Molecular Dynamics Modeling of
Orientation-Induced Nucleation in Short Alkanes:
Toward Molecular Modeling of Flow-Induced
Crystallization in Polymers
by
Predrag Duranovidf
Submitted to the Department of Materials Science and Engineering
in partial fulfillment of the requirements for the degree of
CHIVES
AR_
MASSACHUSES INSTTJE
Master of Science
OF TECHNOLOGY
at the
MAY 20 2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LIBRARIES
September 2012
@ Massachusetts Institute of Technology 2012. All rights reserved.
Author ......
Department of Materials Science and Engineering
August 17, 2012
Certified by.
>///
6
CJ
I
Gregory C. Rutledge
Lammot du Pont Professor of Chemical Engineering
Thesis Supervisor
Certified by .......................................................
Samuel M. Allen
POSCO Professor of Physical Metallurgy
Thesis Supervisor
A ccepted by .............
.............
Gerbrand Ceder
R. P. Simmons Professor of Materials Science and Engineering
Chairman, Department Committee on Graduate Students
2
Molecular Dynamics Modeling of Orientation-Induced
Nucleation in Short Alkanes: Toward Molecular Modeling of
Flow-Induced Crystallization in Polymers
by
Predrag Duranovid
Submitted to the Department of Materials Science and Engineering
on August 17, 2012, in partial fulfillment of the
requirements for the degree of
Master of Science
Abstract
The enhancement of the primary flow-induced nucleation rate in short chain alkanes
(C20 and C150) has been examined for different levels of orientation by atomistic
molecular dynamics simulations. The nucleation rate has been found to change drastically by varying average molecular orientation and temperature. For example, it is
possible to accelerate nucleation kinetics by three orders of magnitude at the same
temperature, but varying the average level of orientation (P 2 (cos 0)) . The size of
the critical nucleus has been found to increase with the level of undercooling Tm - T
decrease, consistent with the classical nucleation theory. Our atomnistic molecular
dynamics simulation model is even tractable at the small levels of undercooling, thus
clearly demonstrating the effects of orientation (melt anisotropy) on nucleation kinetics when thermal nucleation is expected to be negligible. Furthermore, we calculate
the influence of melt anisotropy on the growth rate. As expected, the growth rate
is also altered by melt anisotropy. Furthermore, the growth rate maximum always
occurs at the temperature above the nucleation kinetics maximum.
Thesis Supervisor: Gregory C. Rutledge
Title: Lammot du Pont Professor of Chemical Engineering
Thesis Supervisor: Samuel M. Allen
Title: POSCO Professor of Physical Metallurgy
3
4
Contents
1
2
3
Introduction
11
1.1
B ack To Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.2
Experimental Background: Measuring the Nucleation Rate . . . . . .
12
1.3
Atomistic Modeling of Nucleation in Polymer Melts . . . . . . . . . .
13
1.4
Thesis Approach
15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Molecular Dynamics Simulation
17
2.1
System ...
2.2
M ethod and Force Field
. . . . . . . . . . . . . . . . . . . . . . . . .
17
2.3
Introducing Orientation in Molecular Dynamics Simulation . . . . . .
19
2.4
Order Param eter
19
.......
. . . . . ...
. . . . . .. . .....
. . ... . ..
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.4.1
Global Order Parameter
. . . . . . . . . . . . . . . . . . . . .
19
2.4.2
Local Order Parameter . . . . . . . . . . . . . . . . . . . . . .
19
2.4.3
Defining an Ordered Phase in Molecular Simulation . . . . . .
21
2.5
Characterizing Nucleation
. . . . . . . . . . . . . . . . . . . . . . . .
21
2.6
Characterizing Growth . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Results and Discussion
27
3.1
Cluster Search Algorithm Parametrization
3.2
Molecular Dynamics Simulation of Uniaxially Oriented Systems
. . .
28
3.3
Nucleation from the Isotropic State . . . . . . . . . . . . . . . . . . .
30
3.4
Molecular Dynamics Simulation from Anisotropic n-eicosane Melts . .
32
3.5
Temperature Dependence of the Nucleation Rate
36
5
. . . . . . . . . . . . . . .
. . . . . . . . . . .
27
3.6
4
Temperature Dependence of the Growth Rate . . . . . . . . . . . . .
Conclusions and Future Work
41
45
A Local Order Parameter Properties
47
B Supplementary Data
49
6
List of Figures
2-1
An analytical example of mean-first passage time curve (MFPT) indicating the following parameters: critical nucleus size n*, average induction time <
3-1
T*
> and growth regime Ni < N < N 2 . . . . . . . . .
Spatial distribution of the local order parameter S for an oriented and
mixed system, containing about 50 % of crystalline phase.
3-2
25
. . . . . .
28
The largest cluster in the simulation plotted for three different values
of local connectivity parameter Rc. The trajectories were shifted by
+50 along y axis for clarity purposes. . . . . . . . . . . . . . . . . . .
3-3
29
(P 2 (cos 0)) versus different strengths of biasing parameter y corrected
for temperature of equilibration for two C20 and C150. . . . . . . . .
30
3-4
Example quiescent C20 and C150 MD trajectories at 20% under-cooling 32
3-5
An example on one successful nucleation trajectory from the state
point (p = 0.12 kT , T = 295 K), which corresponds to 6% undercool-
ing. (P 2 (cos 0)) maintained a steady value from the quench to a critical
nucleation event. After the critical nucleation event approximately at
70 ns, the largest cluster begins to grow rapidly, with the corresponding
expected increase of (P2 (cos 0)) . As in the quiescent case, nucleation
appears to be a stochastic process, just like in the quiescent case at
much larger undercooling levels. This is illustrated in Figure B-7.
3-6
. .
33
Visualization of the nucleation and growth process for the state point
(p = 0.12kT ,T = 295K).
. . . . . . . . . . . . . . . . . . . . . . . .
7
34
3-7
MFPT fitting of the state point (t = 0.12 kT, T = 295 K).
tracted parameters are: (T*) = 37 ns and n* = 125.
3-8
. . . . . . . . . .
35
An example of low barrier crossing process, when the MFPT curve
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
MFPT curves for p = 0.17kT at different temperatures . . . . . . . .
38
. . . . . . . . . . . . . . . . . . . . .
39
does not plateau.
3-9
The ex-
3-10 Summary of all nucleation data
3-11 Critical nucleus size as a function of temperature and bias strength.
As the undercooling AT decreases, the critical nucleus size increases,
consistent with the classical nucleation theory. Data is plotted for all
y values except for - = 0.02 kT. . . . . . . . . . . . . . . . . . . . . .
40
3-12 MFPT for the state point (p = 0.12 kT , T = 295 K) and identification
of the growth regim e . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-13 Calculated linear growth rates in the growth regime 600 < N < 1000
42
42
3-14 Calculated linear growth rates vs. simulation temperature and biasing
factor p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
B-I Visualization of uniaxially oriented C20 molecules. Different chains are
represented by different colors . . . . . . . . . . . . . . . . . . . . . .
50
B-2 Visualization of uniaxialy oriented C150 molecules. Different chains are
represented by different colors. Some chain segments were continued
outside of the simulation box for clarity purposes. . . . . . . . . . . .
51
. . . . . . . . . . . . . . . . . .
52
B-3 Histogram of induction times in C20
B-4
Example quiescent C20 MD trajectories at 20% under-cooling
. . . .
53
B-5
Example quiescent C150 MD trajectories at 20% under-cooling . . . .
