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POLYMER NANOPARTICLES AND NANOPARTICLE ARRAYS
By
JIWU LIU

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment. of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Physics and Astronomy

2009

ABSTRACT
POLYMER NANOPARTICLES AND NANOPARTICLE ARRAYS
By

J iwu Liu

The manufacturing of polymeric nanoparticles by intramolecular crosslinking is
studied by molecular dynamics simulation. Firstly an overview of the intramolecular
crosslinking process is obtained by the simulations of benzocyclobutene(BCB) / styrene
copolymers using an atomistic model. Then various coarse grained models, including
Freely Jointed Chain (FJ C), Freely Rotating Chain (FRC) and stiff chain models, are
adopted for studying general properties of intramolecular crosslinking of polymers.
A temperature series simulation on the FJ C model reveals that the change of ambi-
ent temperature results in the formation of nanoparticles with distinct morphologies.
To describe their structures, a quantity referred to as chemical distance density is
introduced, with a quantitative relation between it and the radius of gyration being
found. The subsequent study of rigidity effects adopts FRO and stiff chain mod-
els. It is found that in the rigid regime, the crosslinking process leaves a substantial
number of crosslinkers unlinked, and forms nanoparticles that are significantly larger
than their non-rigid counterparts. The Maxwell constraint counting method is used
to determine the rigidity thresholds, which yields good agreements with the simula-
tion data. In the last chapter, the atomistic model for polystyrene in the previous
crosslinking simulations is employed for a study of polystyrene chains on attractive
substrates. The phase diagram and a rough overview of chain dynamics on substrates

are obtained.

To my family

iii

ACKNOWLEDGMENTS

At the time when I finish my PhD thesis, I would like to thank those who have
helped me to accomplish this important goal in my life.

First of all, I want to give my deepest thanks to my advisor, Professor Phillip
Duxbury, for guiding me through my PhD study with his expertise, insights and
great patience. My future work will continuously benefit from what I have learned
from him.

I am grateful to all professors in my PhD guidance committee, Subhendra D.
Mahanti, Simon Billinge, Chong—Yu Ruan, Gregory Baker and Michael Mackay, for
all the help and advice they have offered me.

I would also like to acknowledge all my officemates. Radu Cojocaru gave me
suggestions on my thesis writing. I learned many things about scientific research and
computer techniques from working with Chris Farrow and Erin McGarrity. And Dan
Olds, Saurabh Gujarathi, Luke Granlund and Kanokkorn Pimcharoen make the office
an enjoyable place to work in. Saurabh Gujarathi has proof read my thesis and given
me valuable advice on my writing.

Thanks to all my friends in graduate school, Teng Yang, He Huang, Wenduo Zhou,
Yuping Huang, Zhensheng Ding, Gaoming Zhang, Zhiyi Bao and many others, for
standing by me and giving me help whenever I need.

Most importantly, my great gratitude to my parents, my brothers and my sister.

You have given me enormous support and love. This work is dedicated to you.

iv

Table of Contents

List of Tables ................................. vii
List of Figures ................................. viii
1 Introduction to polymer physics ................... 1
1.1 Basics of polymers .......................... 2
1.2 Chain models ............................. 6
1.3 Phases for single and bulk polymers ................. 8
2 Molecular dynamics simulation of polymers ............ 13
2.1 Introduction .............................. 14
2.2 Difficulties in the simulation of polymers .............. 16
2.3 Basics of molecular dynamics simulation .............. 17
2.4 Force field ............................... 22
3 Intramolecular crosslinking of BOB-styrene copolymers ..... 29
3.1 Introduction .............................. 29
3.2 Experiment .............................. 31
3.3 Model ................................. 32
3.4 Simulation ............................... 36
3.5 Crosslinking reaction ......................... 40
3.6 Rigidity effect .............................. 42
3.7 Summary ............................... 50
4 Temperature series simulation of intramolecular crosslinking . . 51
4.1 Introduction .............................. 51
4.2 Simulation ............................... 52
4.3 Contraction of polymers ....................... 54
4.4 Chemical distance density ...................... 58
4.5 Summary ............................... 65
5 Rigidity effect in the crosslinking of polymers ........... 67
5.1 Introduction .............................. 67
5.2 The freely rotating chain model ................... 69
5.3 The stiff chain model ......................... 73
5.4 Ring-weighted (TOSSllIlkll'lg ...................... 83
5.5 Conclusion ............................... 83

6 Polystyrene on attractive surfaces .................. 87

6.1 Introduction .............................. 87
6.2 Phase diagram ............................ 88
6.3 The dynamics of polymer adsorption ................ 94
6.4 Conclusion ............................... 97
7 Conclusion ................................ 100
Bibliography .................................. 103

vi

3.1

3.2

List of Tables

The parameters of the force field for the united atom model of a
BCB/styrene copolymer. Energy is in kcal/mol. and distance is in

Table of bonded potentials in a crosslink ................

vii

39

1.1

1.2

1.3

1.4

1.6

2.1

2.2

3.1

3.2

3.3

3.4

3.5

3.6

List of Figures

Isotactic(top), syndiotactic(middle), and atactic(bottom) polystyrenes.
The polymer backbone is in the plane of the paper, and the solid and
dashed wedges indicate groups above and below the plane, respectively. 3

Conformations of n-butane ........................ 5
Rotational potential of n—butane ..................... 5
Polymer models at different time and length scales. .......... 9

Coil-globule transition of polystyrene (Mu, = 2.6 x 107) in cyclohexane.

S is the radius of gyration. R H is the hydrodynamic radius[7]. . . . 10
Temperature dependence of Young’s modulus of bulk polymers [5]. . 12
Dihedral angle (1 defined for the particle sequence 1-2-3-4. ...... 24
Improper angle X defined for the particle sequence 1-2-3—4. ...... 26

Crosslinking experiments in the coil regime can be done above To by
using good solvents. ........................... 35

The crosslink between two BCB monomers. .............. 39

The number of unlinked crosslinkers for BCB / styrene copolymers with
750 monomers decreases as the crosslinking simulation proceeds. From
the top down, the three panels are for 80%, 40% and 20% BCB monomers

respectively. ................................ 43
The parameter k* from fitting to eq. 3.3. ............... 44
The parameter B from fitting to eq. 3.3. ................ 45

The parameter 100(unlinked crosslinkers) from fitting to eq. 3.3, show-
ing a rigidity effect at around 60% BCB crosslinkers. ......... 46

viii

3.7

4.1

4.2

4.3

4.4

4.5

4.6

4.7

5.1

The radius of gyration of crosslinked BCB / styrene copolymers, show-
ing a rigidity effect when the crosslinker concentration is above 60%. 48

The average contraction ratio 7, namely the ratio of the radius of
gyration of the polymer after and before crosslinking, as a function
of the crosslinking temperature. The four curves correspond to four
different concentrations of crosslinkers. ................. 55

The simulation data for the contraction ratio '7 = R905) / 129(0) are

fitted to the expression 72 = 1 — fipe, yielding values for the prefactor
,8 and the exponent e. .......................... 57

The chemical distance density Pd and radius of gyration R9 are plotted
for different temperatures. This data is for F J C samples with N = 500
and x = 0.2 ................................. 59

Eq. 4.6 is fitted to the simulation data for FJC samples at differ—
ent temperatures. Starting from the top the data is for crosslinking
temperatures T = 8, 4 and 2 respectively. ............... 60

The radius of gyration before crosslinking found from fittings of crosslinked
F JC samples to eq. 4.6 is compared to the simulation data, showing a
good agreement. ............................. 62

The exponent g in eq. 4.6 for the F J C model is obtained from fittings
at all temperatures. Note that the exponent is independent of the
crosslink density. ............................. 63

The radius of gyration as a function of chemical distance density for
intramolecularly crosslinked BCB / styrene copolymers is plotted. From
the top down, the three panels are for samples with 750, 500, 250
monomers respectively. In each panel, the data (open symbol) and the
fit to eq. 4.6 (solid line) are shown. ................... 64

The number of unlinked crosslinkers in the FRC model as a function of
the simulation time. It shows that the crosslinking reaction goes very
fast in the beginning, while it basically stops after about a quarter of
a million steps. .............................. 71

ix

5.2

5.3

5.4

5.5

5.6

01
\1

5.8

The percent of crosslinkers in F J C samples that. can find their partners
to react. For the crosslinking with angle constraints, a considerable
decrease can be observed when the crosslinker concentration is above
40%, which agrees with the estimate of the Maxwell constraint count-
ing that 40% is the rigidity threshold for this model. On the contrary.
for the other crosslinking the linked percent changes little when the
crosslinker concentration grows, while the Maxwell counting also pre—
dicts no rigidity effect for this model. The inset shows the difference
between two crosslinking curves. ....................

R9 of FRC samples crosslinked with angle constraints shows an abrupt.
change at the crosslinker concentration of 40%. Below this thresh-
old, Rg decreases greatly with the increasing of crosslinks. But above
the threshold, it is almost flat. On the contrary, R9 of F RC sam-
ples crosslinked without angle constraints lacks a rigidity effect, and
decreases over the full range of crosslinker concentrations. ......

The crosslinking reaction rate for a stiff chain with 1000 beads and
a segment length m = 4 indicates that the reaction saturates after a
quarter of a million steps. ........................

R9 of stiff chains with segment length m = 2,3,4 are plotted in
separate panels from top down. By the Maxwell constraint count-
ing method, when they are crosslinked with angle constraints, the
rigidity thresholds are 50%, 36.36% and 28.57% respectively, while
crosslinking without angle constraints can not cause the rigidity effect
in these chains. It can be seen in each panel that two curves corre-
sponding to two types of crosslinks nicely overlap with each other at
lower crosslinker concentrations, and split above the rigidity thresh-
olds, showing good agreement with the theoretical estimate. .....

The linked percent of crosslinkers in stiff chains with segment length
m = 2,3,4 are plotted in separate panels from top down. In each
panel, the results for two types of crosslinks are plotted together for
comparison. ...............................

The difference between linked percents of two types of crosslinks. They
exhibit considerable declines at their respective rigidity thresholds,
which are 50%, 36.36% and 28.57% as found by the Maxwell constraint
counting method. ............................

The difference between van de VVaals volumes of stiff chains crosslinked
by two types of crosslinks shows a similar transition for all segment
lengths. ..................................

75

76

77

78

79

81

5.9

5.10

6.1

6.2

6.3

6.4

6.5

6.6

AVvdw for long chains (5000 beads) and short chains (1000 beads).
Both of them have a segment length m = 4. The difference in van de
Waals volume is approximately proportional to the size of the chain.

R9 for freely rotating chains and stiff chains (in = 4) crosslinked by
ring-weighed crosslinking. When the angle constraints are employed,
crosslinking still leads to a larger R9 for both models. However, in this
case all By curves are smooth without abrupt changes at theoretical
rigidity thresholds. ............................

The phase diagram of tethered polymers on substrates. It was obtained

82

by a Monte Carlo simulation study using the bond fluctuation model[105]. 89

The z—coordinate of the center of mass of a polymer is plotted against
[33. Temperature decreases from the top curve to the bottom one. It
can be seen that approximately at [35 = 0.25, the z-coordinate is the
same for all temperatures .........................

Hg of polystyrene chains (with 125 styrenes) on substrates. W hen
Eeff is greater than 1.6Kcal/mol, all chains are adsorbed and show a
transition from AC to AE; when feff is less than 0.6Kcal/mol, they
are desorbed and the transition is from DC to DE. Two solid lines give
average R9 for all desorbed cases and all adsorbed cases respectively.
In addition, below 400K the polystyrene chains are frozen in glassy
states, so that they keep their initial globule morphologies, even for
feff > 1.6Kcal/mol, and assume smaller radii of gyration. .....

The contour plot of the z-coordinate (in .71) of the center of mass of
polystyrene chains with 125 monomers on substrates. .........

The contour plot of R9 (in A) of polystyrene chains with 125 monomers
on substrates ................................

The z—coordinate of the center of mass of polystyrene chains on sub-
strates during the heating process .....................

xi

92

93

94

98

Chapter 1

Introduction to polymer physics

Polymers, alsoreferred to as macromolecules, are large molecules made of re-
peating chemical units connected by covalent bonds. They are primary materials of
modern civilization. The use of polymers by mankind started since ancient times,
but. it was then limited to the naturally available polymers such as wood, fiber, nat-
ural rubber and so on. It was since 19th century when the synthetic polymers were
invented that. the polymer industry started growing rapidly. Thousands of new types
of polymers have been synthesized. and have found tremendous use in almost every
aspect of our daily lives.

The applications of polymers keep expanding owing to the fast evolving poly-
mer technology. Among other things, a recent experiment showed that nanoparticles
can be made by the intramolecular crosslinkings of polymers[4], opening the door for
polymers to enter nanoparticle world. Polymeric nanoparticles have many promis-
ing applications ranging from constituents of new nanomaterials to functional parts
of novel nano—devices. This thesis shall focus on the study of their properties and
syntheses from the point of view of physics.

A brief introduction to polymer physics is given in the following three sections,

with emphases on issues that are relevant to this thesis. Section 1.1 covers some basic

knowledge about polymers, such as their classifications and descriptions. In section
1.2, two basic theoretical models of polymers are introduced. The last section de-
scribes different phases of polymer chains and of bulk polymers. More about polymer

physics can be found in many good textbooks[5, 6, 7, 2].

1.1 Basics of polymers

The elementary units of a polymer are termed monomers. A polymer is called
a homopolymer when all monomers in it are of the same type. On the contrary, a
polymer is called a heteropolymer or copolymer if its constituent monomers are of
different types. Copolymers can be classified by the orderings of their monomers.

Some typical types of copolymers are:
0 Block copolymer: -—A——A-—A—~-A—--A—A—-—B—B—B-~B——B—-—B—
0 Alternate copolymer: —»A—B—-—Ar—B —'A——B——A— B —-A-—B —A— Be“
0 Random copolymer: -~—A——BA—~B-—B ——A—B~—A—A-—B—B——A—~B——

Biopolymers such as DNA and proteins are well known examples of copolymers. DNA
consist of 4 types of nucleotides (denoted as A,T,C.G) whereas proteins are made
from 20 different amino acids. The ordering of monomers in a biopolymer carries rich
information. It makes biopolymers complex enough to serve as fundamental units of
life.

Polymers show vast diversity in their chemical and physical properties due to
differences in their constitutions, syntheses as well as processings. In order to fully
describe a polymer, three levels of charatersiscs are required, namely constitution ,
configuration and conformation.

Constitution of a polymer specifies all the chemical units of the 1i)olymer as well

as their connections. It thus defines the size, the type and the basic structure of a

 

Figure 1.1: Isotactic(top), syndiotactic(middle). and atactic(bott01n) polystyrenes.
The polymer backbone is in the plane of the paper. and the solid and dashed wedges
indicate groups above and below the plane. respectively.

polymer.

Configuration refers to the spatial arrangement of atoms in a polymer, in partic—
ular the orientations of its substituents. so that polymers of the same constitution,
namely isomers, can be distinguished. A polymer can not change from one con—
figuration to another without breaking its chemical bonds. Figure 1.1 shows three
configurations of a polystyrene chain. i.e.. isotactic. syndiotactic and atactic config-
urations. The difference among them resides in the locations of the phenyl groups.
In the isotactic configuration. phenyl groups are located on the same side of the
backbone. In the syndiotactic configuration. phenyl groups distribute alternatively
on both sides of the backbone. In the atactic configuration. the locations of phenyl
groups are random. Different tacticities of polymers can lead to their different physi—

cal and chemical behaviors. For example, an isotactic polymer is easier to crystallize

due to its ordered structure.

Conformation specifies the dihedral angles about covalent bonds that connect
monomers. To change from one conformation to another no chemical bonds need
to be broken. Instead it is achieved through the rotation of atoms or side groups
about covalent bonds. Therefore the conformation of a polymer is also referred to as
Rotational Isomeric State (RIS).

N -butane provides a starting point for the analysis of conformations of polymers.
In fig. 1.2, three representative rotational isomeric states of nebutane are shown,
namely trans, gauche and eclipsed. The potential energies of polymers at different
conformations are different, too. Fig. 1.3 shows the potential energy of a n-Butane
as a function of its dihedral angle. It has three minima, corresponding to trans(t),
gauche+(g+) and ganche’ (9—) conformations, respectively. Among them, the global
minimum is achieved at the dihedral angle of 180°, which is the trans conformation.
On the other hand, the global maximum occurs when the dihedral angle is 0°, i.e.,
eclipsed conformation.

