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Molecular Simulation Study of Homogeneous Crystal Nucleationin n-alkane Melts
by
Peng Yi
B.S., Physics, Tsinghua University (1999)M.S., Physics, Tsinghua University (2002)
Submitted to the Department of Physicsin partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHYat the
MASSACHUSETTS INSTITUTE OF TECHNOLOGYSeptember 2011
Signature of A uthor ........................................... ..........Deptfnent of Physics
August 31, 2011
C ertified by .................................................... .........Gregory C. Rutledge
Lammot du Pont Professor of Chemical EngineeringThesis Supervisor
C ertified by ....................................... ...............Mehran Kardar
Francis Friedman Prof sor of PhysicsTjesis Supervisor
A ccepted by ........................................................... ...................lAcceptd bya Rajagopal
Professor of PhysicsAssociate Department Head for Education
Molecular Simulation Study of Homogeneous Crystal Nucleation
in n-alkane Melts
by
Peng Yi
Submitted to the Department of Physics on August 31, 2011
in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
Abstract
This work used molecular dynamics (MD) and Monte Carlo (MC) method to study thehomogeneous crystal nucleation in the melts of n-alkanes, the simplest class of chain molecules.Three n-alkanes with progressive chain length were studied, n-octane (C8), n-eicosane (C20),and C150, using a united atom force field, which is able to reproduce physical quantities relatedto the solid-liquid phase transition in n-alkanes.
Using a 3D Ising model, we proved that the size of the largest nucleus in the system, nmax, is thecontrolling reaction coordinate during the nucleation process. We have made direct observationof the homogeneous crystal nucleation using MD simulation at as small as 15% under-cooling.We calculated the nucleation rate and identified the critical nucleus through a mean-first-passagetime (MFPT) analysis. At about 20% under-cooling, the critical nucleus size n* is around 100united atoms, and is slightly decreasing as the chain length increases. Abnormal temperaturedependence of n* against classical nucleation theory was found in C150 system. This behaviorcould possibly be explained by the high viscosity of the melt formed by long chain molecules.
The crystal nucleus has a cylindrical shape. We have observed the change of the structure of thecrystal nucleus as the chain length increases. For C8, the chains attach to and detach from thecrystal nucleus as a whole, and the chains end at the end surface of the cylindrical nucleus. ForC20, the partial participation of chains in the crystal nucleus became apparent, where the criticalnucleus consists of a bundle of crystal segments with the tails on the same chains extending intothe amorphous melt. For C150, chain folding was observed during the nucleation stage.
3
A cylindrical nucleus model was adopted to characterize the crystal nucleus. The nucleus freeenergy AG(n) was sampled using MC, and was used to calculate the solid-liquid interfacial freeenergies based on classical nucleation theory. The end surface free energy ae is about 4 mJ/m 2
and the side surface free energyoa is about 10 mJ/m 2 . Their values are insensitive to the chainlength.
Thesis Supervisor: Gregory C. Rutledge
Title: Lammot du Pont Professor of Chemical Engineering
Thesis Supervisor: Mehran Kardar
Title: Francis Friedman Professor of Physics
Thesis committee member: Thomas Greytak
Title: Lester Wolfe Professor of Physics, Emeritus
Thesis committee member: J. David Litster
Title: Professor of Physics
4
Acknowledgements
I would like to first thank my thesis supervisor Professor Rutledge for his guidance of my thesiswork. His enthusiasm for scientific research and teaching has greatly motivated me. Heprovided me with great freedom in research, and he always helped and encouraged me towardevery milestone. He is a good tutor and has a genuine concern for the students of their academiccareers.
I also want to thank my co-supervisor Professor Kardar, for his knowledge in physics and manyvery helpful discussions on my research. He is always ready to offer help whenever I have anyquestion.