54
B-6 The summary of induction time (T), growth rate G, the size of the
critical nucleus n* and logarithm of the nucleation rate I. . . . . . . .
55
B-7 8 simulation trajectories at T=295K and biasing parameter At = 0.12kT
demonstrating stochastic nature of the nucleation process . . . . . . .
56
B-8 8 simulation trajectories at T=280K and biasing parameter y = 0.12kT
demonstrating stochastic nature of the nucleation process . . . . . . .
8
57
B-9
16 simulation trajectories at T=290K and biasing parameter P
0.12kT demonstrating stochastic nature of the nucleation process
9
-
58
10
Chapter 1
Introduction
Since the first direct experimental observations in the 1960s of flow-induced crystalline
phases occurring in polymeric materials, numerous theoretical and experimental research efforts [21, 23, 24, 29, 34, 30, 5] have been directed over the past several decades
to understand the nucleation and growth process and the resulting microstructure
that ultimately determines physical properties of the final product. It is generally
known that the kinetics of nucleation and crystal growth and the resulting morphology are extremely sensitive to experimental conditions such as temperature, pressure,
molecular weight distribution, flow type, imposed strain and strain rate. This complexity often leads to confusion and controversy among researchers. This thesis will
focus only on the early stages of flow-induced crystallization -
homogeneous primary
nucleation and immediately ensuing growth. In the following sections, the basics of
experimental and modeling challenges and ultimately understanding of the motivation that made the approach of this thesis different from all other simulation works,
will be introduced.
1.1
Back To Basics
Any kind of flow (extensional, shear, Couette, biaxial, ...) perturbs a random-coil
polymer configuration. The resulting orientation and deformation of polymer chain
molecules in systems undergoing phase change affect various aspects of thermody11
namics and kinetics of the phase transition. According to the first classical theory of
phase transformation (the famous Johnson-Mehl-Avrami-Kolmogorov equation), the
degree of transformation
X at the
instant t of a new phase is:
In (1 - X (t))
=E
(t)(1)
where
E (t) = jN
where
N
i
(s)
G (z) dz ds
(1.2)
is the rate at which the new phase nucleates. G denote linear growth rates
in various directions. So now we see that accurate modeling of phase transformation
requires the knowledge of the nucleation rate N and generalized growth rate G. In the
case of flow-induced crystallization in polymers, the inability to accurately describe
the effect of molecular anisotropy on both N and G, precludes successful application
of Equation 1.1. In the following section, we will provide a brief overview of relevant
experimental and modeling efforts targeted toward understanding of the first step in
phase transformation -
1.2
nucleation.
Experimental Background: Measuring the Nucleation Rate
The most widely accepted assumption is that imposing flow on polymers drastically
increases the nucleation rate upon cooling below the melting temperature. The speculated increase in the nucleation rate is by many orders of magnitude; however experimentally it becomes complex, and often cumbersome, to quantitatively evaluate
the influence of the flow. Flow-induced crystallization (FIC) experiments usually
involve melting the polymer samples above the melting temperature, quenching to
just below the crystallization temperature, usually determined by differential scanning calorimetry (DSC), and applying a short period of flow. These measurements
indirectly measure the increased nucleation rate through a decrease in the crystal12
lization half-time. In situ crystallization following the flow can be monitored with
common experimental techniques such as small-angle X-ray scattering (SAXS). The
authors in [161 notice drastic decreases in crystallization half-time when observing
signatures of thread-like nuclei in SAXS signals. They postulate that crystallization
rate is unaffected and that only the density of potential flow-induced nucleation sites
changes. In situ rheo-optical techniques combined with wide-angle X-ray scattering
(WAXD) reveal the emergence and structure of oriented nuclei that develop upon
strongly shearing an isothermal melt of polydisperse isotactic polypropylene
[23].
Re-
cent optical microscope observations have provided a more direct measurement of
the polymer nucleation
[6],
by counting the nuclei as they emerge from shear flow
and confirming the speculated increase of the nucleation rate. Janeschitz-Kriegl [32]
plotted the logarithm of the number density of spherulites versus temperature for
different amounts of mechanical work applied to iPP in shear flow. The concept of
flow-enhanced activation of dormant precursors is another speculation to explain the
huge effect of flow on the number density of spherulites. Peters et al.
[17]
derive
the number of point-like nuclei from the rheometry experiments by modeling the
system as a suspension. Alternatively, depolarized light scattering [26] or infrared
spectroscopy can be employed to study the induction period prior to crystallization.
1.3
Atomistic Modeling of Nucleation in Polymer
Melts
While flow-induced enhancement of nucleation and growth measurements is more than
abundant in experimental literature, very little simulation work has been done to understand the kinetics of nucleation on the atomistic level in polymer melts. Molecular
dynamics simulations are a powerful tool for this task. While they already have provided information about growth [35, 36] and crystallization behavior under uniaxial
extension [22] their utility in examining the nucleation process is just beginning to
emerge. Quite recently, Yi et al. [39, 40] dispelled a tremendous doubt that atomistic
13
simulations can capture nucleation events in quiescent polymer melts, by fully characterizing the nucleation process for short alkanes. While this work demonstrated the
ability to characterize the nucleation rate at 20% undercooling in short alkanes, it is
the first account of detecting nucleation events in complex fluids. With experimentalists' findings about tremendous nucleation rate enhancement under the influence
of flow, it appears that simulating the nucleation process should become even more
feasible.
Here is where molecular simulations reached stumbling roadblocks.
Non-
equilibrium molecular dynamics (NEMD) simulations, which simulate polymer melts
in flow fields, are currently under development
[2, 11, 18, 10]. These simulations are
usually constrained to planar elongation or shear flows and can be run indefinitely
because of the existence of appropriate boundary conditions, at extremely high shear
rates in order to stay within the limits of available computational time. Such shear
rates are beyond the limits of linear response theory and the extrapolations of the
results by many orders of magnitude is currently under debate.
Unfortunately, in
uniaxial flow, which potentially has a higher degree of orienting power, the size of the
simulation box sets the time limit to MD simulations, as the smallest dimension will
continue to diminish indefinitely. Jabbarzadeh et al. [19, 20] provide significant insight into crystallization behavior by making the comparison between the total shear
and shear rate.
While their simulations could have potentially yielded nucleation
rates as function of temperature, strain rate and total strain, it is unclear why the
authors have not identified nucleation events. In addition to NEMD simulations, a
notable novel coarse-grained kinetic Monte Carlo method has been reported
[141.
Despite scarcity in atomistic modeling, microrheological modeling of FIC is abundant. Many of these microrheological models demonstrated great success; however,
the nucleation rate is usually either assumed or parametrized from experimental data,
thus often failing to predict the behavior or properties that are not obvious from experimental data. While FIC microrheological modelling is of tremendous value, it
does not provide any new information about the nucleation process on the atomistic
level.
14
1.4
Thesis Approach
Macromolecular non-equilibrium states, commonly encountered in polymer process-
ing, can be characterized with great precision on the most fundamental molecular
level -atomic
bond level - regardless of the processing history, flow field and other
numerous experimental conditions.
Experimental techniques such as birefringence
and infrared dichroism give the aggregate averages (cos 2 0) of the polarizable or IR
active structural units. X-ray diffraction spectra can fully resolve any structural unit
in the crystal phase. Raman spectroscopy gives both (cos 2 0) and (cos 4 0) of the scattering unit. In addition to these moments, NMR can determine even higher moments,
such as (cos 6 0) and (cos 8 0) of structural units. Therefore, spectroscopic techniques
can already provide structural information regarding orientation distribution function
of local bond vectors in non-equilibrium states even without performing any molecular simulations. Colhoun et al.