The probability for a n-butane to take on the trans or gauche states are significant
because of their low potential energies. Likewise, a rotational isomeric state of a long
polymer chain consists of predominantly trans and gauche. So it can often be written
as a sequence of them, e.g., tg'l'ttg’ ------ . The number of total rotational isomeric
states grows exponentially with the number of monomers, and can become enormous
for a long polymer.

Radius of gyration (R9) is the primary parameter to describe the dimension of

a polymer. A simple definition of R9 is:

 

R“ = i (1.1)

 

Potential energy (Kcal/mol)

 

gauche eclipsed

Figure 1.2: Conformations of n-butane

 

I ‘ l l ' I ' I
5 eclipsed eclipsed

gauche

 

 

 

 

J l . l l_ I
3 60 120 180 240 300 360

Dihedral angle(degree)

Figure 1.3: Rotational potential of n—butane

where R0 is the center of mass of the polymer, and m,- and R,- are the mass and

position vector of the ith atom in the polymer. Another equivalent definition is

 

N N
-' e 9
Z Z (ml-R,- — ijj)“
. '21':' 1
123:2 3 2+ 2 (1.2)
(Zine)

The mass factors in the definitions of R9 are not always essential. W hen all types
of atoms are evenly distributed in the polymer (which is true for homopolymers and
random copolymers), all mass factors can be set to unity with little change to the
value of R9.

Persistence length. of a polyn‘ier is the parameter to describe its stiffness. A

polymer with persistence length lp satisfies the equation below:
< .. _ —l/lp
(30.50! >—- 6 (1.3)

where 01 is the angle between the tangents at two points on the chain separated by a.

contour distance of l.

1 .2 Chain models

A polymer can be modeled as a chain of many segments connected together,
where a single segment may represent one or more monomers. The chain model
can be applied not only to linear polymers, but also to complex polymers consisting
of branches, loops or crosslinks, for they can be regarded as multiple linear chains
connected together. This brief introduction is limited to linear polymers only.

The simplest chain model of polymers is the ideal chain (also known as Gaussian

chain, unperturbed chain or Brownian chain) model. In this model the segments of a

chain are chosen ”to be appreciably longer than the persistence length of the polymer
such that they are independent of each other. One segment can acquire arbitrary
orientation with respect to other segments. Furthernmre, there is no interactions
among the segments.

Consider an ideal chain with N segments, 13,13, "ul—iV- The mean square of its

end to end distance is defined as:
< R2 >=< (25,92 > (1.4)

As there is no correlation between two segments, the sum of ensemble average <
12'. - lj > vanishes when 2' # j. If each segment has an average length I) = V < [i >, it
can be easily derived:

< R2 >2 N <12 >= N122 (1.5)

Analogy can be drawn between an ideal chain and a random walk. The contour of
an ideal chain can be directly compared to the trajectory of a random walk. It can
be seen that the above result agrees with the random walk model as expected.
The mean square of R9 of an ideal chain is [5]:
< 122 > Nb?

2 .
<R >=—-—————=-——- 1.0
9 6 6 ( )

So the size of ideal chain shows a power law dependence on the number of monomers
N in a polymer:

R ~N1/2 (1.7)

The real chain (or perturbed chain) model is derived on the basis of the ideal chain
model. It takes into account the hard core repulsion between the segments, so that it.
is a more realistic description of polymers compared to the ideal chain. Mathematical

treatment of the perturbed chain is more difficult. It turns out a similar power law

\1

dependence of R9 on the number of monomers can be found:
R ~ N3/5 (1.8)

The exponent is bigger than that of the ideal chain model due to the excluded volume
effect.

The chain models for polymers are the so—called coarse-grained models. In addi-
tion to them, polymers can also be modeled in other ways, as illustrated in fig. 1.4.
The ab initio method employs quantum mechanics to study the processes in polymers
that take place at very small time and length scales. Atomistic and coarse-grained
models are often adopted in the molecular dynamics or Monte Carlo simulation of
polymers. They rely on classical force fields to describe polymers. And the properties
of large polymer systems, such as bulk polymers, can be studied with fluid models or

finite element methods.

1.3 Phases for single and bulk polymers

A long polymer can exhibit two distinct phases, namely coil and globule, which
have been extensively studied in the past [8]. Basically, for a polymer at low tempera-
tures, the attraction between monomers is dominant, so that it forms a globule, with
R9 approximately scaling as N1/3. While at high temperatures, a polymer tends to
expand to acquire a larger conformational entropy, therefore it takes on the coil state,
where the thermal effect completely overcomes the attraction between monomers, and
leads to a net excluded volume effect in the polymer. In this case polymers behave like
perturbed chains, with R9 being proportional to N3/5. In fig. 1.5, R9 of polystyrene
chains in cyclohexane is plotted against temperature, where a transition from the
globule to the coil state can be observed.

In the transition from the globule to the coil state. there is a special ten‘iperature

Time scale

 

Finite Elements

 

>

Fluid models

    
 
    

Coarse-grained

Atomistic models

 

 

 

 

 

>
Length Scale

Figure 1.4: Polymer models at different time and length scales.

 

 

 

 

S
esool— —
2000- n

.2
5 l500l— RH __
13
O
a:
{000” -1
500-— -
0 .1 l_ 1 _l _L L _1
20 30 60

40 5
Temperature ( °C)

Figure 1.5: Coil-globule transition of polystyrene (All-w = 2.6 x 10’) in cyclol'iexane.
S is the radius of gyration. R H is the hydrodynamic radius[1].

10

point where the attraction between the monomers exactly cancels the effective hard
core repulsion, so that the excluded volume effect vanishes. It appears that the
monomers do not interact with each other, and polymers behave as if they were ideal
chains. This special temperature is called the @(theta) temperature.

In addition to the temperature, the solvent quality can induce the coil globule
transition, too. Since the attraction between the monomers and the solvent molecules
offsets the attraction between the monomers themselves and results in an expansion
effect, the solvent plays a similar role as the temperature. The solvent that causes a
polymer to expand is called a good solvent of this polymer, otherwise a poor solvent.
The solvent satisfying theta condition is called 6') sol-vent, accordingly.

The quality of a solvent is changeable. A solvent can be poor at low temperatures
and become good at high temperatures. If a solvent. is made of the same molecules
as the monomers, it will always offset the attraction between the monomers. Con-
sequently, polymers stay in the perturbed chain state at all temperatures. Such a
solvent is called an atherrnal solvent.

Bulk polymers have different phase behaviors, which can be indicated by the
Young’s modulus as in fig. 1.6. Firstly, at very low temperatures, most bulk polymers
do not crystallize. Instead, they are trapped in the glassy state (region 1). When the
temperature rises, they undergo a transition from the glassy state to the rubbery state
(region 3). The transition is marked by region 2. And at very high temperatures,
bulk polymers start melting and change into polymer melts eventually, which are

indicated by region 4 and 5 in the figure.

11

Log Young‘s Modulus. Pascals

 

 

 

Temperature

Figure 1.6: Temperature dependence of Young’s modulus of bulk polymers [2].

12

Chapter 2

Molecular dynamics simulation of

polymers

Computer simulations have been extensively used to study polymers for the past
few decades. They are applied to a wide variety of polymer systems, such as polymer
chains, polymer melts, polymer membranes and polymer solutions etc. They are also
adopted in the emerging areas of polymer science, e.g., polymer nanocomposites and
polymeric nanoparticles. It can be foreseen that computer simulations will play a
more important role in polymer science in the future.

Molecular dynamics (MD) and Monte Carlo (MC) are two major simulation
techniques in the study of polymers. Both of them are capable of simulating a polymer
system at equilibrium, which is done by firstly generating an ensemble of microscopic
states for the system being studied and then evaluating its properties by averaging
over the whole ensemble. These two methods yield results with same accuracy and are
equally efficient. However, they differ from each other in their ways of generating their
ensembles of microscopic states. A Monte Carlo simulation does it in a stochastic way,
where the microscopic states are generated randomly; while a molecular dynamics

simulation does it by integrating the equations of motion of the system, so that the

13

microscopic states it generates also follows the trajectory of the system in its phase
space. Therefore, the study of the system dynamics can only be done by molecular
dynamics simulations.

This chapter focuses on the molecular dynamics simulations of polymers only.
Firstly, section 2.1 provides an overview of molecular dynamics. Then in section 2.2
its application to polymer systems is discussed. Finally, in section 2.3 and section 2.4

some basics of molecular dynamics and force fields are introduced.

2. 1 Introduction

As an input to a molecular dynamics simulation, the initial configuration of the
system under study has to be created prior to the simulation. An initial configura-
tion specifies the coordinates and velocities of all particles in the system. It can be
prepared in many different ways, which can be realistic or not. Sometimes the initial
configurations may be created far from equilibrium, so that the subsequent molecular
dynamics simulation fails. To cope with this problem, an energy minimization usu-
ally is applied before the real molecular dynamics simulation. It drives the system to
a stable state, corresponding to a local minimum in the potential energy landscape.
Another approach is to introduce a so called “soft potential” for a short period of
time. Unlike physical potentials such as the Lennard—Jones potential, soft potentials
remain finite even when particles overlap with each other. Thus they are able to push
particles apart gradually and leave the system in a safe state.

A typical molecular dynamics simulation comprises of two parts. The first part
is the equilibration run where the system is brought to its equilibrium. The second
part is the production run where a number of snapshots of the system are taken to
calculate its properties. In some cases, an energy minimization may be performed on

those snapshots prior to the calculation to eliminate thermal noises.

14

Molecular dynamics employs classical mechanics to describe the motion of a
molecular system being studied. Though quantum mechanics is a more accurate
description of such a microscopic system, it is not feasible because of the tremendous
difficulty in solving the Schrodinger equation that involves a large number of atoms.
Since classical mechanics can only provide approximate descriptions of microscopic
systems, it inevitably causes errors in molecular dynamics simulations. Consider a
simple oscillator with a resonant frequency V. When its energy quantum hu is greater
than kBT, it should have a large probability of staying at the ground state according
to quantum mechanics. However, classical mechanics allows the oscillator to absorb
heat continuously so that it can rise to higher energy levels. Hence corrections have to
be made to the kinetic energy and specific heat of the system[9]. Basically, when there
are strong quantum effects such as electron excitation/ transfer/ tunneling, molecular
dynamics has to be corrected or replaced with its quantum mechanics counterpart.

A molecular dynamics simulation is divided into discrete time steps. At each
time step, coordinates and velocities for all atoms are computed by integrating the
equations of motion. However. since the integration algorithms neglect higher order
terms in Taylor’s expansions of coordinates and velocities, large errors can arise if the
time step is too big. To reduce this error, it is usually required that. the integration
time step is appreciably shorter than the period of the fastest oscillation in the system.

On the other hand, the computing time of a simulation is not proportional to
the real time period being simulated, but. the number of integration steps to run,
which simply is the real time period divided by the integration time step. Thus,
the longer each integration time step is, the shorter computing time the simulation
takes. Basically there are two ways to enable a. longer time step. One is to replace
fast harmonic oscillators, (2.9., that of the bond stretching and bond bending, with
hard constraints. Another way is to use relatively less rigid harmonic potentials to

describe those oscillators.

In a molecular dynamics simulation, the force field that describes the system is of
the central importance. Force fields are designed “artificially” in order to reproduce
by computer simulations the real dynamics of molecular systems, using parametrized
classical potential functions . So a force field by its nature is not a precise account of
real systems, even though its parameters have often been carefully fit to the experi-
mental data. Furthermore, an additional “transferability” error can occur[10] when a
force field is applied to systems that it is not trained for. And it can also occur when
calculating some physical properties whose experimental data are not incorporated
in the fitting of the force field. More about force fields are discussed in section 2.4.

Overall, molecular dynamics is an effective technique to study complex systems,
though with certain approximations. Usually, the larger the system is, the more
approximations one has to introduce. So the result of a molecular dynamics simulation

should always be checked with experiments.

2.2 Difficulties in the simulation of polymers

The simulation of polymers has intrinsic difficulties due to their complex nature[11].
Firstly, polymers are inhomogeneous systems which exhibit distinct structures at dif-
ferent length scales. To study their properties the samples have to be large enough
to take into account all the inhomogeneities. For example, to study the properties
of a polymer chain, the sample has to be of a size equal to the full chain, which is
typically of tens of thousands of atoms. The samples have to be even larger when it
comes to bulk polymers such as polymer melts. Compared to simple liquids which are
homogeneous above a length scale about. 10 A (so that a. small number of atoms are
sufficient to show its properties), polymer systems are significantly more complicated.

Secondly, the simulation of a polymer system usually takes a. long computing

time because the time scales of its processes are vastly different. Vibrations of chem-

16

F13 second, while it takes

ical bonds in a polymer take place at a time scale of 1(
about 10""11 second for a polymer to overcome the torsional potential barriers. The
equilibration time for a large polymer system is significantly longer, ranging from
10’8 to 10—5 second. As mentioned in section 2.1, the time period of a single inte-
gration step has to be appreciably shorter than the period of the fastest oscillation in
the system. Therefore, a time step of 10’15 second is usually adopted for a realistic
polymer simulation. Consequently, the whole simulation of a polymer system may
take tens of millions of steps to reach equilibrium, making it a CPU intensive task.
Thirdly, polymers have large configurational varieties such that their physical
properties vary greatly. To better account for this diversity, it is often needed to
study a number of polymers with various configurations, or a very long chain. The
use of a long sample or multiple short samples can also improve the ergodicity of a.
simulation. Ergodieity ensures that the phase space of a system is faithfully sampled
according to its equilibrium distribution, but is often hard to achieve for a molecular
dynamics simulation which typically corresponds to a very short period of real time.
In answering these difficulties, the rapidly evolving computer technology as well
as the advances in simulation techniques greatly improve our ability of handling the
simulation of polymers. A comprehensive introduction to the simulation of polymers

can be found in many good books[11, 12].

2.3 Basics of molecular dynamics simulation

2.3.1 Units

Real units and Lennard-Jones reduced units are two widely adopted units for
molecular dynamics simulations. The real units are often used in atomistic models of
polymers. In real units, energy is measured in Kcal/mol, mass in g/mol, temperature

in K and distance in A . The units for other quantities can be determined accordingly.

17

The Lennard-Jones reduced units are more suitable for theoretical models of

polymers. They are based on the parameters in the Lennard-Jones potential:

ULJ = 4r (3)12 — (3)6 (2.1)

e and o are chosen to be the unit. for energy and distance respectively, and the mass
of a proton is adopted as the third basic unit. Other units are derived from these

three. In particular, the unit of temperature is T = t/er, and the unit of time is

t: y/m02/6.

2.3.2 Boundary condition

In a molecular dynamics simulation, the system to be studied is enclosed in a
simulation box for which the boundary condition has to be set.

The simplest way is to use the free boundary condition. It is good for an isolated
system, such as a single polymer chain in vacuum. A direct consequence of the free
boundary condition is the lack of pressure because pressure is exerted by the walls of
the simulation box, which, however, are not fixed under the free boundary condition.

Most real systems exist in nontrivial environments to which the free boundary
condition does not apply. In this case, the periodic boundary condition is often used.
It assumes that a simulation box is surrounded by translated images of itself in three
directions. And particles can interact with their image particles across the boundary.
Those particles that exit from one side of the simulation box should re-enter from the
opposite side.

The periodic boundary condition is more realistic compared to the free boundary
condition. But it has one side effect, 'i.e., the fake periodic correlation. So it is
important to use a large enough system such that the fake correlation is negligible.

And many trial simulations may be performed in order to find the appropriate size

18

of the simulation box.

2.3.3 Verlet algorithm

Verlet algorithm is the most popular algorithm of integrating the equations of
motion in molecular dynamics. It updates the positions and velocities of particles by

the following formula]13]:

r(t. + At) = 2m) — r(t — At) + m2 + 0(At4) (2.2)
v(/.) 2 [r(t + m) — r(t — At)]/(2At) (2.3)

This Verlet algorithm has been n'iodified to form. two slightly different integration
algorithms. They are referred to as leap-frog Verlet and relocity Verlet respectively.
In the leap-frog Verlet algorithm, velocities are computed at the midpoint of a time

step and are used to update positions:

v(t+%) --v(t— i)+i‘At (2 4)
At .
r(t + At) = r(t) + v(t + 7)At (2.5)

In the velocity V ‘,1'l(‘t algorithm, the equations for updating positions and veloc-

ities are:

.. At? .
r(t + At) = r(l) + v(t)At + r0)? (2.6)
- .. .. At
v(t + At) = v(t) + [r(t) + r(t + an]? (2.7)

It can be verified that. all those Verlet algorithms are consistent.
The Verlet algorithms are not the most accurate ones to integrate the equations

of motion. For example, Gear algorithm yields better accuracy. However, V Jrlet

19

algorithms obey the time reversibility and maintain the conservation of energy. They
are easy to implement, too. So Verlet algorithms are by far the most widely used

ones in molecular dynamics.