I also thank the current and former Rutledge group members, Pedja, Junmo, Ateeque, Sezen,Fred, Ahmed, Vikram, Sanghun, Pieter, Numan and MinJae. We are good companions on ourresearch and we also formed great friendship. I want to specifically thank Matthew for editingand proofreading some parts of this thesis.
I also want to thank Professor Greytak and Professor Kleppner and their former group members,Bonna, Cort, Julia, Kendra, Lia, and Tomo. When I first came to this country, I received warmwelcome from them. They have helped me settle down and start my MIT life. I also thank myacademic advisor Professor John Joannopoulos for his encouragement.
The Church in Cambridge is like my home. The spiritual and practical care I received from themis priceless.
I want to thank my parents. Without their unconditional love and sacrifice, I could have neverbeen here today. I also want to thank my wife Esther. She is always loving and supporting me.My parents and my wife have embraced me and carried me through many difficult times.
Last but not the least; I want to acknowledge the financial support from NSF (National ScienceFoundation) through CAEFF (Center for Advanced Engineering Fibers and Films) andExxonMobil for this project.
provides a summary of this thesis and gives recommendations to the future work.
The content of this chapter will appear in a review article submitted to Annual Review of Chemical andBiomolecular Engineering2 The content of this chapter was published in Journalof Chemical Physics, 131, 134902 (2009)3 The content of this chapter was published in Journal of Chemical Physics, 135, 024903 (2011)
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Chapter 2 Fundamentals of homogeneous crystal nucleation
Homogeneous nucleation is a process that governs a broad spectrum of physical-chemical
phenomena. The theoretical formulation of nucleation process has been covered by numerous
textbooks and reviews, e.g., Zettlemoyer [26], Skripov[27], Oxtoby [28], Debenedetti [29] and
Kashchiev [30]. However, due to the difficulty in the experimental observation, the microscopic
mechanism of homogeneous nucleation, especially crystal nucleation from the liquid/melts,
remains poorly understood. Computer simulation became a useful tool in the study of
homogeneous nucleation in the past 20 years, during which new concepts and methods were
developed, often by trial and error. This chapter therefore will review the development of
simulation methods of homogeneous nucleation. To give a complete picture, a brief recap of the
basic theories and common experimental methods will also be presented.
Although the focus here is the homogeneous crystal nucleation, there are also many interesting
topics closely related, including heterogeneous crystal nucleation, crystal nucleation induced by
external forces, nucleation under confinement, cross-over of homogeneous nucleation to spinodal
decomposition, homogeneous nucleation in glasses, etc. These topics are also in the research
frontier of the science community.
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2.1 Nucleation Theories
2.1.1 Classical Nucleation Theory (CNT)
Classical nucleation theory (CNT) [31] has been widely applied to study homogeneous
nucleation. It was first developed by Gibbs [32], Volmer and Weber [33], Becker and Ddring
[34], Zeldovich [35] and others based on the condensation of a vapor to a liquid, and this
treatment can be extended to the crystal nucleation from melts and solutions. Based on CNT, a
crystal nucleus consisting of the thermodynamically most stable phase is separated from the
surrounding liquid by a sharp, infinitely thin interface. For temperatures below the melting point,
the competition between the free energy gain of the interior of the nucleus and the free energy
cost of the interface creates a free energy barrier. For a spherical nucleus of radius r, the free
energy of formation AG can be written as
AG= 4rr 2- 47rr' AG, (2.1)3
where o- is the crystal-liquid interfacial free energy per unit area and AG, is the Gibbs free energy
difference per unit volume between the liquid and crystal phases at the under-cooling
temperature. The top of the free energy barrier corresponds to the critical nucleus with radius r*,
and the critical free energy AG* is the free energy of formation for the critical nucleus. AG* and
r* are obtained by maximizing AG with respect to r in Eq.(2.1):
AG* = AG(r*) = 16;7 a 2 (2.2)3 (AG,)
16
and
r 2 (2.3)AG,
Correspondingly, the size of the critical nucleus for the spherical nucleus model is
n 32 (2.4)3 3 AG,
According to some conversions, the term "nucleus" is only used to refer to a nucleus of size
equal to or greater than n*; and "embryo" is used to refer to a nucleus of size smaller than n*.