[71
performed NMR studies of sheared polystyrene
to fully reconstruct the orientation distribution function of the relaxing polystyrene
sample in real time. This study further inspired Colhoun [8] to employ SGMC formalism developed by Rutledge
[33] to generate
the atomistic configurations that fully
reconstruct the moments measured by NMR experiments. Then, Bernardin et al.
[3]
generated atomistic configurations to reproduce a desired value of (P 2 (cos 0)) . This
quantity is particularly chosen because it is related to birefringence and stress-optical
coefficient, both of which can be measured. In this way, molecular simulations can
be used to predict properties, which is the ultimate goal of modelling.
Mavrantzas et al.
[27]
However,
developed a Monte Carlo scheme to model uniaxial orientation
through a tensorial orienting field acting on the end-to-end distance conformation
tensor. Instead, Bernardin
[3]
in his approach used the SGMC framework to bias the
local bonds, as well as the end-to-end vector. These two approaches are intimately
related but SGMC approach is designed to reconstruct atomistic configurations given
the information on the bond level, whereas Mavrantzas' approach relies on evoking
a rheological model (such as elastic Hookian dumbbell or FENE model) to connect
the biasing tensorial field with the flow rate. Therefore, depending on the model they
15
use, the biasing tensorial field will be different. This is not the case with the SGMC
method, which only relies on the moments of the orientation distribution function.
According to [3], the probability density of the species in the semi-grand canonical
ensemble can be transformed in terms of Legendre polynomials:
Inp oc -
U U (r) ..
P P2k (cosO)
(1.3)
k=1
For uniaxial orientation, the sum is truncated at the first non-vanishing Legendre
polynomial P 2 (cos 0) as such moment can be obtained from birefrigence measurements.
The SGMC approach has also inspired this thesis, in a way of adding an extra
biasing potential to a very well known force field, to introduce uniaxial orientation
in molecular dynamics simulation. By using the biasing potential, we can introduce
macromolecular deformation and effectively bypass any knowledge about the flow
field. This approach makes this thesis different from all other NEMD approaches to
the same problem. The ultimate goal of this thesis is to determine how molecular
deformation, as described by (P 2 (cos 0)) , influences the nucleation and growth rates
of chain molecules with varying degree of undercooling and molecular anisotropy.
16
Chapter 2
Molecular Dynamics Simulation
2.1
System
We simulated a system of 400 n-eicosane (C 20 ) chains and a system of long chain
alkane C 15 0 in the isothermal-isobaric (NPT) ensemble at one atmosphere. Periodic
boundary conditions were employed in all three directions while the simulation box
was maintained isotropic and orthogonal.
Molecular dynamics simulations in the
NPT ensemble enable realistic system contraction, which is indispensable in accommodating density changes of the melt during nucleation and crystallization processes.
Initially, we prepared an oriented system at a temperature higher then the melting
point of the material T > Tm, and then suddenly quenched the system below the
melting point T < Tm. Since nucleation is a stochastic process, we repeated multiple
simulations starting from different initial configurations.
2.2
Method and Force Field
We used molecular dynamics simulation to create non-equlibrium oriented states at
temperatures higher than the melting point of n-eicosane and longer chain alkanes.
We run MD simulations by using open source code LAMMPS (Large-Scale
Atomic/ Molecular Massively Parallel Simulator) package 1 . The equations of motion
1 http: //lammps.sandia.gov
17
were integrated with the timestep At = 5fs
Every simulation was performed in
2.
the isothermal-isobaric (NPT) ensemble. Pressure and temperature were maintained
at their set points by the Nose-Hoover thermostat and barostat, each with damping
frequencies wt = 1/(1000At) and w, = 1/(100OAt), respectively. While temperature
tracked the set temperature very well, pressure showed greater deviations from the set
point, yet it averaged precisely at the pressure set point, as expected with the systems
with large internal energy virial contributions. We used the united aton force field,
originally proposed by Paul, Yoon and Smith [31] and then modified in [35, 36, 25] in
studies of the crystal-amorphous interface and growth in n-alkanes. The functional
form and parameters of the force field are given as:
-- Kbend(/
Ebo
Eangie =Kangie(0
Etorsion =
1
-
2
[ki (1 + cos #) + k 2 (1
where Kbond = 350kcal/(mol
or = 0.401nm,
E
A),
-
1o = 1.53
lo)2
(2.1)
-
00)2
(2.2)
cos 2#) + k3 (1 + cos 3#)]
-
6
EU =4e
--
- (
-
A,
7
(2.3)
12~
-)
(2.4)
Kangie = 60 kcal/rad 2 , 60 = 1.23 rad,
= 0.469kJ/mol, k, = 6.69kJ/mol, k 2 = -3.63kJ/mol,
k3
13.56 kJ/mol.
2
Waheed et al. [35] have previously evaluated the effect of using larger integration time steps
during the MD simulation of n-alkanes with this PYS force field. They found that changing At
from 1 to 5 fs increased the bond energy, the angle-bending energy, and the relaxation time for the
chain-orientation autocorrelation function by approximately 10%. The consequence of these errors
is mitigated through the use of a thermostat, such that no detectable difference in crystallization
kinetics was observed for simulations with 1fs < At < 5 fs
18
2.3
Introducing Orientation in Molecular Dynamics
Simulation
We introduced uniaxial anisotropy in the simulation box by adding a physical biasing
potential that acts on the orientation of the local chord vector ui = ri_1 - rjsi
3
Ubias
=
2
(2.5)
pCOS2 2
where 02 is the angle between ui and biasing (flow) direction z. A single parameter
y determines the strength of the biasing field, hence driving the overall orientation
within the simulation box. The biasing forces acting on united atoms updated the
LAMMPS force calculation by the expression:
Fbias
2.4
2.4.1
-V
=
(2.6)
Ubias
Order Parameter
Global Order Parameter
In order to characterize the level of anisotropy of the entire simulation box, we monitored the global order parameter of the entire simulation box, defined as:
(P 2 (cos 0)) =
K
2
-
1
(2.7)
where O6,2 is the angle between the chord vector connecting the (j - 1)th and the
(j + 1)th bead (uj = rj+ - rj_1) and the direction of the uniaxial anisotropy, z.
Averaging is performed over all chord vectors in the simulation box.
2.4.2
Local Order Parameter
In order to identify the local crystalline nuclei and differentiate between oriented
and crystalline phases, we used the nematic order parameter. The degree of order
19
is characterized by a scalar S ranging from -1/2
to 1, where 0 indicates complete
disorder and 1 indicates a perfectly ordered uniaxial phase. In order to calculate the
local order parameter at the beads site ri, first we generate a set of local chord vectors
j = f{uj :
Iri -
r
l<
R,}, which are within a certain distance R, from the bead
site ri. The distance R, should be at least as large as the first neighbor distance
and certainly smaller than the dimensions of the simulation box. This value is set to
R, = 10
A'.
