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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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