Therefore the term "nucleation" refers to the formation of one nucleus that serves as a stable
center for further crystal growth. However, we do not make such distinction here. Any nucleus
in our conversion can have size as small as 1.
The rate of nucleation, i.e., the number of critical nuclei formed per unit time per unit volume,
can be expressed in the form of the Arrhenius equation:
I = Ie-AG*/kT, (2.5)
where Io is a kinetic prefactor, and kB is the Boltzmann constant. Io is given as
I4 ~ Nv, (2.6)
where N, is the molecule number density in the melt state, and v is the frequency of molecular
transport at the nucleus surface. Furthermore, v can be approximated using the Stokes-Einstein
relation
17
SkB (2.7)
where ao is the molecular diameter and r7 is the viscosity. [5]
For a small degree of under-cooling, AG, can be approximated by
AG, ~ pFAHf AT / T (2.8)
where AHf is the heat of fusion per molecule at the equilibrium melting temperature Tm, AT
(equal to Tm - ) is the under-cooling, and p, is the molecule number density of the crystal phase.
For deeper under-cooling, more precise approximation is needed,
AG,~ pAHfATT / T,,2 . (2.9)
Using the first order approximation, the temperature dependence of the critical nucleus size and
the critical free energy is given by
r* oc (A T)-',(2. 10)
and
AG* oc (AT)- 2 . (2.11)
When the under-cooling AT increases, AG* decreases but the viscosity r7 increases. The
combination of these two factors results in a maximum of the nucleation rate I at a temperature
Tma somewhere between the melting point Tm and the glassy transition temperature Tg. Tmax in
general depends on the material.
18
Assuming that the molecular diffusion to the surface of nucleus is also an activated process, we
can further express Eq.(2.5) into
I = Ae-E /kBT e-AG*IkBT - A-(Ed+AG*)IkBT (2.12)
where A is a temperature independent factor and Ed is the diffusion free energy barrier.
2.1.2 Density Functional Theory (DFT)
One of the main criticisms to CNT is the capillary approximation, that is, small portions of the
new phase are treated as if they represent macroscopic regions of space, and that material at the
center of the nucleus behaves like the new phase in bulk; and that the surface free energy of a
small cluster is the same as that of an infinite planar surface. These assumptions can only be
deemphasized when the nuclei are big enough, making CNT best for nucleation near
coexistence. Far away from the coexistence, especially close to spinodal decomposition, CNT
could become very inaccurate. As a matter of fact, the nucleation rate obtained in experiments
and predicted by CNT, using independent measured thermodynamics quantities such as
interfacial free energies and heat of fusion, often differs by orders of magnitude. Under such
circumstance, more sophisticated theory is needed.
Density functional theory is a quantum mechanical modeling method used in physics and
chemistry to investigate the electronic structure of many-body system. It has also been proved a
powerful approach to study nonclassical nucleation of the gas-to-liquid transition by Cahn and
Hilliard [36], Abraham [37], and Oxtoby [38]. Their approach expresses the free energy as a
19
functional of radial density profile p(r). The density varies from the center of the nucleus
outward, and the density at the center of the nucleus does not have to be the same as that of the
bulk new phase, nor does it have to behave like a planar interface.
Oxtoby [39] has shown that DFT yields more accurate results than CNT. For the gas-liquid
nucleation, DFT has clarified issues for weakly polar liquids and has given rise to explanations
of the behavior of non polar fluids. In binary condensation, Oxtoby and Kashchiev [40] have
proven the nucleation theorem, a relationship between the effect of pressure, or chemical
potential, on the free energy of the critical nucleus and its size and composition.