Then, we construct the local second rank ordering tensor at every bead
site i:
Si =
-UOU 3 - - 6!3)
2
2
)
(2.8)
fj
By construction, this tensor is traceless and symmetric. Since any symmetric secondorder tensor has three eigenvalues and three corresponding orthogonal eigenvectors,
a Cartesian coordinate system can be found in which it is diagonal. The eigenvector
corresponding to the numerically largest eigenvalue is a director n which points along
the average orientation, in the case of uniaxial symmetry. For the proof of the stated
theorems, as well to learn more about properties of the local order parameter tensor,
refer to Appendix A. For a uniaxial nematic phase, this tensor further reduces to:
S,3 = S
(2.9)
n(rn3 - 1 6,3)
If n is parallel to the z axis of uniaxial orientation, the three non-zero components of
S are:
2
1
S22 = -S and S, = SY = -- s
3
3
(2.10)
In the isotropic case S = 0 and in the nenatic phase 0 < S < 1. These properties
indicate that S is a good candidate for the local order parameter in our simulations
3
We determined this number by evaluating the number of united atom sites which would be
contained in a sphere of radius R,. As indicated in the following chapter, our C20 simulations
contain 8000 united atoms with the simulation box size of 61.27A in equilibrium and 59.71A in the
completely crystalline state. A sphere of a radius of 10 A would approximately encompass 12 united
atoms, which is a good number for determining the local order parameter value S. Any increase of
R, parameter value would coarsen the numerical value for the local order parameter. Conversely,
any decrease in R, would lead toward erratic changes in the local order parameter values and poor
estimates of local order
20
with uniaxial anisotropy -
it is sensitive to the direction and magnitude of the
average local orientation.
2.4.3
Defining an Ordered Phase in Molecular Simulation
Identifying an imperfectly ordered crystalline phase can be computationally challenging. First, the value of the local S order parameter must be evaluated at every site.
Then, if two beads are neighbors (within some certain preset distance Rc) and their
local directors n (which are obtained from orthogonalization of the local nematic tensor S) are nearly parallel (within some tolerance angle 0c) and their local S values
exceed some critical value Sc), then they belong to the same cluster. By sweeping
recursively through the entire simulation box outputs, a list of clusters and their
respective sizes is constructed. However, parameters S, and Rc must be sensibly determined for optimal cluster search outcomes. These parameters will be determined
in the next chapter.
2.5
Characterizing Nucleation
The crossing of a free energy barrier in a thermally activated process is typically a
rare event and thus, difficult to observe and quantify in a simulation. Nucleation is a
stochastic process: it may occur after a short or a long time, or not occur at all in the
limited time of a simulation. In addition to this, the location of the transition state
is usually not known a priori. Several methods have been developed to estimate the
critical nucleus size and average induction time from brute-force MD simulations
[11.
In this work, we followed the procedure from [38] based on a very well known concept
of first-hitting-time models, which are a subclass of probability survival models. The
first-hitting-time is defined as the time when the stochastic process first reaches a
certain threshold. In this work, first-hitting-time will be rephrased to first-passage
time. The procedure of extracting the mean-first passage time (MFPT) directly from
MD simulation trajectories is relatively straightforward. For each simulation, the size
of the largest cluster n in the system is recorded at regular intervals. After the end of
21
each simulation, the first-passage time curve is constructed by noting the time when a
cluster of size n appears for the first time. Subsequently, the mean first-passage time
for each cluster size nrma, is simply obtained by averaging first-passage time curves
obtained from multiple molecular dynamics simulations starting from different initial
conditions. The strength of the method, as fully elaborated in
the the barrier to nucleation is sufficiently high (AG
[38],
is that as long as
kT), the mean-first passage
time of the largest nucleus in the system takes the following analytical expression:
Tr(n) -
r*
+(1erf (b(n - n*)))
2
(2.11)
where n* is the critical nucleus size and T* is the average induction time. The parameter b is related to the Zeldovich factor b =
fiZ:
d 2 AG(n)
dn2
Zd=
\(2.12)
2,rkbT
MFPT curves obtained from molecular dynamics trajectories are then fitted by using
the Levenberg-Marquardt algorithm with bisquare weights to obtain the numerical
values of n* and T*. This fitting minimizes a weighted sum of residue squares, where
the weight given to each data point depends on how far the point is from the fitted
curve.
For a sufficiently high energy barrier in a thermally activated process, we
expect that the distribution of induction times follows Poisson statistics. Since we
are interested in determining the mean first-passage time, we report the standard error
of the mean, which is obtained by block averaging of the mean. This is different than
reporting standard deviation, as that becomes relevant in describing the dispersion
of individual induction times.
For example, if N simulation trajectories yield N
induction times, the standard error is computed by the following formula:
0'(T)
ZTI
()
22
(2.13)
where Q is any k-combination subset of {T1,
T 2 ,. - -, TN}.
If k = 1 or k = N, the
standard error reduces to standard deviations of individual observations
Ti.
In this
work, where N = 16, k is chosen for the block averaging of 4 independent induction
times. The average nucleation rate I is calculated as:
I =
<
T*
(2.14)
> V
where V is the volume of the simulation box. The error AI is given by the expression:
A (T*)
IT*)
2.6
I
-(T)
I
Tx
(2.15)
Characterizing Growth
Molecular dynamics trajectories that have nucleated during the simulation time can
also give valuable information regarding the growth rate G. In MD simulations, the
growth rate is the velocity v by which the crystalline cluster is expanding radially.
In the case of very short alkanes, the critical nucleus has a cylindrical shape
[40].
By
assuming that short chains are aligning in the orientation direction and simply adding
onto a nucleus, the numerical growth rate can be easily extracted from molecular dynamics trajectories. For a visual confirmation of the cylindrical supercritical nucleus
assumption, refer to Figure 3-6c in the next section, in which the supercritical nucleus
appears as a slab of alkanes, with the thickness corresponding to the maximum chain
contour length4 :
0
~ 20 10 sin 2
where lo = 1.53
A is the length between
(2.16)
two beads, and 0 is the angle between the two
adjacent bonds in the pure trans conformation of an alkane. By assuming cylindrical
growth, the number of united atoms in a cylinder of length I and radius r is simply:
N = r 2 r /p
4
(2.17)
According to Yi et al. [40], the critical nucleus contains dangling ends, thus its shape deviates
from fully extended all trans configuration
23
where p is the number density of beads (for example, in units of beads per nm 3 ) and r
is the cylinder radius. While p is a direct simulation observable (density), the radius
r is not. Since the cluster search algorithm identifies the size of the cluster, the radius
r is simply:
r
(2.18)
Taking the time derivative of Equation 2.17 gives:
dN
dN = 2rrlpG
dt
where the radial growth rate (velocity) is G = g.
(2.19)
Simple substitution yields the
expression for the growth rate:
G
2 (Np7lr)
dN
/ dt
1 2
(2.20)
where N is the largest cluster size (simulation observable) and dN/dt is the inverse
slope of the mean-first passage time, which can be calculated as a numerical derivative.
Namely, for every mean-first passage time curve, the growth part of the curve is
determined. For example, in Figure 2-1 the growth regime corresponds to a region
between N 1 and N 25 . The numerical derivative is calculated at all points Ni < N <
N 2 by linear fitting of the subregion N - AN and N + AN. The standard error of
the mean is calculated by block averaging.
5Although the plateau appears completely horizontal, it does have a finite slope. The presence of
this plateau indicates that the rate limiting step in the formation of a large cluster is the activation
time to overcome the critical size and that the time required for the subsequent growth of the cluster
is negligible compared to that. According to 138], in the case of low-barrier crossing, this plateau
interferes with the sigmoidal shape of the mean first-passage time, indicating that nucleation is not
a rate-limiting step
24
M
CU
U_
C
CU
The Largest Cluster n
N.