2.3.4 Thermostat and barostat

The integration of the equations of motion in molecular dynamics updates a
system according to Newtonian equations. It generates microscopic states that follow
the microcanonical distribution. To study more general cases, namely the canonical
and grand canonical distribution, thermostat and barostat have be incorporated.
They ensure a system to stay at a constant temperature or pressure. In below two
major thermostat / barostat techniques are briefly introduced, i.e., Nose-Hoover and
Berendsen.

In 1984, Nosé proposed a method to couple a dynamical system to a heat bath[14],
which later was improved by Hoover[15, 16] and thus was referred to as Nose—Hoover
method. Basically the method adds an extra degree of freedom 3 to the Hamiltonian
of the system to describe the effect of the heat bath, which leads to one additional

kinetic energy term TS and one additional potential energy term V3:

9

P5 . -
T» — 2.8
. ,Q <. >
V3 2 ngTln.(s) (2.9)

where g is the number of the degrees of freedom of the system, and Q is the effective
mass for the new degree of freedom 3. Based on the new Hamiltonian it can be

derived that[17]:

f 2-5—40 (am)
771.
. 1
= _T_T an
C Q( a ( l

where T and T0 are the temperature of the system and the heat bath respectively,
and C is essentially a friction parameter. It can be seen that Nose-Hoover method
basically adds a frictional force CU to the system to achieve the coupling to the heat
bath. Its time derivative is proportional to the difference between current system
temperature and the temperature of the heat bath.

Nose-Hoover method generates the canonical ensemble of the microscopic states
of the system in a continuous, deterministic way. And in the mean time, the time
reversibility is followed. Due to its nice features, Nose—Hoover method is widely
employed in molecular dynamics simulations.

The algorithm introduced by Berendsen et al. [18] implements the thermostat / barostat
in a simpler way. The coupling of a system to its environment is made explicit with-
out any change of the Hamiltonian. Berendsen method requires that the temperature

and pressure of the system obey two equations in below:

@- = ——-—T — T0 (212)
(ll, TT '
(ff. : fl (2 13)
alt rp '

where T and P are the current temperature and pressure of the system, while T0 and
P0 are those of the environment. The system temperature and pressure are changed
by rescaling velocities and positions of all atoms directly.

Berendsen algorithm allows a desired coupling strength to be chosen by selecting
appropriate values for 7T and rp, which determine how frequently the positions and
velocities of atoms are rescaled. Usually in the equilibration part of a simulation a
short TT(TP) is chosen for the system to reach equilibrium quickly, while in the pro—
duction part a. longer rT(rp) is used to ensure that the perturbation to the dynamics

of the system is negligible when its properties are evaluated.

2.4 Force field

2.4. 1 Overview

A force field defines the interactions between particles. Depending on the level
of abstraction, a particle can represent a real atom, a united atom or a group of
atoms. Accordingly, the force. field is referred to as an all atom force field, a united
atom force field or a coarse grained force field respectively. An all atom force field
defines the interactions between real atoms in the system. A united atom force field
treats a carbon atom with all hydrogen atoms bonded to it as a single particle, i.e., a
united atom. It only defines the interactions between those united atoms. In a coarse
grained force field, a large number of atoms are grouped together to be considered as
a whole body, c.g., the bead spring model. It leads to a highly simplified force field
without chemical details. Only the interactions between those abstract “particles”
are considered to determine the primary conformations of polymers.

Force fields can be derived by either the empirical method or the ab initio com-
putation method. The empirical method fits the force fields to experiment data to
determine their parameters. Sometimes data. obtained from quantum chemistry com—
putation of small molecules are also used in the fitting. Most popular force fields are
obtained through this approach, such as CHARMM[19, 20, 21], AMBER[22, 23, 24,
25, 26], and MM2/lV'IM3/MM4[27, 28, 29]. On the other hand, the ab initio com—
putation method derives force fields by direct quantum chemistry computations. It
is utilized in the derivations of MMFF94[30] and CFF[31] force fields. The ab initio
computation method builds a force field 011 a consistent ground. It is not affected by
errors or incompleteness of experimental data. Due to this advantage, the ab initio
computation method is very promising in the future derivations of force. fields.

A short introduction to force fields is given in the following two subsections.

Further readings can be found in some review papers [32. 33, 34] and references cited

22

therein.

2.4.2 Implementation

A force field consists of different types of potentials. In general, they can be
separated into two classes, bonded interactions and pair (non-bonded) interactions.
Bonded interactions consists of bond, angle, dihedral and improper interactions, while
pair interactions include van de Waals and Coulomb interactions. The total potential

energy of a system can be written as:

U = Ubonded + Upair (2.14)
Ubonded : U bond + Uangle + Udihedral + U improper (2-15)
UPair : Upande W'aals + UCoulomb (2'16)

The potential functions in a force field can take on various forms. The simplest
one is denoted as the class-I additive potential energy functions, which are supported

by most force fields. They are defined as the following:

()

.d

U bond : ZKTfr _ 7'0)

U

angle

= 21",,(9 — 60)2
U - = EK(1+ cos(ncb — 6))
dihedi al 4?) .

x . 2
Uimproper : SAX“ — X0)
0' 12 a 6
Umrndel/Vaals : 246 [(7) — (—,T) ]
(ff/1(1)

('7‘

 

UCoalomb : E

The class-I functions contain only harmonic terms. which are adequate in the vicinities

of the potential minirna. But in order to study states far from equilibrium, a class-II

‘23

 

 

 

 

Figure 2.1: Dihedral angle a defined for the particle sequence 1-2—3—4.

force field is often needed since it contains higher order terms so that it can treat large
deviations from equilibrium more accurately. A class-II force field is more elaborate
and provides a better description of the potential energy[34].

Four types of bonded interaction are bond, angle, dihedral and improper inter-
actions. Bond interaction defines the bond stretching motion between two chemically
bonded atoms, and angle interaction defines the bond bending motion for three con-
secutively bonded atoms.

Dihedral interaction is described by dihedral potentials. A dihedral potential
applies to two atoms that are separated by three consecutive bonds. In fig. 2.1, the
dihedral angle for four consecutively bonded atoms 1,2,3 and 4 is defined, which is
the angle 01 between plane 123 and 234. The corresponding class-I dihedral potential

energy function is simply:

UDihedral = 1\'(l + (308$?) (2.17)

It. has three minima at 60°, 1800 and 3000 corresponding to one trans and two gauche

24

conformations, with the trans conformation at o : 180°. This definition of a dihedral
angle is used in the LAMh‘IPS[35] software. Other molecular dynamics software may
use a different definition, where the trans angle becomes 00 and the dihedral potential
energy function form changes accordingly.

Improper interaction can maintain chemical stericity. Typically, an improper po-
tential should be enforced about. a sp2 hybridized carbon atom to ensure its planarity.
Furthermore, in united atom force fields, an improper potential can also be enforced
about a sp3 hybridized carbon atom that is bonded with one hydrogen atom and
two other atoms. Because the hydrogen atom is not explicitly represented in united
atom force fields, an improper potential is required to prevent those four atoms from
collapsing into a plane.

The definition of an improper angle involves 4 atoms 1', j, k and l. It is the angle
between plane i jk and plane jlcl. Usually atom i is the center atom, to which atoms
j, k and l are bonded, while there are usually no bonds between atoms j , k and l. In
fig. 2.2, the improper angle X for four atoms labeled as 1,2,3 and 4 is illustrated. It
is the angle between plane 123 and 234.

According to this definition, improper angles in some common cases can be found.
For a sp2 hybridized carbon atom, the improper angle is 0°. For a sp3 hybridized
carbon atom, the improper angle is approximately 35.260.

Those bonded potentials have different strengths. Dihedral potentials are impor-
tant in determining the conformation of a polymer. They are usually weak such that
polymers can overcome dihedral potential barriers to arrive at other conformations
easily. On the contrary, other bonded potentials, i.e., bond, angle and improper po—
tentials are typically strong. They are key to maintain the configurations of polymers
but have little to do with their conforn‘iational changes.

There are two types of pair interaction, van de Waals and Coulomb interaction.

Van de Waals interaction is con'imonly described by the Lennard-Jones 12-6 potential,

[\D
01

 

 

 

 

 

.4

 

Figure 2.2: Improper angle X defined for the particle sequence 1-2-3-4.

but subtle differences exist between different force fields. Firstly, the so called 1-2,1—3
and 1-4 pair interactions may be treated differently. Those pairs refer to two atoms
that are separated by one, two and three bonds, respectively. The atoms separated
by one bond have a direct bond potential between them, while atoms separated
by two bonds usually have an angle potential defined. Bond potentials and angle
potentials are much stronger than Lennard-Jones potentials, thus the 1-2 and 1-3
pair interactions are often excluded from the total potential energy. As for the 1-
4 pair interaction, usually it is included but scaled by a factor '7 (3v < 1), with 7
assuming different values in different force fields.

The second difference resides in the mixing rules for the 1;)arameters of various
Lennard-Jones potentials. The parameters in Lennard—Jones potentials are typically
defined on a per-type basis, namely, each type of atom has the same parai'neter set.
The mixing rules are en'iployed to. determine the parameters for the van de Waals

interaction between two different types of atoms. A commonly used set of mixing

26

rules is:

6 = m (2.18)

U = (01+02)/2 (2.19)

Thirdly, the cut—off distances of Lennard—Jones potentials vary from one force
field to another. To save computing time, all force fields exclude from the potential
energy calculation those pairs of atoms that are too far apart, i.6., beyond the cut-
off distances. The cut—off distances are basically artificial choices, but. they may be
slightly correlated with the parameters of Lennard-Jones potentials because force
fields are fit to experiment data as a whole.

The last term in the potential energy is the Coulomb interaction, which is im-
portant for charged polymers. Since this thesis does not involve charged polymers,
Coulomb interaction is not to be discussed here. Details about it can be found in the

aforementioned references.

2.4.3 Comparison

The designs of force fields are far from converging. They differ not only in the
potential function forms and parameters, but also in the subtle issues such as the
treatment of the 1-4 pair interaction[34]. However, the apparent discrepancy between
them is largely due to the fact that each force field is designed for a particular type of
systems, and is thereby only fit to the relevant experiment data. Basically, all force
fields perform well in their own fields. as was verified by numerous simulation work
in the past.

A force field should only be applied to what it is designed for. In general, the
use of a force field for a. system that it is not trained for can cause excessive er-

rors, which is referred to as the “transferability” issuellO, 36]. It is thus important

27

to know the applicable area of each force field[34]. In short, CHARMM, AMBER,
GROMACS[37, 38] are primarily designed for macromolecules in condensed phases
or in solutions. However, since AMBER took into account the transferability issue
when it was developed, it has a wide range of applications that include the gas phase.
MM2/MMB/MM4, MMF F 94, COMPASS[39, 40], CFF[31], Dreiding[41] and UP F [42]
deal primarily with small molecules in the gas phase. Except for MMF F 94 and COM-
PASS, all of them focus on the local structures of molecules. Since little experiment.
data of condensed phases were used to derive them, the nonbonded interactions are
described with less accuracy.

In addition to the applicability, the performance of each force field is another
factor to consider when choosing a force field. Various popular force fields have been
assessed in some previous works [36, 43, 44, 45]. It turns out that all force fields
have their own strengths and weaknesses. Among all the force fields, MMFF 94 force
field stands out, partly being attributed to its full quantum chemistry foundation.
Furthermore, AMBER force field is good in finding the correct ground state structure
but produces relatively large errors in the energy calculation. MM2/MM3/MM4 force
fields are fit to most comprehensive experiment data, so that they can serve as a
touchstone for evaluating other force fields. UFF and DREIDING force fields are two
universal force fields. They are designed to work with all systems. However, since
they are not optimized by experiment data, when studying a specific system, they
yield less accurate results than a parametrized force field that is fit to the particular

data.

28

Chapter 3

Intramolecular crosslinking of

BCB-styrene copolymers

3.1 Introduction

The intramolecular crosslinking of a polymer is a process in which chemically
active monomers react with one another to form additional covalent. bonds. Active
monomers are referred to as “crosslinkers” to distinguish them from monomers that.
can not crosslink. The intramolecular crosslinking process permanently alters the
structure of a polymer. As the density of intramolecular crosslinks increases a poly-
mer becomes more compact, and eventually transforms into a particle-like object.
The feasibility of synthesizing nanoparticles by intramolecular crosslinking of poly-
mers has been demonstrated in several recent experiments[4. 46. 47, 48], opening the
door to fabrication of designed nanoparticles by crosslinking of carefully controlled
unimolecular precursors.

Nanoparticles have abundant applications, such as bio—integration [49], polymer
mane—composite materials[50]. drug delivery[51], stabilization of polymer films[52, 53],

facilitation of polymer processing[54. 55] and so on. Those applications typically rely

29

on nanoparticles with specific chemical and physical properties, which entails the
ability to customize nanoparticles. h'lor'eover, industrial production of nanoparticles
requires an effective control over the manufacturing process. Therefore, a good under-
standing of the intramolecular crosslinking process is indispensable for its application
to nanoparticle mamifactm‘ing.

The percentage of crosslinkers in a precursor polymer is a first factor that can be
utilized to direct intramolecular crosslinking processes, as it is easily tuned yet has a
direct impact on the structure of the polymer nanoparticle end product. A polymer
with more crosslinkers is more rigid and usually, though not always, it is also more
compact. The percentage of crosslinkers also affects other aspects of the crosslinking
process including the reaction rate and and the fraction of crosslinkers that are never
able to find a partner, i.e. the reaction completeness.

Intramolecular crosslinking of a polymer bears some resemblance to the folding
of a protein as both of them involve the collapse of chains due to the interaction
between their segments. In particular, the rigidity effect has been studied in protein
folding processes[56]. One thus can expect that the rigidity of a polymer has a
similar dependence on its intramolecular crosslinking density, as demonstrated here
by the studies of the conditions under which the rigidity transition may occur due to
irreversible crosslinking.

In this chapter, the intramolecular crosslinkings of BCB / styrene copolymers with
a full range of crosslinker concentrations were studied by molecular dynamics simula-
tions using the LAMMPS software[35]. It is organized as follows. In section 3.2, the
experiment on the intramolecular crosslinking of BCB / styrene copolymers is intro—
duced. In section 3.3 a united atom model of BCB / styrene copolymers is presented.
And in section 3.4 the procedure of the crosslinking simulation is described in detail.
In section 3.5 the crosslinking reaction rate of BCB / styrene copolymers is studied and

the fraction of BCB monomers that are unable to crosslink is also found. In section

3.6 the rigidity effect on the intramolecular crosslinking of BCB/ styrene copolymers
is discussed. Simple theoretical models are provided to help understand the system-
atics of the data. In the last section, the simulation studies are summarized and the

overall understanding derived from them is given.

3.2 Experiment

The simulation work on the intran’iolecular crosslinking of BCB/ styrene copoly-
mers was motivated by a recent experiment[4], where it was demonstrated that
nanoparticles with radii of gyration ranging from 511m to 20 nm can be manufactured
by the intramolecular crosslinking of BCB / styrene copolymers. The experiment was
performed in a good solvent so that the precursor polymers achieved expanded mor—
phologies. A typical precursor polymer in the experiment had 1000 monomers, among
which about 20% were BCB monomers. When heated to 250°C, the BCB monomers
started reacting with one another to form crosslinks. The crosslinked polymers ex-
hibited considerable contraction and formed nanoparticles eventually.

In practice, two different types of crosslinking can occur in a crosslinking ex—
periment, namely intramolecular and intermolecular crosslinking. Intramolecular
crosslinking connects monomers within a single. polymer, while intermolecular crosslink-
ing connects monomers from different polymers.