Harrowell and Oxtoby [41] extended DFT method to the solid-liquid transition. Since solid-
liquid transition involves not only the density change, but also an order change. The free energy
is a functional of not only the density profile, but also the Fourier components of the lattice
structure [42]. They have observed that the properties of a critical nucleus can differ
significantly from those of the stable bulk phase that eventually forms. They have applied this
theory in metal alloy crystallization. Another application is to protein crystallization from
aqueous solution in which protein concentration and crystal structure evolve together but not at
the same rate. The density functional approach applied to these situations should yield more
information on these systems and resolve many other problems inherent in the classical
approaches. However, due to the simplicity of classical nucleation theory, the discussion that
follows will be limited within the scope of classical nucleation theory.
20
2.2 Experimental methods
Homogeneous crystal nucleation occurs in the interior of an under-cooled liquid, making it
difficult for any experimental equipment to detect. Moreover, impurities induce the
crystallization and quickly drive the whole system to crystallize. The droplet technique was
proposed by Vonnegut [43] to address this problem, and was used by Turnbull with much
success [44]. The sample liquid is dispersed into a large number of tiny, normally micron in size,
droplets, exceeding the number of impurities in the liquid. A significant number of droplets are
therefore impurity-free and could be used for homogenous nucleation. The volume of each
droplet is so small that the one nucleation event automatically precludes other nucleation events
in the same droplet. Once a nucleation event happens in a droplet, since the crystal growth is
much faster than nucleation rate, this droplet almost immediately crystallizes completely. The
crystallization process can be monitored by using X-ray scattering, dilatometry, differential
scanning calorimetry (DSC), or visual method. Thus we are able to estimate the homogeneous
nucleation rate L
Crystal nucleation can still occur on the surface even the droplets are impurity free. Special
procedures thus must be taken to ensure that the nucleation happens in the bulk. One simple test
is to check whether the nucleation rate is proportional to the volume of the droplets or to the
surface.[45]
Unlike the gas-liquid transition, where the interfacial free energy is equal to the interfacial
tension and it is easy to measure. The solid-liquid interfacial free energies are very difficult to
21
measure, especially away from coexistence. The available data are rare. [46] Homogeneous
nucleation experiment allows us to measure the solid-liquid interfacial free energy. There are
several different methods, depending on the under-cooling condition, isothermal or with a finite
cooling rate.
For the isothermal nucleation experiments, combining Eq.(2.8),(2.2) and (2.5) we have
1 6,c T2 o3In I= n Io 1(2.13)
3kB p,2AH2 AT2 (1
Therefore if the nucleation rate I is measured as a function of temperature, the interfacial free
energy o can be obtained from the slope of a plot of InI against 1/AT2T and the kinetic prefactor
1o is the intercept. This method has been adopted by Turnbull and his coworkers to measure the
interfacial free energy o-for some organic and inorganic materials.
For nucleation experiments with finite cooling rate, an alternative approach is available. [47] The
temperature at which any given fraction of the droplets are solidified can be related to the
cooling rates ri, r2 by the equation
r 2AG*AT(
r2n kBTAv(2
where ATav is the average undercooling at the chosen fraction crystallized and ATd is the
difference in this temperature at the two rates.
22
2.3 Simulation studies
The development of computer technology has made numerical simulation a very useful tool to
study condensed systems, allowing scientists to work with nanoscale time and space resolution.
In practice, Ising model [48], Lennard-Jones model [49], soft-sphere model [50] and hard-sphere
model [51] are among the most studied because of their simplicity and representation for a large
group of real systems. Simulation studies in 2D systems, e.g., 2D Ising model and hard-disk
system were also available, although we will focus on 3D systems. There are two main
approaches to the nucleation problem, kinetic and thermodynamics. Molecular dynamics
method belongs to the former; Monte Carlo method the latter, often being used to sample the free
energy. Monte Carlo method was also used to "flip" the spins in an Ising model to study the
"kinetics" of a lattice system.