N2
Figure 2-1: An analytical example of mean-first passage time curve (MEPT) indicating the following parameters: critical nucleus size n* average induction time < T* >
and growth regime N, < N < N2 .
25
26
Chapter 3
Results and Discussion
3.1
Cluster Search Algorithm Parametrization
In the previous chapter, three parameters were left to be determined: Sc, the critical
value of the local order parameter S above which an united atom belongs to a crystalline phase, and Rc, which determines the minimum distance between two united
atoms belonging to the same cluster. The tolerance angle Oc was set to 100 to ensure
nearly parallel alignment of the local orientation directors n. Figure 3-1 shows the
spatial distribution of the local order parameter S calculated 'at every united atom
site in the simulation box for two melts: one consisting of a mixture of about 50%
crystalline phase, and another one in which no nucleation has occurred. As expected,
the first distribution is bimodal. The two peaks in the distribution correspond to
a disordered but oriented melt phase and ordered crystal phase. The S value that
corresponds to the minimum of the first distribution, serves as the dividing point to
distinguish united atoms in the melt phase from those in the crystal phase. Therefore, this critical local order parameter value was set to Sc = 0.7.
For the other
parameter Rc, any two crystal beads that are within a distance 1.5c- are considered
to be in the same nucleus. The numerical value of 1.5a- is slightly larger than the
first nearest-neighbor distance for united atoms belonging to different chains in the
crystal phase. Figure 3-2 demonstrates one trajectory of the largest crystal size versus
simulation time, plotted for several values of the connectivity parameter Rc. While
27
0.5
Local Orientation P2
Figure 3-1: Spatial distribution of the local order parameter S for an oriented and
mixed system, containing about 50 % of crystalline phase.
the cluster size changes, the location of the critical nucleation event does not change
significantly. Since MD trajectories were analyzed to extract induction time, the application of different numerical values of Rc would not change the numerical value for
induction time. Yet, it changes the cluster size, hence we kept the value of Rc = 1.5o
consistent throughout all MD simulations.
3.2
Molecular Dynamics Simulation of Uniaxially Ori-
ented Systems
The system was equilibrated by positioning C20 (C150) molecules randomly over a
large volume and applying compressive pressure of 1 atm, thus allowing the system
to equilibrate for 10 ns at the melt density.
28
Both C20 and C150 equilibrated at
700
600
C
R =2 ay
R =2.5 a
500N
(D)
400 -
2 300 -
200
-
0
0
I
I
I
I
ii
2
4
6
8
10
12
I
I
I
I
14
16
18
20
22
Simulation time
Figure 3-2: The largest cluster in the simulation plotted for three different values of
local connectivity parameter Rc. The trajectories were shifted by +50 along y axis
for clarity purposes.
920 kg/m
3
and 940 kg/m
3
, respectively.
Then, the biasing potential introduced in
Equation 2.5 was applied to local chord vectors ui.
The global order parameter
(P 2 (cos 6)) was monitored during simulation to determine the equilibrium value of
(P 2 (cos 0)) . C20 was equilibrated at T = 320 K , while C150 was equilibriated
at T = 500 K . Figure 3-3 shows the equilibrated (P 2 (cos 0)) for both systems as a
function of local chord vector biasing strength p. The standard error numerical values
of the equilibrated (P 2 (cos 0)) were calculated by block averaging. The (P 2 (cos 0))
values calculated by this method are in a good agreement with the results of Bernardin
et al.
[3].
During uniaxial orientation the system density increased to densities close
to 988 kg/m
3
for both chain lengths, which indicates closer packing of chain molecules
than in the isotropic melt. Consequently, the percentage of trans dihedral states also
29
0.4
0.35 -
0.3-
0.25-
S0.2-
CIA
1
0.15-
20
0.1 -
-
0.05-
0
0
0.05
0.1
0.15
0.2
0.25
p/ikT
Figure 3-3: (P 2 (cos 0)) versus different strengths of biasing parameter y corrected for
temperature of equilibration for two C20 and C150.
increased, while no crystalline clusters were observed. It is interesting to notice the
difference in (P 2 (cos 0)) at higher bias strengths. According to
[3],
these differences
can be traced to the lower fraction of chain ends in longer chain molecules. Chain
ends tend to reduce the system density while exhibiting greater orientation freedom
[4]. Figures B-1 and B-3 in Appendix B provide visualization of the system anisotropy
for the two chain lengths.
3.3
Nucleation from the Isotropic State
Before investigation of the effect of anisotropy on the nucleation rate, several quiescent
(p = 0) MD simulations were performed after the quench to estimate the nucleation
rate in C20 in the isotropic state at T=250 K, which corresponds to 20% undercooling.
30
Figure B-4 in Appendix B shows the time evolution of the largest cluster in the
simulation box after the quench for 36 trajectories. In a typical MD trajectory, the
size of the largest cluster fluctuates before a successful nucleation event occurs, a
finding consistent with the classical nucleation theory. Then, the critical nucleation
event occurs, and the largest cluster continues to grow rapidly until it consumes the
entire simulation box. Figure B-3a clearly demonstrates the stochastic nature of the
nucleation process. The average induction time for the quiescent C20 melt at 20%
undercooling is calculated by fitting the simulation MFPT plots with Equation 2.11
and extracting KT*).
(T)
= 75 ns ± 10 at T= 250K. MD runs were also performed on
entangled longer C150 chains at a similar level of undercooling. Since the melting
point of polyethylene is 420 K, the simulations were run at 340 K. Figure B-5 shows
only 16 uncorrelated runs, with much longer induction times. MFPT analysis yields
(T) = 220±20 ns, as shown in Figure 3-4. Also, there has been an attempt to simulate
quiescent melts at lower levels of undercooling, such as 260 K in C20 and 380 K in
C150, however not a single critical nucleation event was detected within 500 ns of
simulation time. At that time limit, MD trajectories were terminated.
For more
detailed analysis about quiescent nucleation in n-eicosane systems, refer to
[401.
31
200-
-
C150
C1 5
LL 150-
100-
50-
o
50
100
20
150
b
so
30 0
blusier Size
350
400
450
500
Figure 3-4: Example quiescent C20 and C150 MD trajectories at 20% under-cooling
3.4
Molecular Dynamics Simulation from Anisotropic
n-eicosane Melts
After the global order parameter
(P2 (cos 0))
has equilibrated above the melting point,
the system temperature was dropped bellow the melting point to study nucleation.
16 simulations were performed for each state point (p, T). A complete summary of
all simulations performed is given in Table B-6 in Appendix B. In a typical simulation, there is a brief period of relaxation to a new metastable equilibrium 1 , in which
the system stays until the critical nucleation event occurs. During this period, many
small clusters appear and melt, which was observed in the case of isotropic melts as
well. During this stage, the level of anisotropy, (P 2 (cos 0)) , also remains constant.
1 Orientational relaxation to a new metastable state is in the order of 100ps, while the longest
Rouse relaxation time is in the order of a nanosecond. Since nucleation can be strongly accelerated
by orientation, it may occur even before the longest Rouse relaxation time
32
(P 2 (cos 0))= 0.21
0.2
1000U
0
020
30
40
Time (ns)
50
W
10W
Figure 3-5: An example on one successful nucleation trajectory from the state point
(p = 0.12 kT , T = 295 K), which corresponds to 6% undercooling. (P2 (cos 0)) maintained a steady value from the quench to a critical nucleation event. After the critical
nucleation event approximately at 70 ns, the largest cluster begins to grow rapidly,
with the corresponding expected increase of (P2 (cos 9)) . As in the quiescent case,
nucleation appears to be a stochastic process, just like in the quiescent case at much
larger undercooling levels. This is illustrated in Figure B-7.
