work In the knot algorithm, we just calculated entanglement of linear chains. We could implement the

more complex situation which takes entanglements of branched polyethylene also into account.

We could also develop the knot algorithm further to ensure that it can handle even more complex

situations and topologies.

Summary

57

6. Summary

It is known that many mechanical properties of semi-crystalline polymers are dependent on the

fraction of chains linking the crystalline regions together. This is particularly true for slow crack

growth. We developed an advanced computationally efficient computer program for modeling

crystalline layers and for calculating the amount of tie-chains and trapped entanglements. The

amorphous inter-layers were modeled with the Flory Monte Carlo ghost-chain concept, which has

proved accurate on polymer melts, while the crystalline regions were assumed to have an

orthorhombic crystal packing. The tie-chain concentration could be evaluated directly with the

Monte Carlo model, while a novel knot algorithm was developed to evaluate the concentration of

trapped entanglement. The concept of algorithm was based on the observation that the only

information needed to approve or falsify a knot was the properties of the intersection points of the

curves when mapped on a two dimensional plane.

Based on the results obtained from our Monte-Carlo model, we investigated how the tie-chain

fraction was affected by relative amorphous density ( /a c ), re-entry probability, temperature

(T ), radius of fetched stem position, number of C in each chain (chain length) and branches. We

also investigated how the entanglement fraction was affected by relative amorphous density

( /a c ), crystal thickness (Lc), amorphous thickness (La), temperature (T ), radius of fetched

stem position and number of C in each chain. The simulations showed that the influence of

entanglements is usually larger than the influence of tie-chains, even though they are often

neglected in other commonly used models, like the famous Huang and Brown’s model. However,

we only considered entanglement of linear chains in our model. In future work, we could also

implement entangled long-branched polymers. From the mathematical perspective, the knot

algorithm in our research handled a complex topology problem. Previous models in this field

have not been able to account for this. The numerical efficiency of our algorithm is

( *log )O n n where n is the filling degree of occupied stems in computational domain which is in

prefect agreement with what we expected.

Furthermore, we did some initial research on small molecule diffusion in semicrystalline

polymers. The simulation program was reliable and efficient.

Reference

58

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Reference

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