The key point of an intramolecular crosslinking experiment is to suppress un-
wanted intermolecular crosslinkings. Typically, this can be. accomplished by reducing
the concentration of polymers in the solution so that. an ultra dilute solution is of-
ten required for the ideal effect. In the ultra dilute solution, it is very unlikely for
two polymers to come close enough to form an intermolecular crosslink. However in
the ultra dilute solution there is only a nominal amount of the reactants, i.e., the

precursor polymers, so this apI'H‘O‘dclfl is not suitable for mass production.

31

The authors introduced another technique to address this difficulty, which was
referred to as “slow addition” n1ethod[4]. In this procedure, the precursor polymers
were added into the reaction container slowly to allow enough time for each indi-
vidual polymer to crosslink by itself. At any time of the process, the concentration
of the precursor polymers was low enough such that the intermolecular crosslinking
was unlikely to occur. But the concentration of final polymeric nanoparticle prod-
uct increased steadily as the experiment went on. Those nanoparticles could not
react further with each other because nearly all their BCB monomers were already
crosslinked during the addition.

Compared to the approach of crosslinking the polymer in an ultra dilute solution,
the slow addition method has obvious advantages. It requires much less solvent yet
yields highly concentrated products. However, its applicability depends on the num-
ber of crosslinkers that are left unreacted as unreacted monomers in a nanoparticle
may lead to crosslinking with nanoparticles that are formed later. The concept can
not work in the case where a considerable number of crosslinkers are unable to find a

partner.

3.3 Model

The structure of a BCB / styrene copolymer is very similar to that of a polystyrene.
The only difference between them is that the former contains BCB monomers. A BCB
monomer, however, is basically a styrene bonded with two extra CH units. Hence,
it is not difficult to derive an atomistic model for BCB / styrene copolymer based on
one for polystyrene.

In the past, a plethora of simulation works have been done on polystyrene, where
a variety of atomistic models were employed. In 1990’s, Mondello et al.[57, 58, 59]

applied a united atom model to simulate polystyrene. It is shown that the model

32

led to good agreements with X ray and Neutron diffraction data. Recent simulations
of polystyrenes using this model also proved to be successful[60, 61, 62, 63]. As a
comparison, their all-atom model surprisingly failed to yield a result of same accu-
racy. TraPPE- HA is another united atom force field that has been applied to the
simulations of polystyrenes[64, 65]. It is relatively simple yet. has been demonstrated
to produce high quality results for some properties such as density, self diffusion etc.
Boyd ct al.developed an anisotropic united atom model [66, 67, 68], where they pro—
posed that centers of united atoms should be shifted from carbon atoms towards
hydrogen atoms. This modification was found to lead to better results for PVT sim—
ulations. Jorgensen ct al.designed an all-atom model [69] which has been utilized in
a number simulations of polystyrenes [70, 71]. It assigns partial charges to phenyl
groups in order to reproduce their electric quadruple moments. The universal force
field DREIDING [41] has also been used in the simulations of polystyrenes[72, 73].
Though it is not particularly optimized for macromolecules, it was found to pro—
duce results with a reasonable accuracy. In addition, some coarse grained models for
polystyrene are also developed[65, 74]. They aim to generate facile representations of
polystyrene as well as their mappings to atomistic models, in order to provide efficient
approaches for the large scale simulations of polystyrenes.

Among all those models, the united atom model by Mondello et al.[57] was
chosen for the crosslinking simulations of BCB/ styrene copolymers in this chapter.
This model treats a carbon atom with its bonded hydrogen atoms as a single particle,
i.e., a united atom. A widely used united atom force field[22] is incorporated in
this model. Though the force field has been considerably updated in the past few
decades[23, 24, 25, 26], the original version is still in use and is valid for molecular
dynamics simulations in vacuum.

The crosslinking simulations in this chapter were also performed in vacuum.

However, the experiment [4] was done in a. good solvent with temperature T = 523K.

33

Consequently the precursor polymers assumed expanded coil morphologies. In order
for polymers to achieve similar morphologies without solvent, a high temperature
(T = 900K) was used in the crosslinking simulations. The temperature is located
in the coil globule transition region for the atomistic model in vacuum. In reality
polystyrene can not survive such a high temperature. But it is doable in the sim-
ulation and the temperature can compensate for the absence of a good solvent. In
this sense, the temperature can be viewed as an effective temperature which incor—
porates the effects of both the solvent and the temperature. In an experiment, the
morphologies of the precursor polymers can be adjusted by either the temperature or
the solvent quality. Therefore, in a temperature—sensitive crosslinking experiment, a
desired morphology can be achieved by choosing an appropriate solvent. For example,
fig. 3.1 shows that a crosslinking experiment in the coil regime can be done above
the reaction temperature threshold Tc by using good solvents.

The original united atom model was slightly modified to adapt to the simula-
tion software LAMMPS[35]. Firstly, harmonic potentials were used to describe bond
stretching interaction between united atoms, while in the original model the bond
lengths were constrained by the SHAKE algorithm. The strengths of the potentials
for bond stretching interaction were taken from experimental data, but they were
reduced by a factor of 4 to increase the time step [68]. Secondly, the original model
incorporated dihedral potentials between atoms of a phenyl in order to maintain its
planar configuration. But in the crosslinking simulations a phenyl was constrained
as a rigid body so that its six united atoms moved uniformly as a whole. The total
force on a phenyl was calculated as the sum of all external forces on its six united
atoms. This treatment of phenyls has two advantages. Not only can it maintain a
strict planar configuration for a phenyl, it can also speed up the computer simulation
because all the internal forces within a rigid body do not need to be considered. A

similar rigid body constraint was also used for benzocyclobutenes in BCB monomers.

34

 

  

Crosslinking Reaction Region

 

Temperature

 

 

 

 

 

>
Solvent Quality

Figure 3.1: Crosslinking experiments in the coil regime can be done above TC by using
good solvents.

35

Each of them consisted of eight united atoms. Other details of the united atom model
in this work were taken from the literature [57].

The force field in the crosslinking simulations took the following forms:

0' 12 0' 6
= .. <—> — <—)
L‘] i 7‘ 'r
F =K(-' —- ')2+K(l—1)2
Jbond 7’ r 70 ' ‘0
,, 2 2

Eangle = [\9(9 — 90) + KQ-(Q’ — 0(0)
Edihedral : quU 'l' C056“) 'l' Ky)“ _ C05(2¢’))
. 2 , 2
Eimproper : leX _ X0) + hwiw — 0’0)

The parar‘neters in the above equations are listed in table 3.1. Lennard-Jones
interactions between united atoms separated by four or more bonds are considered in

the potential energy calculation.

3.4 Simulation

The crosslinking simulations were carried out for BCB / styrene copolymers with
250, 500 and 750 monomers separately. Unlike the experiment where the concen-
tration of BCB monomers was below 30%, the percentage of BCB monomers in the
simulations was varied from 5% to 100% to investigate the full range of crosslinker
concentrations. Ten samples were studied for each size and each crosslinker concen-
tration and the results were averaged to get better estimates of properties.

The initial configuration for a BCB/ styrene copolymer was generated in three
steps. First, its backbone was created by a self avoiding walk (SAW) on a diamond
lattice. Each site of the walk represented a GHQ or a CH united atom alternatively.
The diamond lattice auton’iatically ensured that two neighboring bonds made an angle

of 109.50. In the second step, the side groups were attached to the CH united atoms.

36

Table 3.1: The parameters of the force field for the united atom model of a
BCB/ styrene copolymer. Energy is in kcal/mol, and distance is in A

 

Lennard-J ones

 

 

 

 

 

CH2 (5 = 0.12 a —— 3.80
CH 6 = 0.09 a = 3.7
C((M'), C(a.7’)H‘ e = 0.12 0 3.7
bond

C — (7 KT 2 80 TO 1.54
(7((1.7‘)— C Kl : 80 10 - 1.50
angle

C — C((u‘) — C(aa") Ka :2 70 (to = 1200
C — (.7 — 0 K9 = 60 ()0 = 109.50
dihedral

backbone It’d) = .

phenyl rotation K113 = 1.0

improper

C(a'r) Kw = 80 rug 2 0

(3' KY 2 50 )(0 = 35.260

 

37

There were two types of side groups with different populations, phenyls and BCBs
and they were added in a totally random manner, with concentration a: of BCB units.
Each side group can be attached to any CH united atom and can be on either side
of the backbone, so it essentially generated an atactic random copolymer. However,
since it was possible that newly added side groups were too close to other particles
of the polymer, a short preliminary NVT simulation was then run for the polymer
to resolve any dangerous configurations. In this simulation a soft pair potential was
temporarily imposed for all particles to push them apart. The preliminary NVT
simulation lasted for only a few picoseconds and led to expanded morphologies for all
polymer samples.

Simulation of intramolecular crosslinking of BCB / styrene copolymers began with
the initial configurations generated as described above and consisted of three stages,
equilibration, crosslinking and re-equilibration. In the equilibration stage, NVT simu-
lations of all samples were run for 3.2 us at 900K. In the subsequent crosslinking stage,
the same NVT simulations were run, however they were paused every 0.8 picoseconds
to check if the crosslinking criteria could be met. At each check, the distances between
all pairs of BCB monomers were examined. in particular, the distances were measured
from 01/02 of one BCB monomer to (73/04 of another BCB monomer, as shown in
fig. 3.2. To establish a ('1 bond to (’3 and (72 to C4 their distances have to be less
than 5.21, or approximately 1.30, where 0 is the hard core radius of the Lennard—Jones
interaction between two CH united atoms. This distance requirement is consistent
with previous bulk crosslinking simulations[75]. The relative orientation of two BCB
monomers was also checked, so that new chemical bonds were only formed when the
four active carbon united atoms C71,(,1'2,C'3,C4 approached each other in a way that
their orbital electron clouds overlapped significantly. To impose this condition four
angles were examined as listed in table 3.2. All of these angles have equilibrium values

of 109.50, and in the crosslinking simulations their values have to be more than 90°

38

Figure 3.2: The crosslink between two BCB monomers.

Table 3.2: Table of bonded potentials in a crosslink
bond (.71 — C3 C2 - C4
angle C(ar) — C1 — (73 Cl — C3 -— C(ar)
C(m') — (72 — C4 C2 — C4 — C(m')
dihedral C(ar) — C71 — C73 - C(ar)
C(m‘) —— (7’2 — C4 — C(a'r)

 

 

 

 

 

and less than 1300 in order for the crosslinking reaction to proceed.

As shown in fig. 3.2, the crosslink between two BCB monomers is a floppy eight-
membered ring that introduces a number of extra bonded potentials to the model,
which are listed in the table 3.2. In total two bond potentials, four angle potentials and
two dihedral potentials are added for each new crosslink and its potential functions
and parameters are the same as those of the backbone of a BCB / styrene copolymer.

The check for the reaction criteria was performed every 0.8 picoseconds in the

crosslinking simulations. This interval was long enough to allow for the configuration

39

of a polymer to undergo a nontrivial change, yet it was short enough for the simulated
crosslinking to be finished in a. reasonable computing time. Once a pair of BCB
monomers was found to meet the reaction criteria, the crosslink formed innnediately
between them. The crosslinking stage lasted until there were less than two BCB
monomers left in a polymer or until 17 nanosecond of simulation time had elapsed,
whichever occurred first.

Once the crosslinking ceased the last stage consisted of a re-equilibration process

for another 3.2 ns. The crosslinked polymers were then analyzed.

3.5 Crosslinking reaction

The simulated reaction rate of the intramolecular crosslinking of a BCB / styrene
copolymer is greatly restrained by both orientation and distance requirements for two
BCB monomers. For instance. the simulation shows that it takes about 5 checks on
average to form a crosslink in a polymer with 750 unlinked BCB monomers, which
corresponds to a reaction rate at the picosecond time scale. Moreover, the reaction
rate drops quickly as the number of unlinked BCB monomers decreases and is at the
nanosecond scale when there are only tens of unlinked crosslinkers left.

Due to its slow kinetics, the simulation of the intramolecular crosslinking of a
BCB/ styrene polymer can not be completed in a practical computing time. A time
limit of 17 nanosecond is thus set on the crosslinking simulations. It is found that
a majority of crosslinkers are able to crosslink within this time period though a
substantial number of them remain unlinked. In addition, there are no obvious signs
that the crosslinking reactions are complete even though the reaction rates finally
become extremely slow.

Theoretically it is interesting to know if all crosslinkers can be crosslinked, which

also has significance to real applications. For example, unreacted crosslinkers can

40

function as docking points in drug delivery and they also affect the applicability
of the slow addition method to synthesize the polymeric nanoparticles. Note that
the number of unlinked crosslinkers was recorded at every check in the crosslinking
process (see previous section), i.e. every 0.8 picoseconds. So the theoretical number
of final unlinked crosslinkers can be obtained by extrapolation.

The crosslinking of two BCB monomers is an elementary reaction that is approx-
imated by the law of mass reaction, namely the reaction rate is proportional to the

product of concentrations of reactants. To leading order, it becomes
(ma/d: = 4.1%) (3.1)

where :r(t) denotes the number of unreacted crosslinkers in a polymer at time t, and
k is a rate constant. Note that in the equation above the number of reactants, i.e.,
crosslinkers, is used instead of its concentration. As a consequence, the coefficient I:
should contain an additional factor of 1 / V with V being the reaction volume. Since
the intramolecular crosslinking occurs only in the vicinity of a polymer, the reaction
volume is proportional to the size of the polymer. By multiplying by the number
of monomers N, a normalized coefficient can be obtained which is ks“ = k - N. The

solution to eq. 3.1 is :
117(0)

t:——
mfl 1+At

(3.2)

where A = kr(0). This solution predicts that the number of free crosslinkers decays
slowly with time and goes to zero at. long times. However this simple model omits
the influence of steric hindrance on the reaction rate. In particular if crosslinking
produces a highly constrained 0r rigid nanoparticle it may be impossible for some
crosslinkers to find a partner. As a result, there can be some free crosslinkers left
unreacted even if the crosslinking runs for an infinitely long time.

To approximate the effect. of the steric hindrance on crosslinking. eq. 3.1 is

41

generalized to:
17(0)

where $00 is the number of crosslinkers that are left unreacted eventually. The unity
in eq. 3.2 is replaced with a parameter B in order to maintain its validity when t
goes to zero.

This equation provides an excellent representation of all of the simulation data
that have been obtained, except at really low crosslinker concentrations where strong
statistical variations render the fitting less reliable. Examples of fits to the data for
three different crosslinker concentrations, 80%, 40% and 20% are presented in fig.
3.3. In these graphs all samples have 750 monomers.

From these fits, values for the parameters ls:*, B and $06 can be found for all
crosslinker concentrations but 5%, which has large variations. The normalized coef—
ficient 13* is shown in fig. 3.4, demonstrating that k* is approximately the same for
all concentrations and all sizes. The parameters B and 1700 are plotted in fig. 3.5
and fig. 3.6 respectively. The parameter B assumes values close to unity, which is
reasonable. However the most. interesting result is that 1700 assumes larger values
when the concentration of crosslinkers is above 60% and rises roughly linearly with
crosslinker concentration beyond this point. In the next section this behavior is found

to be related to a rigidity effect.

3.6 Rigidity effect

It is not surprising that higher levels of crosslinking yield more rigid nanopar-
ticles. As noted in the previous section this can lead to some unpartnered BCB
crosslinkers even at very long reaction times. The radius of gyration of the crosslinked
nanoparticles has been studied and exhibits some rigidity effects in its behavior.

Firstly, it is noticed that the radius of gyration of a crosslinked BCB / styrene copoly-

42

 

b'UU

 

0 Simulated
400 — Fitted T

 

 

 

 

 

N
O
O

..L
O
O

 

Unlinked crosslinkers
s

_L
O
0

U1
O

 

 

 

0 l L I i 4 l l
0 5 10 15

Time(ns)

 

Figure 3.3: The number of unlinked crosslinkers for BC B/ styrene copolymers with
750 monomers decreases as the crosslinking simulation proceeds. From the top down,
the three panels are for 80%, 40% and 20% BCB monomers respectively.

43

 

 

 

 

 

 

 

 

Percent of crosslinkers

Figure 3.4: The parameter 19* from fitting to eq. 3.3.

44

A N=250
”A n N=500 1
O N=750
—EI —+
O A
L. A -l
O O A n
D
_ '3 6 0 g Q 6 6
o 0 ii
J i l i l i l I l
20 40 60 80 100

 

 

 

 

 

 

 

 

I T T T T I I I I
A N=250
1,4_.A El N=500 _
O N=750
CI
0
CD12— a
n a A a A
6 g 9 o 8 93
1_ O O O B _
l I L I L I l l l
20 40 60 80 100

Percent of crosslinkers

Figure 3.5: The parameter B from fitting to eq. 3.3.