Then, a critical nucleus appears and initiates the growth phase, which occurs rapidly,
compared to the long induction times. This process is quantitatively demonstrated
in Figure 3-5 and visually shown in Figure 3-6.
The induction time, hence the
nucleation rate, as well as the size of the critical nucleus can be quantitatively determined by fitting MFPT curves to Equation 2.11. Figure 3-7 demonstrates a fitting
example of the state point (p = 0.12 kT, T = 295 K). However, in cases when induction time approaches a few nanoseconds, such as at the higher level of undercooling
and smaller bias, the MFPT curves take a slightly different shape, which still can be
33
(a) Visualization of the onset of the critical (b) Critical nucleus
right after 70 ns.
nucleation event at 70 ns.
becomes supercritical
(c) Supercritical nucleus continues to grow
and forms a lamella of almost completely extended all-trans C20 molecules 2 ns after the
critical event.
Figure 3-6: Visualization of the nucleation and growth process for the state point
(p = 0.12 kT, T = 295 K).
34
45
40-
--
35-
. n130 -
25 -
-- MFPT
a FIT
L2015 -
|E----
-_-
1050
0
100
200
300
'
'
400
500
600
700
Cluster Size
Figure 3-7: MFPT fitting of the state point (p = 0.12kT , T
295 K). The extracted
parameters are: (T*) = 37 ns and n* = 125.
fitted to Equation 2.11 but with some caution. Figure 3-8a shows one such example.
Wedekind et al. encountered similar problems with their U systems at high supersaturation [37]. This might be a case of low-barrier crossing problem -
the MFPT
still increases noticeably for higher cluster sizes but without reaching a clear plateau,
indicating that the growth of the cluster occurs on a similar time scale as nucleation.
For low nucleation barriers, however, the overcoming of the barrier is no longer the
rate limiting step for the formation of a supercritical cluster. This is what is observed
as well in some of our simulations at lower temperatures. For a low barrier crossing case, the critical nucleus size decreased to n* = 47, from 125 at T=295K. Even
though the MFPT does not reach a plateau, the fit is nevertheless reasonably good
around the inflection point of the MFPT curve and provides a very good estimate
of the critical size. In these cases, the inflection point is visually identified and the
fitting parameter range is tuned such that the fitted MFPT curve does plateau near
the inflection point, as it is shown in Figure 3-8a. Wedekind et al. details this ap35
proach in [37]. At the point when the barrier to nucleation diminishes to AG ~ kT,
Wedekind et al. provides a controversial argument that the phase transformation
might be dominated by spinodal decomposition. However, this discussion is beyond
the scope of this thesis, as it would require further analysis in order to identify critical
density fluctuations wavelengths. Gee et al. [12] reported polymer melt simulations
with fluctuation critical wavelengths on the order of 20 nm. This approach would
entail building simulation boxes consisting of approximately several million united
atoms, as it was the case in [12].
3.5
Temperature Dependence of the Nucleation Rate
A strong dispersion of induction times appears to apply only to low levels of undercooling, such as for the quench temperatures 290-295 K. For temperatures lower
then 290 K at the bias strength p = 0.17kT, nucleation events appear almost immediately after the quench, thus yielding almost unmeasurable changes in induction
time, which is illustrated in the Figure 3-9 as almost overlapping mean-first passage
time curves at T=270-280 K. Figure 3-10a shows the calculated nucleation rate for
four different values of bias strengths p and different quench temperatures. For example, at low biasing strength p = 0.02 kT, we could only observe nucleation events
at temperatures 250-270K. The quoted nucleation rates were calculated by running
16 independent trajectories.
For temperatures above 270K (280 and 290 K), not
a single trajectory nucleated within 500 ns of simulation time, hence the induction
time for those particular temperatures are certainly higher than 500 ns. Similarly,
for the bias strength t = 0.12 kT, no nucleation occurred within 500 ns above 295 K.
Since more interesting orientation effects on nucleation rate appear at lower levels of
undercooling for a given biasing strength p, or when induction times become longer
and disperse, it is beneficial to re-plot Figure 3-10a on a semi-log scale. Figure 3-10b
clearly demonstrates the effect of the bias strength pt on nucleation rate -
for exam-
ple, at T=280K, the nucleation rates changes approximately 3 orders of magnitude
by increasing the bias parameter y = 0.02 kT to t = 0.17 kT. Although there is
36
('
0.5-
0
0
100
50
(a) MFPT fitting of the state point
are: (r*) = 1.5 ns and n* = 47.
150
200
Cluster Size
250
350
300
400
= 0.12 kT , T = 270 K). The extracted parameters
3.5F-
3
2.5 -
2
HH-
1.5
0.5 H
0
0.5
n1.5 2
2.5
3
Induction Time (ns)
(b) Histogram of induction times demonstrates very narrow distribution of induction times/
Figure 3-8: An example of low barrier crossing process, when the MFPT curve does
not plateau.
37
45
40-
C/,
a>
E30
---
----
-F-
-
295K
-
25 -
20-7-
LJ.
15 -
-
290K
210
5
--
0
280K
100
200
300
400
500
600
Cluster Size
Figure 3-9: MFPT curves for y = 0.17 kT at different temperatures.
very little experimental investigation of FIC in flowing n-eicosane, it is worthwhile
to compare the results from Figure 3-10b with some emerging rheological model predictions. One of the earliest theoretical investigations of the mechanisms of FIC was
by McHugh [28]. McHugh makes a relatively simple argument that the difference in
nucleation rate from the extended chain configuration, relative to that of the random coil configuration, is proportional to the exponential of the free energy change
of deformation. Under this assumption, he obtains the exponential flow induced enhancement as a function of chain extension. This can be qualitatively related to our
measure of anisotropy (P 2 (cos 0)) , as chain molecules become more oriented and
extended compared to their random-coil configuration, with application of the bias
p. Graham et al.
[13]
invoke the GLaMM [15] model and theoretically derive the
expected exponential enhancement of the nucleation rate as a function of the stretch
38
x 1027
10r-
pl 1=0.02kT
12 =0.07kT
9
-
8
g4=0. 17kT
-- M-
p13 =0.12kT
E
cCZ
0
i
z
+
I,,
1
4
Nj
i~
N
-~
~.
0t
240
250
260
280
270
290
-
320
310
350
Temperature (K)
(a) Summary of nucleation rates as reported in Table B-6.
28.5 r_t =0.02kT
S2=0.O7kT
28 F
. - --
--
-
27.5
F-
$----
1 3=0.12kT
*t 4=0.17kT
.
U
27F
-1
0) 26.5 F
26k
25.5 F
25
F
'241
240
I
I
250
I
260
I
270
280
I
290
I
300
Temperature (K)
(b) Logarithm of the nucleation rate
39
Figure 3-10: Summary of all nucleation data
I
310
140 r
-
2=0.07kT
* 3=0.12kT
_Ap=0. 17kT
120 F
N
ciD
N
C/5
100k
C/)
0
U
80 F
4'
4-a
60 F
-a
4
U
20
250
4
'
255
260
265
270
275
280
285
Quench Temperature (K)
290
295
300
Figure 3-11: Critical nucleus size as a function of temperature and bias strength. As
the undercooling AT decreases, the critical nucleus size increases, consistent with the
classical nucleation theory. Data is plotted for all [y values except for y = 0.02 kT.
ratio of entanglements. Grizzuti et al. [9] recalls the theory of Lauritzen and Hoffman
and uses the Doi-Edwards model to calculate the change in free energy upon orientation, which is simply added to the barrier to quiescent nucleation. And again, they
derive the exponential enhancement of the nucleation rate as a function of shear rate,
which shows the same qualitative trend of the nucleation rates as in Figure 3-10b.