Ising model has been used in physics community for many years to study phase transition. The
discrete positions of spins and the simple form of interaction make computation much less costly
compared to off lattice models. In addition, analytical results are often available. Binder [52]
has carried out extensive theoretical and simulation study on nucleation using the Ising model.
Even in the past two decades, it is still a very useful model to examine the nucleation process.
Ising model is particularly suitable for Monte Carlo not only to sample the free energy landscape,
but also to study the real dynamics, e.g. the Kinetic Monte Carlo (KMC) method [53-55] was
developed to estimate the real transition time if the transition rate for all possible directions are
known and can be tabulated. Nevertheless, in the study of solid-liquid transition, realistic off-
lattice models still draw more attention.
23
2.3.1 Dynamical approach
Nucleation is intrinsically a non-equilibrium, dynamic process, and the most convincing
simulation study is molecular dynamics simulation. The first molecular dynamics study of
crystal nucleation (in a Lennard Jones system) was reported by Mandell et al. [49]. They used a
small system of only 108 particles with periodic boundary condition, which raised the question
of finite size effect to nucleation. This finite size effect was examined later by Honeycutt and
Andersen [56], and a further simulation used 15,000 and then 106 Lennard Jones particles was
reported by Swope and Andersen [57], where a Voronoi analysis was adopted to define the
crystal region and thus crystal nucleus. Compare to gas-liquid transition where only
densification is involved, the translational ordering for solid-liquid transition requires a more
sophisticated definition of crystal phase. There have been different ways in practice to define a
crystal nucleus [58-60], and the effect of different choices of definition needs to be considered in
each individual numerical study.
With the identification of crystal nuclei, Swope and Andersen [57] were able to measure the
steady state nucleus size distribution Pst(n). They fitted Pst(n) to a polynomial, and by finding
the maximum of Pst(n) they estimated the critical nucleus size n*. It is not a rigorous approach, a
more systematic procedure to identify the critical nucleus size and induction time through MD
simulation was later introduced as a mean-first-passage time (MFPT) method[61-63].
Swope and Andersen [57] claimed to have observed nucleation. However not only did they use
a doubtful ensemble, i.e., canonical ensemble that in principle prohibits a phase transition; they
also did not observe a clear induction period. The induction period is the signature of a
24
nucleation event, or a rare event in general. An under-cooled liquid is in a metastable
equilibrium state and the crystal phase is in the stable equilibrium state. These two states are
separated by a free energy barrierAG(x), where x is the reaction coordinate, and the top of this
barrier is called the transition state, corresponding to the presence of one critical nucleus. The
waiting time for a system to produce a fluctuation big enough to overcome that free energy
barrier is the induction time. The timescale accessible to computer simulations is normally
between ns to ps, far shorter than the induction time in real experiments. Furthermore, the
induction time scales inversely proportional to the system volume, making the direct observation
of nucleation in MD simulation very difficult. Accelerated molecular dynamics methods, e.g.,
metadynamics [64], were proposed to help the system find and overcome the free energy barrier.
They are summarized in ref.[64].
The induction time r measures the ability of a system to stay in metastable equilibrium state. It
depends on the observables used to determine whether the system still remains in the metastable
state. During the induction period, all, not just some, system variables should be fluctuating at a
level corresponding to the metastable equilibrium. However, lacking an induction period does
not always mean the absence of nucleation because r* is a system size dependent quantity.
When the system is big enough, the chance of finding one critical nucleus could becomes so high
that the induction time is too short to catch by an observer.