 

 

 

 

 

 

 

 

40 I I I I T T I I I (D
, A N = 250 _
El N = 500
e O N = 750
g 30 — —
.E ED
(3) o
L. _ El _
0 20
8 . o D A i
‘E‘ A
._ D
E 10 -— B 2 _
:3 El D O
_ A a
g a 3 D A
O l I I I l I l I l
20 40 60 80 100

Percent of crosslinkers

Figure 3.6: The parameter 2790(unlinked crosslinkers) from fitting to eq. 3.3, showing
a rigidity effect. at around 60% BCB crosslinkers.

46

mer in the simulation is comparable to that in the experimentl4]. For instance, in
the experiment a precursor polymer with molecular weight 11200 amu and 10% BCB
monomers yields a nanoparticle with hydroradius Rh 2 6.2mm after crosslinking.
This polymer has roughly 100 BCB monomers and 900 styrenes. In the simula-
tions the largest samples have 750 monomers. With 10% BCB monomers, they form
nanoparticles with average radius of gyration R9 = 5.372772.

In fig. 3.7, the average radius of gyration found in simulations is plotted for
BCB/ styrene copolymers with three different sizes. It shows a decreasing trend with
the increasing fraction of crosslinkers. However, when the BCB monomer concentra-
tion is above 60%, the R9 curves cease to show any decreases. This unusual behavior,
as well as large numbers of unlinked crosslinkers observed before suggest that when
the crosslinker concentration is above 60%, the nanoparticles are quite rigid.

In a previous work, the rigidity effect on the intramolecular crosslinking of poly-
mers was studied using coarse grained models[76). It was found that a rigidity tran-
sition occurred in some models with increasing crosslinker concentration. For exam-
ple, the freely rotating chain model (F RC) exhibited a rigidity transition at 40% of
crosslinkers. Above the rigidity transition, the HQ of crosslinked polymers showed lit-
tle change with the crosslinker concentration, and the number of unlinked crosslinkers
is higher. The Maxwell counting method[77] was employed to determine the rigidity
transition threshold for the coarse grained models and a nice agreement with the
simulation result was achieved.

To apply the Maxwell counting technique to BCB/styrene copolymers, both
the chain stiffness and the crosslinker concentration need to be incorporated. Stiff
polymers can be modeled as freely jointed chains in which each segment represents
m(m > 1) monomers. The segment length m is known as Kuhn length. Usually in
the freely jointed chain (F JC) models, each segment contains only one bead which

represents m monomers. But in order to describe the crosslinking, one bead should

47

 

70 I r I I I I I I I

 

 

 

 

 

 

 

20 l I l I l I l I I

 

Percent of crosslinkers

Figure 3.7: The radius of gyration of crosslinked BCB / styrene copolymers, showing
a rigidity effect when the crosslinker concentration is above 60%.

48

represent a single monomer, be it a styrene or a BCB monomer. So one segment
contains m beads in the new model for stiff polymers. They are distributed evenly
along the whole segment, and are strictly aligned in a straight line all the time.
Before crosslinking, the number of degrees of freedom in the chain is 2N / m since
the chain is equivalent to a simple FJC with N /m beads with two unconstrained
degrees of freedom for each bead. The number of crosslinkers is IN where y is the
crosslinker percentage. In the atomistic model, a crosslink brings about 7 additional
constraints. However, the only effect of those 7 constraints on two monomers is to
limit their distances. Therefore a crosslink adds only one effective constraint in the
new coarse grained model. Based on the Maxwell counting, the number of floppy

modes F of the model can be written as :

F (31) = ‘ —- — —— (3.4)

The rigidity transition should occur when F vanishes, yielding :rc = 4 / m. Polystyrene
has a Kuhn length m = 7 [5]. Thus the Maxwell counting estimate of the rigidity
threshold gives me = 4 / 7 x 57.14%.

On the other hand, based on the simulation data, the rigidity threshold for a
crosslinked BCB / styrene copolymer is around a: = 60% (see fig. 3.6 and 3.7), which
is in good agreement with the Maxwell counting estimate, indicating that simple
constraint counting ideas provide a good starting point for understanding rigidity
effects in unimolecular crosslinking. It should be noted that the crosslinks are flexible
in the sense that they can rotate somewhat, however they impose limits on the ability
of the chain to fluctuate and in particular to extend. In technical terms they are closer

to tensegrity constraints rather than pure rigid constraints.

49

3.7 Summary

A simulation study of intramolecular crosslinking of BCB / styrene copolymers is
carried out using an atomistic model. In addition, freely jointed chain models were
adopted for a rigidity analysis. Various factors that can affect the crosslinking and
the size of the crosslinked end state were considered, including the concentration of
crosslinkers, the ambient temperature, and the rigidity effect.

Simulations of the atomistic model of BCB / styrene copolymers show that over a
wide regime, the crosslinking rate has a quadratic dependence on the number of free
crosslinkers. However, steric hindrance plays an important role in the intramolec-
ular crosslinking where it. may prevent a finite fraction of crosslinkers from being
crosslinked even at long times. An equation. incorporating both the quadratic depen—
dence and the steric hindrance Provides a successful description of the crosslinking
reaction.

A rigidity transition occurs in the intramolecular crosslinking of BCB/ styrene
copolymers when the BCB monomer concentration is above 60%. In this region, the
number of unsatisfied crosslinkers goes up considerably. Moreover in this regime,
R9 of the crosslinked copolymers exhibits little change with increasing number of
crosslinks. hv'laxwell counting was applied to the F] C model of a BCB / styrene copoly-
mer, and found the quantitative dependence of its rigidity threshold on its Kuhn

length.

Chapter 4

Temperature series simulation of

intramolecular crosslinking

4. 1 Introduction

Polymers assume various morphologies under different environmental conditions,
which can also play a role in the intramolecular crosslinking process. A single polymer
can have two distinct morphologies, i.e., that of a compact globule or of an expanded
coil. It can transform from one state to the other on changing either the ambient
temperature or the quality of the solvent. Intuitively, when a polymer is in the glob—
ule state, the crosslinks are more likely to form between well separated crosslinkers,
resulting in a more compact nanoparticle. However, when the polymer is in the coil
state the crosslinks tend to occur between neighboring crosslinkers so that a more
loosely crosslinked nanoparticle is formed. This fact was briefly acknowledged in pre—
vious works[78, 79), however a. systematic study of the effect of initial morphologies
on intramolecular crosslinking of polymers is not. available.

In this chapter, the FJC model is adopted to study the effect of initial morpholo-

gies on intramolecular crosslinking, by performing a. temperature series of simulations.

51

The quality of the solvent can also induce a change in the morphology of the precursor
polymer, so that the result of the temperature series simulation is also meaningful
to the crosslinking experiments in various solvents at least to a first order approx-
imation. The chapter is organized as follows. Firstly, the simulation procedure is
described in section 4.2. Then in section 4.3, the contraction ratio of crosslinked
polymers are calculated and are compared to a previous work[78]. In section 4.4 a
variable chem/ical distance density is introduced to account for the contraction ratio.
It is applied to the simulation results of both freely jointed chains and BCB / styrene

copolymers. The last section provides a summary of this chapter.

4.2 Simulation

A temperature series of simulations of intramolecular crosslinking were performed

on a number of FJC samples using the standard interaction potential,

E: 4. (2)13 (2)6] + Afr—T 2 4.1)
pgs [ T 7' 53213 ( 0) (
where 6 = 1, 0 = 1, K = 1000 and 7‘0 = 1. All of them are in the Lennard-Jones
units.
Each F.) C sample comprised of 500 beads, of which a fraction, 37, were crosslink-
ers while the remainder were normal beads. Aside from its ability to crosslink, a
crosslinker is the same as a normal bead, with both interacting through the force
field defined in eq. 4.1. Four different concentrations of crosslinkers were studied,
namely 5%, 20%, 50% and 100%.
The temperatures being considered were in the range from T = 0.5 to T = 11
(also in the Lennard-Jones units), which covered all the temperature points of interest
for the FJ C model of the simulations. In particular, more points were taken between

T = 3 and T = 5, which lie in the coil globule transition regime and in total 44

temperature points were considered. At each temperature point, 50 F JC samples
were studied for each of the four crosslinker concentrations mentioned previously, so
the total number of FJC samples was 50 x 4 x 44 = 8800.

The temperature series of simulations started from high temperatures to low
temperatures. Firstly. the initial configurations of the FJC samples at the highest
temperature T = 11 were generated. The chains were constructed by self avoiding
walks on a FCC lattice with lattice constant a = 1 and the crosslinkers were ran—
domly distributed over the chains. The initial configurations for the samples at lower
temperatures were simply copied from the equilibrated samples at the next higher
temperature, 6.9., the initial configurations of the FJC samples at T = 10 were ob-
tained from the FJC samples after equilibration at T = 11. Note that in this copy
procedure the locations of crosslinkers were rearranged randomly, so that the initial
configuration for a sample at lower temperature had a new monomer sequence.

A molecular dynamics simulation of intramolecular crosslinking of the F J C model
was similar to that of the atomistic model of BCB/ styrene copolymers. It also con-
sisted of equilibration, crosslinking and re—equilibration stages. In the equilibration
process, N VT simulations were run for 10000 Lennard—Jones time units. Subsequently,
the crosslinking process was carried out, which consisted of 9000 Lennard-Jones time
units, with the interval between checks for the crosslinking criteria being 10 Lennard-
Jones time units. Unlike the atomistic model, there was no need to check for the
orientation between two crosslinkers. The only condition to be met for two crosslink-
ers to react was that their distance should be less than 1.30. The final re—equilibration

process consisted of NVT simulations for 10000 Lennard-Jones time units.

4.3 Contraction of polymers

The contraction factor is defined as the ratio of the radius of gyration of a
polymer after crosslinking to that of it before crosslinking, denoted as Rg(2:) and

R9 (0) respectively:

 

(4.2)

In fig. 4.1 the contraction factor against temperature is presented and it shows that
the largest contraction occurs between T = 3.5 and T = 4, which is just above the
coil globule transition region and close to the 9 temperature of the F JC model.

Intramolecular crosslinking results in a wide structural diversity in the final prod—
ucts because there are numerous ways to crosslink a chain with 500 beads. This is
reflected in the wide range of values of R9 especially at higher temperatures, leading
to relatively big fluctuations in 7 above the 9 temperature. But at lower tempera-
tures, the starting state is already a globule so the effect of intramolecular crosslinking
on the size of the polymer decreases and the fluctuation in 7 is smaller.

The contraction of an intramolecularly crosslinked polymer is naturally related
to its density of crosslinks, and since all crosslinkers can find a partner in the F JC
model, which has no rigidity effect, the density of crosslinks in the final state is
directly proportional to its crosslinker concentration. As evident in fig. 4.1, larger
concentrations of crosslinkers lead to greater contractions in the final states of the
FJC samples at all temperatures. This is counter to the effects observed in systems
exhibiting a rigidity effect where the contraction of polymers ceases to increase beyond
the rigidity threshold.

An early work [78] studied the quantitative relation between Hg and the den-
sity of crosslinks, in cases where the precursor polymers were made completely of
crosslinkers, i.e. :r = 1. Using the ideal chain model they argued that for a chain

with N monomers and m crosslinks, the ensemble average of its contraction can be

 

0

0
0
0
0
v

 

 

‘ '-' = =
_ A F. _
0.7 \r“ '5 .1
, l" T
l.“ n ._ t" u .
v .‘I . . -
0.6 — ‘ “JL' M A. ‘ “
\v“ I. ‘- i _“ :-
L- 'A\ ’ ‘_ _I‘ —
‘VA 4 -
.\ A '1 I ‘ ‘ p >
0.5 — ‘\ 5‘ ‘. . f \ f“ v v ——
“. 4: . I
_ O . a ’ o ’v G v 1
G—O Percent - 5/0 .. ’ A ,l . .,
0.4 “ E—El Percent= 20% ‘ ~ " ' " '4

A—A Percent = 50%
f 0—0 Percent = 100%

 

 

 

 

 

1 4f 1 I l I 1 I L I
0'30 2 4 6 8 10
Temperature

Figure 4.1: The average contraction ratio 7, namely the ratio of the radius of gyration
of the polymer after and before crosslinking, as a function of the crosslinking temper-
ature. The four curves correspond to four different concentrations of crosslinkers.

described by
(9(N, 712)) = 1 — x3p1/2 (4.3)

where g :2 R§(1) / 123(0), and p = m. / N. This prediction was compared to experiments
where the crosslinking reaction could be turned off before it was complete, enabling
access to the full range of values of crosslink density. The density of crosslinks p can
then assume any value from 0 for a chain with no crosslinks to 0.5 for a fully crosslinked
chain. The theory agreed well with their experiments. However, the authors pointed
out that their theory can not be applied to the intramolecular crosslinking of poly-
mers with only a fraction of randomly distributed crosslinkers. The fundamental
reason, they argued, is that the density of crosslinks by itself is insufficient to de—
scribe the crosslinks, and instead the distribution of crosslinking rings is the key. The
radius of gyration of a crosslinked polymer depends on detailed information about
all crosslinking rings, particularly their number and their sizes. The more rings there
are or the larger each individual ring is, the greater the contraction of the polymer is.
The density of crosslinks, however, only accounts for the number of crosslinking rings
but not their sizes. It suffices in the case of crosslinking a fully crosslinkable polymer
because the size distribution of its rings can be evaluated[78]. But in the case where
the chains are made of randomly distributed crosslinkers, the crosslinking rings with
various sizes can arise and are distributed in a totally random way. Therefore, even
if two crosslinked polymers have the same crosslinker concentrations, they can have
drastically different R9 due to differences in their crosslinking rings.

The FJC samples in this simulation work consisted of randomly distributed
crosslinkers. In order to compare the simulation result. of the contraction ratio to

the theoretical work, the data was fitted to a generalized form of eq. 4.3, namely:

“1’2 = 1 — .396 (4.4)

 

 

 

 

 

 

 

 

 

0.8 —
fl _
5
C 0.6 —
o
O. -
><
LIJ

0.4 —-

0.2 —

1 l 1 l 1 l 1 l 1 l
0 2 4 6 8 10
Temperature

Figure 4.2: The simulation data for the contraction ratio 7 2 129(1) / 129(0) are fitted

to the expression 7‘2 = 1 — 13/)“, yielding values for the pref-actor I3 and the exponent
c.

"J

C"

The fitting results are presented in fig. 4.2. The theoretical prediction based
on the ideal chain model is "r = 0.75, e = 0.5 and applies to the theta point that
occurs at a temperature of approximately T = 3.5 for the FJC model in this chapter.
It can be seen that the exponent found by the fitting is considerably smaller than
0.5 though the prefactor is close to that predicted by the theory, implying a stronger
contraction effect than predicted by the ideal model. This is most likely due to the
fact that the precursor chains are more expanded than ideal chains because of hard
core repulsion effects. The non-monotonic behavior in the exponent reflects the non—
monotonic temperature dependence of the contraction ratio (see fig. 4.1) and is due
to a subtle interplay between the dimension of the precursor chain and the number

of large rings that are produced by crosslinking.

4.4 Chemical distance density

To further quantify the effect of the sizes of crosslinking rings on the structure
of a crosslinked polymer, the “chemical distance” was introduced as the number of
monomers (along the backbone) that separate two crosslinked monomers. And the

chemical distance density is defined as:

ill d,-

11.1: —,—,— (4.5)
where d,- denotes the chemical distance of the ith crosslink and N is the total number
of monomers in the polymer. The summation runs over all the crosslinks. Chemical
distance density indicates the overall effect of the crosslinking rings, incorporating
both their number and their sizes. It can serve as a useful measure of the structure
of an intramolecularly crosslinked polymer. The largest value of Pd is of the order

of the number of monomers N. For example, when every monomer z' (1 S i _<_

N / 2) is connected to a monomer N/ 2 + i, yielding a chemical distance of N/2, the

58

 

 

 

 

 

 

 

 

 

 

 

 

j I I I T I I I I I
12 F c 16
I .1 0—0 pd I
_ ° ‘ 12
8- i
pd 1 " H R9 precursor _ 8 g
‘11 H Rg crosslinked -
4 l— , ‘2‘“ r -1
+- s‘, ‘11,qu A ,- ‘ __I 4
_ = = 3 : : r
0 a l 1 l 1 L 1 1 1 1 o
0 2 4 6 8 10
Temperature

Figure 4.3: The chemical distance density pd and radius of g rration R9 are plotted
for different temperatures. This data is for FJC samples with N = 500 and a: = 0.2.
chemical distance density of the crosslinked polymer acquires a value of N / 4. It is
also convenient to extend the definition of the chemical distance to a linear polymer,
where each pair of connected monomers contributes a chemical distance of unity, and
its chemical distance density is also approximately unity.