The critical nucleus size can also be extracted from the MFPT curves. Figure 3-11
attests to the applicability of the classical nucleation theory. The size of the critical
nucleus increases as AT = Tm - T decreases.
40
3.6
Temperature Dependence of the Growth Rate
For every state point (p, T) MFPT was constructed in order to identify the growth
regime. Figure 3-12 exemplifies one state point. The growth regime was identified as
the region N > 400 and the interval 600 < N < 1000 was chosen for the growth rate
calculations. Figure 3-13 demonstrates the extracted linear growth rate as a function
of the cluster size. AN is a variable parameter in this analysis, and in this particular
case was chosen as AN = 25, which means that the slope dN/dt from Equation 2.20
was calculated from 50 points at every cluster size data point. This is a variable
parameter and it is the largest source of error in growth rate calculations. If AN is
chosen too small, the calculated growth rate will become more erratic, hence introduce
larger standard error. If AN is chosen too large, it inadequately calculates the local
slope dN/dT. In case of the state points (p, T) which do now have a clear sigmoidal
shape, as discussed in the previous section, the growth regime was identified as any
region after the inflection point on the MFPT curve. The temperature dependence
of the growth rate at every state point (t, T) was calculated and summarized in
Figure 3-14.
It is important that these linear growth rates were calculated based on primary
nucleation events. Waheed et al.
[35]
calculated the same growth rate on the basis of
the secondary nucleation, however arrived at the numerical values of the same order
of magnitude as those reported in this work. It is also interesting to notice the shift
of Tmax, the temperature at which the growth rate peaks, for a given P, which also
demonstrates the utility of Ziabicki's empirical model for crystallization:
G(T) = Gmax -4 ln 2
(T -Tnix
2
+ Cf
2
(3.1)
and accounts for the amorphous orientation prior to quenching, by including Herman's
orientation factor fa = (P2 (cos 0)).
41
45
40
35
30
C,,
-
25
1-
0- 20
LL
400
1000
60
Cluster Size
Figure 3-12: MFPT for the state point (p= 0.12 kT, T
of the growth regime
295 K) and identification
1
E
-I..
0.6--C
4
0
Q
600
II
650
I
I
1
700
750
800
Cluster Size
850
900
950
1000
Figure 3-13: Calculated linear growth rates in the growth regime 600 < N < 1000
42
3_ * 1 ip=0.02kT
_
-
2 =0.O7kT
3 =0.12kT
2.5Sg
4 =0.17kT
2 -/
E//
C1 1.5
T
0.5
0,
240
250
260
280
270
290
300
310
Temperature (K)
Figure 3-14: Calculated linear growth rates vs. simulation temperature and biasing
factor p.
43
44
Chapter 4
Conclusions and Future Work
The modeling and conceptual understanding of flow-induced crystallization presents
enormous challenges across all length and time scales.
The ultimate goal of this
entire research field is rational polymer product design through development of the
quantitative integrative model to predict useful material properties such as strength,
elasticity, optical transparency and permeability. This thesis tackles the first step
of this integrative approach on the most fundamental level. Particularly, the modeling approach developed in this thesis offers a novel way of calculating the primary
nucleation rate in a uniaxially anisotropic melt of a short alkane. Non-equilibrium
molecular dynamics (NEMD) was used to study the dependence of the primary nucleation rate at various levels of undercooling and system anisotropy. The model is even
tractable at low levels of undercooling, where FIC shows the most dramatic effects.
The main achievements of this thesis can be summarized as:
o Development of a tractable atomistic simulation method which introduces uniaxial orientation and explores homogeneous primary nucleation events at any
level of undercooling Tm - T within reasonable simulation time at all levels of
uniaxial anisotropy, quantified by (P 2 (cos 0)) of the system
o Interpretation of the results within classical nucleation theory
The nucleation rate was found to depend strongly (exponentially) on the level
of orientation and temperature.
In contrast to this finding for primary nucleation
45
kinetics, the linear growth rate immediately following the primary nucleation events
was found to be relatively insensitive to both the level of the undercooling, and the
level of uniaxial anisotropy, but still measurable. Furthermore, the critical nucleus
size was found to increase with decreases in the level of undercooling, consistent with
the classical nucleation theory.
The current work, as described in this thesis, is not intended to be the final word
on this subject.
Rather, it represents a first attempt to study, by molecular sim-
ulation, the dependence of homogeneous primary nucleation rate in oriented short
alkane melts, as well as to probe the simulation time scales associated with the nucleation process, in preparation for applying the same method to longer chain molecules.
The results obtained in this thesis are promising for the investigation of long chain
molecules, in hopes of capturing more interesting polymer dynamics, such as chain
folding and entanglements, so far unobserved in this work. However, this extension
introduces significant computational challenges. For example, in order to study chain
folding in long-chain molecules at realistic melt densities, the length of the fold will
dictate the minimum dimension of the simulation box. This can easily increase the
number of particles in the system by one to two orders of magnitude, in order to
avoid finite size artifacts. Also, larger system sizes will allow studying multidimensional linear growth rate, as in more complex systems, a more challenging topological
description of the nucleation and growth process will be necessary.
In addition to a simple extension of this thesis approach, the future work, or the
next step in multi-scale modeling, must provide the connection between the biasing
field, which was introduced in MD to simulate uniaxial orientation, with currently
developing rheological models. These models tend to coarsen the atomic description
from MD simulations to a different set of quantities, such as the (RR) (the so-called
conformation tensor, where R is the end-to-end vector of the chain molecule) or the
stretch parameter A in the Rolie-Poly model, to name just a few.
46
Appendix A
Local Order Parameter Properties
1.
S is traceless.
2.
In the isotropic case, S0p = 0
3.
In a perfectly aligned nematic (all local chord vector pointing along z axis):
-1/3
S
4.
=
0
0
0
-1/3
0
0
0
2/3
(A.1)
Since S is symmetric, a Cartesian coordinate system can be found in which
the diagonal components are
2S,
-jS+B and
jS-B.
are ordered, such as |S| has the biggest absolute value.
These eigenvalues
The corresponding
eigenvector is called a director n. B is biaxiality of the chord vector distribution.
For a uniaxial nematic phase, B = 0 and S simplifies to:
Sao = S (nrno
1
)
(A.2)
Since in our MD simulations n points along z axis, the three nonzero components
of S are:
2
szz = -S and S,
3
47
=SYY =
3
S
(A.3)
The scalar quantity S is a measure of the degree of alignment of molecules.
More quantitatively, if P(O) sin OdO is the fraction of chord vectors which point
between 0 and 0 + dO, then:
S
=
f
-
3 sin2)
P(O)dO
(A.4)
In the isotropic case S = 0 and in the nematic phase 0 < S < 1. In general,
-j
< S < 1, where negative values of S correspond to biaxial orientation
perpendicular to the director (also called perfect oblate phase).