The data of fig. 4.3 shows that the chemical distance density has a clear tem-
perature dependence, assuming large values in the globule region and becoming quite
small in the coil region. In particular, there is a sharp decline of chemical distance
density at the coil globule transition regime.

To find a quantitative relation between the chemical distance density and the
radius of gyration, the simulation data at each temperature point are examined sepa-
rately. By doing so the temperature effect on the radius of gyration can be excluded

from the analysis. At each temperature point there are four different crosslinker

 

 
 
 
  

 

Percent = 5%
Percent = 20% ~
Percent = 50%
Percent = 100% ‘

DIG.

 

 

 

 

 

4.5

II'IirIITT

 

 

 

A

 

Figure 4.4: Eq. 4.6 is fitted to the simulation data for FJC samples at different
temperatures. Starting from the top the data is for crosslinking temperatures T = 8, 4
and 2 respectively.

concentrations, and 50 samples for each concentration. This yields 200 data points

for R9 and chemical distance density. Based on analysis of this data, a power law

dependence is found for R9 on chemical distance density (see fig. 4.4):

R9 = Rg(())(1+ pd)? (4.6)

Unity is added to pd in the equation above to recover the correct. limit for precursor
chains. The fitting parameters in eq. 4.6 are the exponent g and 129(0). Note that

129(0) is the radius of gyration of a polymer before crosslinking. The fitting results

60

are presented in fig. 4.4, where the fits and the data are compared at T224, and
8, providing an overview of the whole temperature range. Overall eq. 4.6 turns
out to be a reasonably good fit to the simulation data, though some deviations can
be seen for large chemical distance densities and low temperatures. A large chemical
distance density implies that there are complicated interpenetrating ring structures in
the intramolecularly crosslinked polymer, while at low temperatures the crosslinking
is a less important factor as the precursor polymers take on compact morphologies
already.

A series of values for 129(0) and for g at each temperature are then obtained from
the fitting procedure, As shown in fig. 4.5, Rg(0) obtained from the fitting procedure
agrees well with that found directly from simulations. The exponent g in eq. 4.6 is
plotted in fig. 4.6. It changes smoothly from the lowest temperature (T205) to the
highest temperature (T211), where g is approximately —0.4. Note that though 9 is

temperature dependent, it is not dependent on the crosslinking fraction (see fig. 4.4).

It is also tested whether the power law relation of eq. 4.6 applies to the in-
tramolecularly crosslinked BCB / styrene copolymers using data generated in chapter
3, where the simulations were performed for samples with 250, 500 and 750 monomers.
For each size, 11 concentrations from 5% to 100% were studied with 10 samples each,
so that in total there were 110 samples for each size. Eq. 4.6 was then used to fit the
data as presented in fig. 4.7. It can be seen that the fitting yields good agreements
except for a few data points with large chemical distance densities.

To compare the F JC results with those for the BCB/ styrene copolymer it is
noticed firstly that one FJC bead represents about 7 BCB/ styrene monomers[5). A
500-bead F JC sample in the temperature series of simulations thus approximates a.
BCB / styrene copolymer with 3500 monomers. Secondly. the crosslinking simulations

of the atomistic model were performed at T 2 900K, which was near the coil globule

61

 

  

 

 

 

 

 

 

 

 

' l I T ' l ‘ l I l
16 2 p
— Fitting
12 — 0 Simulation 2
A i' .1
O
‘61
8 r _
CK
l. _
4 _. _
0 1 l 1 l 1 l 1 l 1 l
O 2 4 6 8 10

Temperature

Figure 4.5: The radius of gyration before crosslinking found from fittings of
crosslinked FJC samples to eq. 4.6 is compared to the simulation data, showing
a good agreement.

62

 

    

 

 

 

1 l 1 r 1 i a i
-o.1—
-o.2 —
D) _
41.3 ._
-o.4 —
_0_5 1 l 1 1 1 l 1 l 1 l
o 2 4 6 8 10

Temperature

Figure 4.6: The exponent g in eq. 4.6 for the FJ C model is obtained from fittings at
all temperatures. Note that the exponent is independent of the crosslink density.

63

 

70
60

40
30

I IIIIIIIIIIII

 

20

A {,3
°$ 50
U) 40

30

IIIIIIIIIIIII

 

 

 

 

 

 

Figure 4.7: The radius of gyration as a function of chemical distance density for
intramolecularly crosslinked BCB / styrene copolymers is plotted. From the top down,
the three panels are for samples with 750, 500, 250 monomers respectively. In each
panel, the data (open symbol) and the fit to eq. 4.6 (solid line) are shown

64

transition temperature for the atomistic model in vacuum, and corresponded to a
Lennard-Jones temperature between T 2 3 and T 2 4 for the FJC model, namely its
coil globule transition regime. The exponent g for the FJC model ranges from —0.25 to
—0.35 in this regime. Data extracted from simulations of the atomistic BCB/ styrene
model lead to values of g of -—0.33, -—0.31, -0.29, for N 2 250,500, 750 monomers
respectively, which are consistent with each other and also with the results of the

FJC model.

4.5 Summary

Freely jointed chain models were adopted for a temperature series of simulations
of intramolecular crosslinking. It is showed that the morphologies of the precursor
polymers greatly affect intramolecular crosslinking. The change of the morphology
of a polymer can be induced by the ambient temperature or by the solvent quality,
providing experimental controls of intramolecular crosslinking. The key qualitative
effect is that crosslinking of polymers in the globule state yields tightly interconnected
particles, while crosslinking of polymers in the coil state leads to relatively loose
particles.

The chemical distance density was introduced as a useful measure of the internal
structure of a crosslinked polymer. It provides a better account of the contraction of
a crosslinked polymer than the density of crosslinks, particularly for a polymer with
randomly distributed crosslinkers. A systematic simulation using FJC samples shows
that there exists a. power law relation between R9 and chemical distance density.
The exponent of the power law relation is determined for a series of temperatures
by fitting, and approaches —0.4 in the high temperature limit. A power law relation
between R9 and chemical distance density is also found for the atomistic model of

BCB/ styrene copolymers. In the coil globule. transition regime of this model and of

65

the FJC model, the exponent g is found to be approximately -—0.3.

66

Chapter 5

Rigidity effect in the crosslinking

of polymers

5.1 Introduction

Polymers are made of a large number of repeating chemical units, namely monomers,
connected by covalent. bonds. The covalent. bonds have fixed bond lengths and bond
angles such that the bond stretching and bending motions in polymers are strongly
inhibited, however the bond rotations are still allowed. In a long polymer, the bond—
rotational degrees of freedom lead to plenty of floppy modes, which makes it highly
flexible.

Intramolecular crosslinking connects different monomers of a polymer together by
forming extra covalent bonds between them. Due to those extra constraints the floppy
modes of the polymer are reduced. As the crosslink density increases, it may happen
that at a certain threshold the number of floppy modes in the polymer vanishes,
resulting in a rigidity effect. Previously the same rigidity effect as a. result of the loss
of floppy modes has been studied in other fields. such as network glasses[80, 81] and

( ' (is '7'. 1) . 1", nor ' 0‘ asses a 1 lat e o e cm s I" ' -. t. '2. t. c ..
protem f)ld1ng [7r r'6] \et k C,l 1e I l f l ent Vt 1th dlffermt V ll‘Il e

67

When the concentrations of high valence elements in a network glass system increase,
more covalent bonds come into being that render it to be rigid. In the folding of
a protein, the changes of environmental conditions, such as the temperature or the
solvent, induce the formation of non-covalent bonds and consequently the reduction
of floppy modes, and result in a rigidity effect. The study of the rigidity effect
in the intramolecular crosslinking process is important to the understanding of the
rigidity effect in general. Moreover, intramolecular crosslinking has been utilized to
manufacture nanoparticles[4], where the rigidity effect has been shown to play an
important role [76]. Therefore the study of the rigidity effect in the intramolecular
crosslinking process also has practical meanings to the nanoparticle manufacturing.
The methods to intramolecularly crosslink a polymer can vary greatly, as was
demonstrated by multiple crosslinking experiments on polystyrenes in the past. In
the early 1970’s, Allen et al. [82] did an experiment to intramolecularly crosslink
chloromethylated polystyrene chains with hexamethylene diisocyanate molecules. The
crosslink units in this experiment are basically aliphatic chains made of more than
ten carbon atoms. Since their conformations can change very easily, the crosslinks
are highly floppy. In a recent experiment by Harth et al. [4], the precursor polymers
are prepared by substituting benzocyclobutene(BCB) for styrene in polystyrenes, and
the crosslinking units are eight-membered rings. each connecting two BCB monomers.
An eight-membered ring is less floppy than an aliphatic chain. But it is still able to
change among multiple conformations and is relatively floppy. Martin et al.[78] and
Antonietti et al.[79] did their crosslinking experiments on polystyrenes using Friedel-
Crafts with p—bis(chloromethyl) benzene. The crosslink units are intermediate phenyls
which connect phenyls of two styrenes together via two carbon atoms. Because the
phenyl rotations in polystyrenes are greatly hindered[83, 84], one can deduce that
the rotations of inttern'iediate phenyls are also considerably hindered. Thus its confor-

mational change is the most difficult. one among all three types of crosslinks. Those

68

experiments show that there are a wide variety of crosslinking protocols available,
with different levels of rigidities.

Another technical difference among these experiments was discussed in the liter-
ature [78], where the authors argued that there were two different types of crosslink—
ing schemes. The first one was adopted in experiments done by Allen et al. and
Harth et al., where polystyrenes were pretreated so that they contained randomly
distributed crosslinkers. Since the locations of crosslinkers on the precursor chains
spread apart, this type of crosslinking led to the formations of large rings such that
polymers contracted considerably after crosslinking. The second one was employed
in the experiments done by Martin et al. and Antonietti et al., where crosslinks were
made via intermediate chemical units. Therefore all styrenes in the precursor chains
were potential crosslinkers. This type of crosslinking favored the formations of small
rings and led to a less significant contraction. The authors referred to the crosslinking
with preset crosslinkers as “random” crosslinking, and the other as“ring—weighted”
crosslinking.

A random network has a continuous rigidity transition, while a protein with its
highly ordered structure behaves more like an organized network and consequently
exhibits a discontinuous rigidity transition[56]. It is possible that these two types
of crosslinks have different rigidity behaviors as well. The random crosslinking of a
polymer with preset crosslinkers can be compared to the folding of a protein so that
the rigidity effect may emerge in a discontinuous manner, while the ring-weighted

crosslinking leads to a. rigidity effect which may occur continuously.

5.2 The freely rotating chain model

Freely Rotating Chain (FRC) model is a simple model that incorporates both

bond stretching and bond bending constraints. Bonds in this model can rotate about

69

their connecting bonds in much the same way as in a real polymer. Therefore it is suit.-
able for the simulation study of the rigidity effect in the intramolecular crosslinkings
of polymers.

The freely rotating bead-spring chain model adopted in this simulation work has
harmonic bond stretching and bond bending potentials. It enforces the Lennard—
Jones 12-6 potential between beads that are separated by at least 3 bonds. The force

field of this FRC model is:

E = Kw _ 7.0)? + 4e [(912 — (3)6] + K609 — 490)2 (5.1)

7

Variables are in the Lennard-Jones units where both a and 6 are set to unity. The
bond stretching and bond bending potentials are significantly stronger than Lennard-
Jones potentials, with K b = 1000, To = 1.0 and K9 = 500, 60 = 1.91 = 10947".

A F RC sample in the crosslinking simulations contains 1000 beads, where a
certain fraction of beads are randomly selected to be crosslinker beads and the rest
are normal beads. A crosslinker can form exactly one crosslink with another one of its
type, and is identical to a normal bead otherwise. The initial configuration of a F RC
sample is generated by a self avoiding walk on a FCC lattice with lattice constant a =
1. After a NVT simulation of 2 million time steps (each time step consisting of 0.005
Lennard-Jones time unit) to reach equilibrium, the PRC sample undergoes another
N VT simulation to crosslink. This simulation pauses every 500 time steps to check all
pairs of unlinked crosslinkers to add new crosslinks. If one pair is within a distance of
1.30[75], a crosslink between them will be added immediately. If there are multiple
such pairs, the one with the smallest distance should be chosen. So the reaction rate
is limited to be no more than one per 500 steps. It ensures the crosslinking occurs in a
controlled manner and also prevents the polymer from overheating. The crosslinking

process has a maximum of ‘2 million steps. If less than 2 crosslinkers are left, the

 

800 |

 

 

 

 

 

\ ----- Percent = 80

600 —i - - - - Percent = 40 —
" Percent = 20

+- .II 1

.b
O
O

Unlirtlked Crosslinkers

 

 

 

 

Time(106 Steosl

Figure 5.1: The number of unlinked crosslinkers in the F RC model as a function of
the simulation time. It shows that the crosslinking reaction goes very fast in the
beginning, while it basically stops after about a quarter of a million steps.
I crosslinking process is terminated earlier to save the computing time. As an example,
fig. 5.1 shows the number of unlinked crosslinkers as a function of the simulation time
steps for three concentrations, 20%, 40% and 80%. One can see that the number of
unlinked crosslinkers drops quickly in the crosslinking process. After about a quarter
of a million steps it becomes almost constant, indicating that the crosslinking process
is basically complete. After the crosslinking process, another NVT simulation of 2
million steps is run for the crosslinked polymers to reach equilibrium again.

The simulations on the FRC model were run for the full range of the crosslinker
concentrations from 5% to 100%. For each concentration 50 samples were studied.
The temperature of the simulation was set to T = 3.5, which was right above the coil

globule transition of the FRC model. All simulations in this work were carried out

using the LAMMPS software[35].

A crosslink can be associated with various bond stretching and bond bend-
ing constraints, as shown by the experiments on the intramolecular crosslinkings
of polystyrenes. But for simplicity this simulation work only employed two types of
crosslinks, one with angle constraint and the other without. This treatment is similar
to that in the rigidity study of protein folding[56]. Protein folding is primarily driven
by effective hydrophobic bonds and hydrogen bonds, where an effective hydrophobic
bond introduces one distance constraint, while a hydrogen bond produces one dis-
tance and four angle constraints. In the crosslinking simulations in this chapter, these
two types of crosslinks were applied to the same model under the same conditions. By
comparing the polymers crosslinked in these two different ways, the rigidity threshold
as well as the rigidity effect can be reliably identified.

In theory, the rigidity threshold in the intramolecular crosslinking process can
be determined by the Maxwell constraint counting method, which was proved to be
successful by the rigidity study of protein folding[56]. It was also employed before for
the intramolecular crosslinking of F RC model with angle constraints and found that
the rigidity threshold was about 40%[76]. At this crosslinker concentration, its mean
coordination is 2 / 5 x 3 + 3 / 5 x 2 = 2.4, which. agrees well with the rigidity results of
network glasses[77] and protein folding[56]. Certainly the Maxwell constraint counting
method can be applied to the FRC model crosslinked without angle constraints as
well. In this case a crosslink brings about only one new constraint. A FRC sample
with N beads initially has N degrees of freedom. Let :1: denotes the crosslinker
concentration. The equation becomes N — (EN / 2 = 0 for the floppy modes to vanish.
It has no solution for at S 1 which implies that no rigidity effect can exists in this
case.

The crosslinking simulations of the FRC model yield nice agreement with the

theoretical analysis. Fig. 5.2 shows the linked percent of crosslinkers for these two

72

types of crosslinks. While the curve for the crosslinking without angle constraints
stays high for the full range of crosslinker concentrations, the one with angle con-
straints exhibits a large decline when the crosslinker concentration goes above 40%,
as is predicted by the Maxwell constraint counting method.

R9 is another important measure of crosslinked polymers. It is plotted for both
cases in fig. 5.3, showing that two Rg curves split at the crosslinker concentration
of 40%. The one crosslinked without angle constraints continues decreasing as the
crosslinker concentration grows, while the other one stays flat thereafter due to the
rigidity effect. Besides, R9 of the rigid one has a larger variation due to the lack of

relaxation in a rigid polymer.