48
Appendix B
Supplementary Data
49
(a) (P2 (cos 0))
(b) (P 2 (cos 0))
=0.00
I
(c) (P 2 (cos 6))
=0.10
17
:0.20
(d) (P 2 (cos 6))
0.25
Figure B-1: Visualization of uniaxially oriented C20 molecules. Different chains are
represented by different colors
50
7
(b) (P 2 (cos 0)) =0.20
(a) (P 2 (cos 0)) =0.00
(c) (P 2 (cos 0)) =0.35
Figure B-2: Visualization of uniaxialy oriented C150 molecules. Different chains are
represented by different colors. Some chain segments were continued outside of the
simulation box for clarity purposes.
51
5-
a
5D
100
150
200
250E
3NS
350
400
450
500
554
(b) Histogram of induction times in C150
(a) Histogram of induction times in C20
Figure B-3: Histogram of induction times in C20
52
1000
.1000
1400
1200
8 00
.4000
800
3000
1000
_
6 00
800
600
4 00
600
400
2 00
400
200
2000
Ca
.-
0
0
50
100
time (ns)
150
5000
1000
200
0
50
0
100
3500
3500
3000
3000
2500
2500
3000
2000
2000
2000
1500
1500
1000
1000
500
500
4000
50
0
0
100
0
20
40
60
20
40
E0
20
40
60
50
100
150
600
500
400
300
1000
0
50
0
100
10
0
(
3000
3500
2500
3000
20
40
0
0
60
200
100
10
20
I
)
0
0
500
2500
400
2000
2000
300
1500
1500
1000
200
1000
1000
500
500
100
500
2000
2500
20000
1500
00
50
100
000L.!0
1000
3000
800
2500
50
100
0
0
20
40
60
4000
0
5000
4000
3000
2000
600
3000
400
1500
1
1000
200
500
0
0''
10
20
30
0
0
2000
2000
1000
1000
10
20
30
0
0
10
20
30
0
0
Figure B-4: Example quiescent C20 MD trajectories at 20% under-cooling
53
4000
3500
4000
3500
3000
3500
2500
3000
3000
2500
5000
4000
2500
2000
-2000
3000
2000
0150020
1500
1500
2000
1000
1000
1000
1000
500
500
500
0
0
100
200
300
400
0
0
100
200
300
400
0
0
100
200
0
300
0
400
time (ns)
3000
3500
2500
2500
3000
2000
2000
2500
1500
200
300
400
2000
2000
1500
2000
1500
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1500
500
500
0
0
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0
0
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0
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3500
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0
0
500
0
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3500
0
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400
3500
10001000
0
0
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0
100
200
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0
100
200
300
0
0
5000
3000
5000
5000
4000
2500
4000
4000
3000
3000
3000
2000
2000
2000
1000
1000
100
200
300
400
100
200
300
400
2000
1000
1000
0
500
0
200
400
600
0
0
100
200
300
0
0
100
200
300
400
0
0
Figure B-5: Example quiescent C150 MD trajectories at 20% under-cooling
54
Bias p
(P 2 (cos 0))
T [K]
p = 0.02 kT
0.05 ± 0.05
250
260
250
0.1 i 0.05
260
270
280
250
270
y = 0.12 kT
0.15 ± 0.05
280
290
295
300
260
270
y=0.15 kT
0.2 ± 0.05
280
290
p
-
0.07 kT
300
(T)
[ns]
8.0 t 0.1
10.0 i 0.1
4.5 + 0.1
3.2 + 0.5
5.0 i 1.0
10.0 i 1.0
1.5 0.5
1.5 0.5
1.8 ± 0.5
6 ± 0.3
37 1.0
200+
1.0 ± 0.1
1.0 ±0.2
2.0 ± 0.5
2.5 ± 0.5
6
±1
(G) [nm/ns]
0.45 t 0.1
0.45 ± 0.1
0.5 ± 0.1
0.8 ± 0.1
0.7 ± 0.1
0.6 i 0.22
1.1 i 0.1
1.3 0.1
1.43 ± 0.2
1.20 t 0.3
0.75 0.3
NA
1.4 ± 0.1
1.7 ± 0.1
2.3 ± 0.2
2.7 ± 0.2
1.8
± 0.25
n*
33
64
40
22
30
80
32
47
55
80
125
NA
32
56
75
85
log (I) [1/(cm3 s)]
26.88
26.76
27.0
27.3646
26.76
25.36
27.48
27.48
27.41
27.2
26.88
25
28
28
27.6
27.0
120
25.36
Figure B-6: The summary of induction time (T), growth rate G, the size of the critical
nucleus n* and logarithm of the nucleation rate I.
55
(D
C,,
6000
2500
6000
5000
2000
5000-
4000,
3000[
4000
4000
1500
3000
CO,
3000
1000
200C
2nnn
2000II.
-J
1000
500
--
1000
0
0
20
40
0
1000
0
100
200
0
0
50
C
0
20
40
time (ns)
4000.
3000
4000
4000.
3000
3000 [
2500
3000
(I.
2000
(4
I'
2000
1500
-j
F
500
0
5
2000
1000
1000
0
0
1000
100
0
2000
1 0 1~
0
0
~
20
40
0
20
40
0
20
2
40
Figure B-7: 8 simulation trajectories at T=295K and biasing parameter p- = 0.12kT demonstrating stochastic nature of the
nucleation process
5000
5000
5000
4000
4000
4000
1400
-
V5
2 3000
c20
Ca
00
-
3000
3000
2000
2000
800
600
/
-
1000
0
0
1000
2
0
4
0
400
1000
-
2
I
0
6
4
1200100
1000
0
4
2
4
time (ns)
6
00
0
5000
6000
6000
6000
4000
5000
5000
5000
4000
4000
4000
3000
3000
3000
2000
2000
2000
1000
1000
1000
3000
*
2
4
6
2
4
6
2
4
2000
1000
0
2
4
6
3500
0
0
2
4
5000
3000
4000
1500
2000
4
6
0
0
5000
3000
C
3000
2
4000
2500
2000
0
0
6
2000
1000
0
0
1000
1000
500
2
0
4
2
4
0
6
2
4
6
5000
3500
3000
4000
30003
2500
2500
3000
3000
2000
2000
2000
1500
5000
4000
j
2000
1500
1000
0
0
2
1000
1000
iiv1
1000
500
4
0
0
0
6
2
4
6
500
0
2
4
0
Figure B-8: 8 simulation trajectories at T=280K and biasing parameter y
demonstrating stochastic nature of the nucleation process
57
=
0.12kT
20 00 -
1400
2000 r
---
1000
|
1200
15 00
800
1500
1000
:3
600
800
10 00
1000
600
JIk
Al)
.1-
400
5c
J54
00
400
500
200
200
0
0
5
10
0L
0
0
2
4
6
0
0
5
0
10
0
time (ns)
1200,
2000
2500
20001
1000
1500
2000
1500
800
1500
1000
1000
600
1000
-,
400
500
500
500
200
-
0
1
2
0
4
2500
2000
1
5
10
0
15
0
10
5
0
0
3000
3000
3000
2500
2500
2500
2000
2000
2000
1500
1500
1500
1000
1000
1000
500
500
500
5
10
15
5
5
10
15
2
4
6
1500
1000
500
0
10
0
0
5
5
10
0
15
35 00
5000
4000
30 00
4000
3500
0
5
10
15
20
0
0
2000
3000
1500
20 00
3000
2500
2000
1000
15 00
2000
1500
10 00
1000
1000
25 00
5 00
0
0
500
500
5
10
0
0
5
10
0
0
5
10
15
0
0
Figure B-9: 16 simulation trajectories at T=290K and biasing parameter P = 0.12kT
demonstrating stochastic nature of the nucleation process
58
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