5.3 The stiff chain model

The F RC model allows a bond to rotate freely with respect to one of its neigh-
boring bonds. It does not apply to stiff polymers where the bond rotations are
significantly hindered. Thus, a different model is required to study the rigidity effect
in the intramolecular crosslinking of stiff polymers.

A stiff polymer can be modeled as a Freely Jointed Chain (FJC)[5]. Normally
in the FJC model each segment contains only one bead which represents a number
of monomers, known as Kuhn length. However, in order to simulate the crosslinking,
every monomer, either a crosslinker or a normal one, has to be explicitly represented
in the model. So a stiff chain model based on the FJC model is arrived, where each
segment contains more than one bead while each head represents only one monomer.
Beads of a single segment are evenly distributed over it, and angle potentials with
equilibrium of 1800 are enforced everywhere of the chain except for the joints of
its segments. Thus all segments maintain linear configurations and can bend freely

with respect to each other. Once again two types of crosslinks were employed for the

73

 

100 I r l I T I I l f

 

 

 

 

 

Linked Percent

 

 

 

 

 

 

 

Percent of Crosslinkers

Figure 5.2: The percent of crosslinkers in F J C samples that can find their partners
to react. For the crosslinking with angle constraints, a considerable decrease can
be observed when the crosslinker concentration is above 40%, which agrees with the
estimate of the Maxwell constraint counting that 40% is the rigidity threshold for this
model. On the contrary, for the other crosslinking the linked percent changes little
when the crosslinker concentration grows, while the Maxwell counting also predicts no

rigidity effect for this model. The inset shows the difference between two crosslinking
curves.

74

 

 

 

 

 

    

 

 

 

12_ Q -
b O-QAngle q

11— — NoAngle ~
L- -

10— a
l- .4

U)
(I 9—- —
_ \ .,
\

8*- . ’...~ _
. * 0 1’

7r
y.

5" . I . 1 . l . I 1
0 20 40 60 80 100

Percent of Crosslinkers

Figure 5.3: R9 of F RC samples crosslinked with angle constraints shows an abrupt
change at the crosslinker concentration of 40%. Below this threshold, R9 decreases
greatly with the increasing of crosslinks. But above the threshold, it is almost flat.
On the contrary, R9 of F RC samples crosslinked without angle constraints lacks a.
rigidity effect, and decreases over the full range of crosslinker concentrations.

 

800

 

 

 

 

 

 

 

 

 

 

‘ '''' Peroent=80
2600b} ----Peroent=40 —
3 -‘ Peroent=20
5 ~'\. ‘
7) ‘.

8400 a
h
0
'0
3
c200 ................................................ 2“.
E
3 x ________ 1
0 —l
1 l 1 J 1 J n
0 05 1 15 2

Time(106 Steosl

Figure 5.4: The crosslinking reaction rate for a stiff chain with 1000 beads and a
segment length m = 4 indicates that the reaction saturates after a quarter of a
million steps.

76

 

 

   
    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l
10 f
t .
9_ .-.Angle —
. — NoAngle ‘
8— __
7: -Q----O--~O----O---O---3
6"; J_ 1 l I I ‘ i I
10— _
9; ln-IAngle —+
0’ _ — NoAngIe ‘
a: 8__ —
6b 1 J I I 1 I 1 l l q
11 7
L _
10__ L-AAngle ‘
9; — NoAngle :1
l. ""- -— .1
8— - ,k‘T" ‘ ”*‘“‘"”‘"-*“‘7i
l. '1
7’— 1 I 1 i I L I I l -‘
20 40 60 30 100

Percent of Crosslinkers

Figure 5.5: R9 of stiff chains with segment length m = 2, 3,4 are plotted in sepa-
rate panels from top down. By the Maxwell constraint counting method, when they
are crosslinked with angle constraints, the rigidity thresholds are 50%, 36.36% and
28.57% respectively, while crosslinking without angle constraints can not cause the
rigidity effect in those chains. It can be seen in each panel that two curves corre—
sponding to two types of crosslinks nicely overlap with each other at lower crosslinker
concentrations, and split above the rigidity thresholds, showing good agreement with
the theoretical estimate.

77

 

t

100

95

90

85
95

90

85

Linked Percent

80
95

85
80
75

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

r- ‘ 4
x.‘
t" \
‘0
f — NoAngle ‘\\ ‘
— O—QAngle ‘0‘ d
I-
; — NoAngle \lx .
_ i-IIAngle "I- -
N '~‘~
_ . I . 1 . l I “T“F'HF
" '1
L’ x i
_ — NoAngle ‘A _
\
y A-AAngle X -1
as“
C‘ (a) “~A--—.i---...——A---j‘
l I l I l I l I l
0 20 40 60 80

Percent of Crosslinkers

100

Figure 5.6: The linked percent of crosslinkers in stiff chains with segment length
m = 2, 3, 4 are plotted in separate panels from top down. In each panel, the results
for two types of crosslinks are plotted together for comparison.

78

 

 

 

 

 

 

 

 

1\j I r I F I I I l
\
0 _ A-\ /.\‘.".\ —
\
\ s
\ ‘1. .\
* \ \
\\ " .\ .‘. m = 2
‘\ \\ \\\ "l m = 3
.. —- \ — = _1
2 ‘ \\ Q\ A A m 4
\\ \ \\
A \ ‘-
0 ~ in
O\ \x \‘ \\
‘78 .4 +— \‘ \\ \\ _
\ \
x \
..c. \\ . .\
_] \ \ \
I \ \
fl- \ \\ \\
-6 — ‘\ \ \\ _.
A\ q 1‘
\
A \ ‘1
\\ \ . I
\\ ! \\\ ’1’
I
-8 '— A\ \\ \\\ '''' ‘ ’Il
\ \ or" ”
\ 1:.I—f. _“—— "f:<t--_—'
(b) ‘1.’ '
-1o 1 1

 

-_. ,,,,
a l u l 1 l 1
0 20 40 60 80 100
Percent of Crosslinkers

Figure 5.7: The difference between linked percents of two types of crosslinks. The)

exhibit considerable declines at their respective rigidity thresholds, which are 50%,
36.36% and 28.57% as found by the Maxwell constraint counting method.

79

 

comparison purpose, one with and one without angle constraints. The angle potential
of a crosslink, if it is applied, has an equilibrium of 90°. Same criteria as in previous
chapter are checked for two crosslinkers to react, namely they have to be within the
distance of 1.30. Besides, because of the chain stiffness, an additional requirement
should be enforced that only beads separated by at least two segment lengths can
form a crosslink.

In the stiff chain model, only beads at the joints of the segments contribute to the
floppy modes. Thus, a N—bead stiff chain model with a. segment length m is equivalent
to a simple FJC model with N/m beads which has 2N/m floppy modes. When the
stiff chain model is crosslinked without angle constraints, it can be derived in a way
similar to the FRC model that a rigidity effect should emerge when 2N / m — ch / 2 =
0, yielding :rc = 4/m. When the angle constraints are enabled for intramolecular
crosslinking, one has to distinguish between the beads at the joints and at the middle
of the segments, for a crosslink causes two angle constraints for the former but only
one for the latter. There are on average N / m beads at the joints of the segments
and N (m — 1) / m beads that are at the middle of the segments. Thus the rigidity

threshold equation becomes:

n_n_i.m_m-1
m 2 m. m

 

--.'rN = 0 (5.2)

which gives a rigidity threshold at 176 = 4 / (3m + 2).

The simulations were run on three stiff chain models with segment lengths m. =
2, 3, 4 respectively. All chains had 1000 beads and their concentrations of crosslinkers
varied from 5% to 100%. The crosslinking simulations were run in the same way
as for the FRC model, and all of them were run at temperature T = 3.5 (in the
Lennard-Jones units).

Because the crosslinking without the angle constraints has a rigidity threshold

80

 

 

 

 

 

 

r I l I r I l I A
lz" ----- k ” ‘l‘:><’
120 — ,A 1”
‘,A” I, ”’
1’ ,’I .’ J
.. K II ’1’
I ' I
F 1”
l
80 *— ,’ i l/ _
g I I .I’
D ’ I, /
> I I I,
> .. II I, 1’ .1
I /
q I I I
I
40 7‘ I# III ,I.’ —1
’ /
I
l, I“ I”
I’ ,’ II .‘. m = 2
,I / I-I m = 3
0 L" ‘ I, i . ‘—A m = 4 —l
‘3 ‘H-."’. g C],
’ \ / ‘ _
. 1,, 1. w .
1 I 1 I 1 I 1 I 1

 

 

0 20 4O 60 80 100
Percent of Crosslinkers

Figure 5.8: The difference between van de Waals volumes of stiff chains crosslinked
by two types of crosslinks shows a similar transition for all segment lengths.

33c = 4/m, all three models with m = 2,3,4 should show no rigidity effect when
crosslinking without angle constraints. On the other hand, the crosslinking with
angle constraints yields a rigidity threshold at .rc = 4/(3m + 2), which equals to
50%, 36.36% and 28.57% for three models respectively.

The theoretical estimate of rigidity thresholds is well supported by the simulation
results. Fig. 5.5 shows the change of R9 with the crosslinker concentration. For
each segment length, two R9 curves for two types of crosslinks overlap with each
other nicely at first, and they split when the crosslinker concentration is above the
theoretical rigidity thresholds. Moreover, the linked percent. of crosslinkers and its
difference for two types of crosslinks also exhibit transitions at the theoretical rigidity
thresholds, as shown in fig. 5.6 and fig. 5.7 respectively.

Van de Waals volume of a crosslinked polymer turns out to be another indicator

81

 

 

 

 

 

 

 

 

 

m I ' I ' I ' I '
600 — ,A ~~~~~ 15 ~12. ,,,,, 1A
I”A”
P- IA” -‘
400 — I“ A-A N = 5000 —
g // A—A N = 1000
>
> 1 x ‘
<1 4‘
I
200 — ,1’ “
I
,1 _____ .A ------ A ------ 4 ------
I- I, ’A“”‘ ...... * "A
II ’A’
II "A,
0 _ A“ ,‘u I —
at“ I
I ‘\‘L 1 I L I 1 l 1
0 20 40 60 80 100

Percent of Crosslinkers

Figure 5.9: AVwa for long chains (5000 beads) and short chains (1000 beads). Both
of them have a segment length m = 4. The difference in van de Waals volume is
approximately proportional to the size of the chain.

of the rigidity effect. In fig. 5.3, the difference between van de Waals volume of stiff
chains crosslinked in two ways are plotted, where a sudden change can be observed for
all three models with different segment lengths, at their respective theoretical rigidity
thresholds, namely 50%, 36.36% and 28.57%.

The rigidity effect was also studied for the intramolecular crosslinking of longer
chains with 5000 beads and a segment length m = 4. Compared to shorter chains, the
changes in their van de Waals volumes are more pronounced as shown in fig. 5.3. The
accurate rigidity threshold of intramolecularly crosslinked stifl chains can possibly be

determined by a. finite size scaling.

82

 

5.4 Ring-weighted crosslinking

A ring-weighted crosslinking occurs in a polymer made completely of crosslinkers.
It can be simulated in a similar procedure as described in section 5.2. However, their
termination criteria are different. A ring-weighted crosslinking simulation should
be terminated once the desired crosslinking density is reached, whereas a random
crosslinking simulation goes until no more crosslinks can be formed.

The ring-weighted crosslinking simulations were carried out on both F RC and
stiff chain models. Their crosslinker concentrations varied from 5% to 100%. 50 FRC
and 50 stiff chain samples with 1000 beads each were studied for all concentrations.
Without loss of generality, the segment length of all stiff chains was set to be 4. The
simulations were performed at T = 3.5 using the same force fields as described before.

Similar to the study of random crosslinking, two types of crosslinks were employed
for ring-weighted crosslinking on both F RC and stiff chain models, one with angle
constraints and the other without. Fig. 5.10 shows R9 for freely rotating chains
and stiff chains with these two types of crosslinks. It can be seen that basically all
By curves are smooth over the full range of crosslinker concentrations without any
abrupt changes at their rigidity thresholds. It indicates that the rigidity effect in the
ring—weighted crosslinking emerges continuously, similar to that of a random network

glass.

5.5 Conclusion

The intramolecular crosslinking of a polymer permanently changes its structure.
It reduces the floppy modes of the polymer by forming new covalent bonds. Thus a
rigidity effect may emerge in this process.

A molecular dynamics simulation study of the rigidity effect was performed using

both freely rotating chain and stiff chain models. And two types of crosslinks were

83

 

 

 

 

 

 

 

 

 

   
    

 

 

 

 

 

 

 

 

14— f
1% \ o-O FRC (Angle) “
_ \ — FRC 1
10— _
8- “+"+”4
If” ” ' d
16 1 I
_ It i
14— _
12.— 1.-.. Stiff (Angle) ;
— Stiff
10— f
8_ “nu-hum
6 l I I I i l l l l
0 20 4O 60 30 100

Expected Crosslinking Percent

Figure 5.10: R9 for freely rotating chains and stiff chains (77). = 4) crosslinked by
ring-weighed crosslinking. When the angle constraints are employed, crosslinking
still leads to a larger R9 for both models. However, in this case all R9 curves are
smooth without abrupt changes at theoretical rigidity thresholds.

84

 

employed in the crosslinking simulations, one with angle constraints while the other
without. Models and parameters in the crosslinking simulations were carefully chosen
so that the crosslinking with angle constraints leads to a rigidity effect at certain
crosslinker concentration, while the one without angle constraints yields no rigidity
effect over the full crosslinker concentration range. Then the rigidity effect can be
identified unambiguously by comparing these two cases.

The Maxwell constraint counting method can be employed to calculate the rigid-
ity thresholds. A simple FRC model has a threshold of 40% when being crosslinked
with angle constraints, while a stiff chain with Kuhn length m has rigidity thresholds
at 4/ (3m + 2) and 4 / m when being crosslinked with and without angle constraints,
respectively.

To detect the rigidity effect, R9 and the percent of linked crosslinkers were eval-
uated for all models. In models where the rigidity effect is absent, R9 exhibits a
continuous decline with increasing of crosslinks. On the contrary, R9 in the rigid
regions shows little dependence on the density of crosslinks and is noticeably larger
than in the corresponding floppy regions. N‘Ioreover, it also exhibits large variations
in the rigid regions due to lack of relaxation.

Compared to R9, the linked percent of crosslinkers of a crosslinked polymer
changes more smoothly with the crosslinker concentration, but a considerable decline
can still be observed at the rigidity thresholds. By taking the difference between two
types of crosslinks, which excludes all other factors but rigidity from the simulation
results, nice transition at theoretical rigidity thresholds can be observed.

The van de Waals volume of a crosslinked polymer is closely related to the linked
percent and appears to be a good indicator of the rigidity effect, too. Because it is
proportional to the size of the polymer, the van de Waals volume of a long chain can
be used to determine the rigidity threshold accurately.

In addition, simulations were also run for another crosslinking scheme, the ring-

85

weighted crosslinking. In this case, the radius of gyration of a crosslinked polymer did
not show any abrupt changes at the rigidity thresholds as in the random crosslinking.
However, because at higher crosslinker concentrations the ring—weighted crosslinking
makes little difference from the random crosslinking, it should also lead to a rigidity
effect as the random crosslinking does. The continuous change of R9 simply implies

that in the ring-weighted crosslinking the rigidity effect emerges continuously.

If

 

1".

Chapter 6

Polystyrene on attractive surfaces

6.1 Introduction

The behavior of a polymer on an attractive surface is dominated by the monomer-
monomer and monomer-surface interaction. In addition, if the polymer is immersed
in a solvent, interaction between the monomers and the solvent molecules also plays
a role. Basically, the interaction between monomers controls the morphology of a
polymer, so that it can be either extended or compact; while the interaction between
monomers and surfaces affects the adsorption of a polymer, so that it can be either
adsorbed or desorbed. Their combination leads to four different states: DE(Desorbed
Extended), DC(Desorbed Compact). AE(Adsorbed Extended) and AC (Adsorbed Com-
pact). Moreover, in the intermediate region complex phases arise because of the
interplay of these two types of interaction.

In the view of statistical mechanics, polymer adsorption is driven by two factors,
the polymer-surface attraction and the entropy of the polymer. It is adsorbed to the
surface when the decrease of the polymer-surface attraction energy can compensate
for its entropy loss due to adsorption, and is desorbed otherwise.

Polymer adsorption is the key to various fields, such as polymer wetting and

87

dewetting[85, 86], organic-inorganic interface adhesion[87, 88, 89] and protein ligand
binding[90, 91]. It has drawn great interest and been extensively studied in the
past few decades[92, 93]. Those researches were fruitful and greatly improved our
knowledge of polymer adsorption, but many questions still remain unanswered in
areas such as thin films, bio-systems, nano—structured interfaces, even the fundamental
adsorption mechanism [94].

In this chapter a molecular dynamics simulation study of polystyrenes on at-
tractive surfaces is presented. It is organized as follows. In section 6.2, the phase
diagram of tethered polystyrene chains on attractive surfaces is studied by molecular
dynamics simulations. In section 6.3, a simple heating simulation is performed to
gain an overview of the dynamic process of the adsorption of a polymer chain. In the

last section a summary of this chapter is given.

6.2 Phase diagram

The phase diagram of polymers on attractive surfaces is critical to the under-
standing of polymer adsorption, which has been studied by various numerical works
in the past based on lattice models [95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105].
Recently an improved simulation study of the phase diagram was carried out [3] which
employed the bond fluctuation model[106]. This research presented the phase dia-
gram of tethered polymers on substrates in terms of two field variables. They are
6,, = eb/kT and I3; = (3 /}1‘.T, where Eb is the Lennard-Jones interaction energy be-
tween the monomers and 63 between the monomers and the substrate molecules. The
phase diagram is shown in fig. 6.1.

The bond fluctuation model allows the bond length to fluctuate within a range
so that the conformations of a polymer can be sampled in more detail. It is basically

an intermediate approach between lattice models and off-lattice models. Along this

88

 

 

 

  

 

 

2-5 I—i—1 1+4 1+1
LS
2-0 ‘ 5" 5" 5"
1.5 - AC
1:? DC a /

   
 

 

/ // /
I / l
’ / ,/ ,./

4 4445‘

 

 

/

 

Figure 6.1: The phase diagram of tethered polymers on substrates. It was obtained

by a Monte Carlo simulation study using the bond fluctuation model[3].

89

 

line the next improvement in the phase diagram study should incorporate true off
lattice models. Moreover, many polymers have large side groups which greatly hinder
their motion at low temperatures, which cause them to be trapped in glassy states.
For example, polystyrene with phenyl groups attached to its backbone assumes a
glassy state under 108K. To better describe those polymers atomistic models are
often required.

Molecular dynamics simulation has been successfully applied to the study of the
dynamics of polymers on surfaces[107, 108] as well as their glassy transitions[109, 110].
A molecular dynamics simulation study is presented below. It investigates the phase
diagram of polystyrene chains on surfaces, and compares it to the results of Monte
Carlo simulation studies. In particular, by employing the molecular dynamics and
an atomistic model of polystyrene. the glassy states can be identified in the phase

diagram.

The samples in this molecular dynamics study were 125-monomer atactic polystyrene

chains. They were represented by the same atomistic model as is described in section
3.3. Initially, each polystyrene sample was tethered to a substrate by one of its ends.
A weak repulsive force was then enforced between the sample and the substrate for
a short period of time to ensure that. it. stayed above the plane 1. = 0 initially.

For simplicity, the simulations were performed in vacuum. Because vacuum is
an extremely poor solvent, high temperatures (up to 1600K) have to be used in order
to study the coil phase of polystyrenes. The temperature here should be considered
as an effective temperature which incorporates the effect of solvent as well.

In the simulation the substrates were not explicitly represented. They were

described by the Lennard-Jones 9-3 potential instead:

(99 - (3)3] (61>

Ito

C31

ULJ93 = Eeff ],

..l
I

where Eeff can be related to the regular Lennard-Jones 12—6 potential well depth 6

by:

I:

27r .
Eeff = -3—na3e (6.2)

n is the number density of the substrate molecules. The Lennard-Jones 12-6 potential
describes the interaction between monomers and substrate molecules. Based on the
equation above, a polystyrene substrate has a Eeff of 0.7Kcal/mol (Recall that the
density of bulk polystyrene is 1050 kg/mg). Eeff of other substrates can be obtained
similarly.

The temperature and surface attraction were varied in a wide range in the sim-
ulation to investigate the phase diagram. A total of 30 temperatures from 200K to
1600K, and 28 surface Lennard-Jones potentials with Eeff from 0.01 Kcal/mol to
4 Kcal/mol were used. Ten samples were studied for each pair of temperature and
surface attraction. Final results were averaged over them. The simulation of each
polystyrene sample consisted of a N VT run for 8 nanoseconds to reach equilibrium,
and a subsequent NVT run for 2 nanoseconds to take 200 snapshots to calculate its
physical properties.

The boundary between AE and DE phases is determined by both the temperature
and the surface attraction. Their effects can be combined in the field variable [33. In
fig. 6.2 the z-coordinate of the center of mass for all samples is plotted against 1'33,
where different curves correspond to different temperatures. Note that there is a
special point 63 = 0.25 where all chains assume approximately equal heights in spite
of temperature difference, qualitatively indicating the boundary between AE and
DE phases. This height is also approximately the persistence length of polystyrene
chains[111].

The extended state and compact state for polymers on attractive substrates are
basically the same as the coil state and globule state for free polymers. The transition

between them can occur either in 2D space (when the polymer is adsorbed) or 3D

91

 

 

 

 

 

 

 

Figure 6.2: The z—coordinate of the center of mass of a polymer is plotted against 1’33.
Temperature decreases from the top curve to the bottom one. It can be seen that
approximately at 135 = 0.25, the z—coordinate is the same for all temperatures

92

 

 

 

 

 

 

 

 

6° ' I ' I ‘ I ' I ' I ' I '
a g D H
50" n a B I ' i a B a
u u E - u - a i
I a D
5 fl . 3 1:1 B a 8
° 9 a B n a >l6Kcal/mol
A 40— Bag 0 g . H eff . __,
o< 8° 3 0 a u o 8m<0.6 Kcal/mol
F—’ 7 n o D 8? § 5 g a "
0:“ 01°35 so so
. B - ,
303°30?-3:Ua En ' 9
DO
.. -B
,Eéngna _
35°11 .
20°13 . - . —
1 1 I . I 1 I 1 I . I 1 I n
300 400 600 800 1000 1200 1400 160C
Temperature(K)

Figure 6.3: R9 of polystyrene chains (with 125 styrenes) on substrates. When Eeff
is greater than 1.6Kcal/mol, all chains are adsorbed and show a transition from AC
to AE; when 66 is less than 0.6Kcal/mol, they are desorbed and the transition
is from DC to DE. Two solid lines give average R9 for all desorbed cases and all
adsorbed cases respectively. In addition, below 400K the polystyrene chains are
frozen in glassy states, so that they keep their initial globule morphologies, even for
Eeff > 1.6K cal / mol, and assume smaller radii of gyration.

space (when the polymer is desorbed). In fig. 6.3, these two transition temperatures
can be roughly determined to be the inflection points of R9 curves, namely T=700K
for the transition from AC to AE and T=1100K for the transition from DC to DE.
The contour of R9 = 201:1 can be qualitatively taken as the boundary of DC
phase. Polymers in DC phase should have the smallest R9 because they assume glob-
ular morphologies. Moreover, the glassy state is marked as the region with larger
z-coordinate of the center of mass at low temperatures. Combining all those phase
boundaries, a rough phase diagram can be obtained for polystyrene chains on sub-

strates. In fig. 6.4 and 6.5, the contour plot. of the z-coordinate of the center of mass

93

 

 

16m10

14m10

Temperature
é

 

    

fi

4000

~~~

. ~

---
-
~
-

I

'0
a)

f
\
I

 

 

 

00

9
m

20 25 33 35 4D
6eff

Figure 6.4: The contour plot of the z-coordinate (in A) of the center of mass of
polystyrene chains with 125 monomers on substrates.

and R9 of polystyrenes are shown, and the phase diagram is overlaid on. it.

6.3 The dynamics of polymer adsorption

The dynamics of polymer adsorption firstly depends on the attraction energy
between the polymer and the surface. It is also affected by the ambient temperature.
An early study on the adsorption of polymer films showed that their adsorptions
exhibit an exponential kinetics at high temperatures, and a stretched exponential
kinetics at low temperatures[112]. Moreover, the lateral diffusion of a polymer on
attractive surfaces also has to do with its adsorption. On the one hand, the diffusion

of a polymer on a surface is suppressed by its adsorption to the surface. On the

94

 

 

1600.0

1400.0

1200.0

d
o
o
.0
0

Temperature
8
P
O

 

600.0

 

 

 

 

Figure 6.5: The contour plot of R9 (in 4) of polystyrene chains with 125 monomers
on substrates.

95

other hand, it is more difficult for the adsorption to reach equilibrium if its diffusion
is restricted by the surface roughness or strong attraction spots on the surface. Such
correlation between diffusion and adsorption was observed in an experiment on poly-
mer films[113]. It was also demonstrated by later molecular dynamics simulations on
polymer chains[108] and polymer films[114].

In the early years, the research of polymer adsorption focused predominantly
on thin film samples. It was since the invention of the technology of direct imag-
ing of single macromolecules, in particular Atomic Force Microscopy (AFM), that
single chain samples have been studied more thoroughly. Molecular dynamics simu-
lations of the adsorption of a single polymer have also been performed on both coarse
grained[107, 115] and atomistic models [73, 116]. The adsorption of a polymer chain
has abundant applications such as single molecule memory device or thermally ac-
tuated drug delivery. It is a stepwise process consisting of three stages[107]. In the
first stage, a polymer approaches to the vicinity of an attractive surface and deforms
into an elongated shape perpendicular to the surface. In the second one, the poly-
mer adsorbs to the surface at an exponential rate. The third one is the relaxation
stage where an adsorbed polymer spreads to the surface to reach the final equilibrium
state. The relaxation stage takes a significantly longer time to complete compared to
previous two stages. In fact, it is often questionable if the true equilibrium is reached.

A rough overview of the dynamics of single chain adsorption was obtained by
molecular dynamics simulations of polystyrene chains on attractive substrates. The
surface attraction was described by a Lennard—Jones 93 potential as before. Its
strength 66 f f varied from 0.5Kcal/mol to 3Kcal/mol. Due to the AC to DC transition,
the surface attraction can not be smaller than 0.5 Kcal/mol otherwise the polystyrene
chains can not be always adsorbed over the full temperature range of the simulations.

25 samples were studied for each Eeff' Their initial configurations were generated

in a similar way as in section 6.2, but initially they were simply placed on substrates

96

 

instead of being tethered to them. A NVT simulation of 12 nanoseconds at 300K
was subsequently performed on each sample to leave it in a compact globule form.
Then a heating process started at room temperature (T=300K) and went all the way
up to T=600K. It took 20 nanoseconds to finish, corresponding to a heating rate of
15K/ns.

In fig. 6.6, the z-coordinate of the center of mass is plotted for polystyrenes on
all substrates. Since the initial temperature is 300K, well below the glassy transition
temperature of polystyrenes, all samples were frozen in their initial states, namely
compact globules, with relatively large heights. As temperature rose, polystyrene
samples became flexible and started to collapse rapidly. Apparently, the temperature
thresholds for polystyrenes to collapse are lowered by increasing of surface attraction.
Since the glassy transition temperature is positively correlated with the temperature
threshold of collapsing, it can be deduced that the glassy transition temperature
for single polystyrene chains is lowered as well. In the end of the heating process,
the heights of polystyrene samples remain constant on almost all substrates, which
implies that they reached a stable adsorbed state. The only exception is the surface
with an attraction energy eeff=0.5Kcal/mol, where small oscillations of the height
at high temperatures were seen. It can be attributed to the transition from AC to
DC, which occurs at around 450K for this surface according to the previous phase

diagram study.

6.4 Conclusion

In this chapter, an atomistic model was adopted to study the behavior of polystyrenes
on attractive surfaces. The phase diagram was firstly obtained by varying the am-
bient temperature and the surface attraction. Owing to the adoption of molecular

dynamics and an atomistic model, a glassy region can be identified at low temper-

97

 

 

20 T ‘ I I 1 I

     
   
    

1 .OKcal/mol 0.5KcaI/mol

 

 

 

 

 

1.5Kcal/mol
A
o< ~ 0.75Kcal/mol
v
N 2.0Kcal/mol
10 +-
2.5Kcal/mol
*__ 3.0Kcal/mol
5;' l" 1. er—-_1 mm»..- ...—
l I l I l
300 400 500 600

Temperature(K)

Figure 6.6: The z-coordinate of the center of mass of polystyrene chains on substrates
during the heating process.

98

atures. At high temperatures the result appears to be the same as previous l\-"Ionte
Carlo simulation studies on lattice models or bond fluctuation models.

The heating simulations were performed on polystyrene chains to study their
adsorption processes. In the beginning of the heating, the temperature was below the
glassy transition temperature for polystyrenes. Consequently all samples were frozen
at the compact globule state. As temperature rose, they gained more mobility and
started collapsing to take on expanded and adsorbed morphologies eventually. It was
found that increasing of the surface attraction can reduce the temperature threshold
for polystyrene globules to collapse, indicating “the decrease of the glassy temperature
of single polystyrene chains. The heating simulation thus provides a good overview

of the dynamics of polystyrene chains on attractive substrates.

99

Chapter 7

Conclusion

Intramolecular crosslinking of polymers is a feasible approach to manufacture
polymeric nanoparticles that have found abundant applications in various fields. In
this thesis, the intramolecular crosslinking process and its end product, i.e., polymeric
nanoparticles, are studied using molecular dynamics simulation. Both atomistic and
coarse grained models are employed in the simulations, providing a comprehensive
understanding of the intramolecular crosslinking process.

The simulation study discovers a variety of factors that are important to the
crosslinking process and, consequently, the properties of final nanoparticles. Those
factors can thus be utilized as experimental controls of intramolecular crosslinking.
The concentration of crosslinkers in a polymer is a first example. It can be adjusted by
polymerization of different numbers of crosslinkable and non-crosslinkable monomers,
or by substitution of crosslinkers for a certain number of normal monomers. \Nith
more crosslinkers a polymer can usually form more compact nanoparticles, unless a
rigidity effect occurs that prevents the complete crosslinking of all crosslinkers.

The morphology of the precursor polymer is another factor in the intramolecu-
lar crosslinking process. In the coil phase, the crosslinking of polymers yields loosely

connected nanoparticles while in the globule phase it yields tightly connected nanopar-

100

 

ticles. The structural difference between these two cases can be described by their
chemical distance densities. For precursor chains with the same crosslinker concen-
tration, the crosslinking in the coil phase leads to small chemical distance densities,
while large chemical distance densities occur in the globule phase. A power law de-
pendence of the radius of gyration on the chemical distance density is fitted to the
simulation data and yields good fitting results.

The rigidity effect plays an important role in the intramolecular crosslinkings of
polymers. It is caused by the loss of floppy modes during the crosslinking process.
The rigidity effect is dependent on the concentrations of crosslinkers. It only occurs
when the crosslinker concentration is above certain threshold. Several other factors
also contribute to the rigidity effect, such as the chain stiffness and the crosslinking
type. A stiff polymer has less floppy modes initially, such that a rigidity effect can
emerge at lower crosslinker concentrations. In principle, the rigidity threshold 2:6 for
a stiff polymer is inversely proportional to its persistence length l, i.e., 1176 ~ 1/[.
Thus a wide range of rigidity thresholds is accessible by using precursor polymers
with different stiffness. Different types of crosslinks utilize crosslinking units with
different levels of rigidity, so that various rigidity thresholds can also be achieved by
adopting different crosslinking protocols.

F urthermore. the analogy between intramolecular crosslinking and protein fold-
ing points out another possible control of the crosslinking process that is to use
designed sequences of crosslinkers in precursor polymers to direct the manufacturing
of nanoparticles with desired shapes and sizes. For example, a recent study[76] shows
that Janus particles[117] can be formed by the intramolecular crosslinking of precursor
polymers with three different types of crosslinkers using the CLICK chemistry[118].

Future researches in the intramolecular crosslinking may include a more thorough
study from theoretical point of view on the reaction rate, the rigidity effect, the

structure size relation and so on. Intramolecular crosslinking simulations of other

101

polymers using atomistic model will also be meaningful. They can provide further
realistic crosslinking examples and possibly lead to new discoveries. Moreover, the
crosslinked polymers from the simulation can also be taken as the basis for the study
of the interaction between polymeric nanoparticles and various other materials, 6.9.,

polymers, solutions and substrates etc.

102

 

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