Depending on the definition, r" might or might not include a transient timer'for the system to
adopt the under-cooling before the metastable equilibrium is reached. This r'could be evaluated
and then subtracted from *, making use of the fact that nucleation is a Poisson process, so that
25
the probability distribution P( r*) should be independent of where to start timing z-*, once the
transient period r' is past. [27]
The MFPT method was proposed to determine the induction time and to extract other useful
thermodynamics information from the MD simulation results.[61-63] This method is presented
in different ways, but they are interrelated.[65] The approach by Wedekind et al.[63] makes
particularly clear the link between the classical theoretical treatment and the quantities available
by MD simulation, and was introduced here. According to this method, the mean first passage
time of a chosen reaction coordinate x, nmax in our case, takes the form
r(nm.) = 0.5r* [1+ erf(Z-(nm. - n*))], (2.15)
where r* is the average induction time, Z is the Zeldovich factor and
1 d 2 AG(nm)Zk = k B" " . (2 .1 6 )2gckBT dnnma
The critical value n*, corresponds to the transition state. Therefore MFPT method allows us to
estimate nm*., Z, and r* from MD simulations. The MFPT method was furthermore extended by
Wedekind et al. [66, 67] to reconstruct the free energy curve of the system AG(nma). Although
in ref. [66] the authors made a mistake by using nucleus size n rather than nma as the reaction
coordinate, it was then corrected later. [67]
The induction time r* is related to the nucleation rate I as
= 1 (2.17)IV
26
where V is the volume of the system. Through this relation, the nucleation rate can be calculated
when r* and V are both known. Eq.(2.17), however, is only limited to the so-called mononuclear
mechanism of nucleation[30], which is applicable to systems undergoing phase transition
through the appearance of only one critical nucleus. As the opposite, the polynuclear mechanism
is for the systems undergoing phase transition through the appearance of statistically multiple
nuclei. An unified formula for the induction time, considering both mechanisms, is (Eq.(29.12)
After correcting the contribution of the linear growth, the MFPT curve was fit to extract the
induction time r* and n' (Table 5.3). Contrary to the classical nucleation theory, the critical
nucleus size n* decreases with decreasing under-cooling. We have also measured the steady
state nucleus size distribution Pst(n). Combining Pst(n) and MFPT results, we used Eq. (2.21) to
constructed the free energy AG(n), shown in Fig.5.6.
106
8
|280K|
4
(k
-2
0 50 100 150 200 250 300
Nuclei size n
FIG 5.6 AG(n) reconstructed by using Eq.(2.21) for four different temperatures.
One possible explanation for this abnormal temperature dependence of n* is that, similar to
previous Ising model simulation (Table 2.1), the nma* estimated from MFPT method is not
consistent with the free energy sampling. Another possible explanation is by the diffusion free
energy barrier Ed introduced by Eq.(2.12). Every attachment of particle to the surface of crystal
nucleus involves a barrier crossing, as pointed out by Turnbull and Fisher[134]. Therefore, the
steady state nucleus size distribution Pst(n) must also reflect the influence of Ed. Ed might not be
important for simple molecules. However, polymer melt is highly viscous and the contribution
of the diffusion barrier is not negligible.
AG(n) reconstructed from Pst(n) includes an accumulation of Ed(1,2), Ed(2,3), ... , Ed(n-l,n),
where Ed(i,i+l) is the diffusion barrier for adding one particle to a nucleus of size i. Since the
exact form of Ed(i,i+l) is unknown, we assume that the cumulative effect of Ed's for a nucleus of
107
size n is represented by AGA(n), which has a simple form k(T)n2 /3. Therefore, we obtained the
true thermodynamics free energy curve AG 0(n) by subtracting k(T)n2 /3 from AG(n), which is
calculated from Pst(n). By tuning amplitude of k(T), the CNT picture, that is, n' and AG* both
increase as AT decreases, was recovered, as shown in Fig. 5.7. The temperature dependence of
k(T) must be related to the viscosity of the melt and the relaxation time of the chain molecules in
the system.
6
4
2
0
-20 50 100 150
Nucleus size n
FIG 5.7 The thermodynamics free energy barrier AG4(n), after subtracting the diffusion
barrier AGd(n) from AG(n), AG0(n)= AG(n)-AGd(n). The pre-factor of AGd(n) is chosen to
be k(T)=0.33, 0.27, 0.18, and 0 for 280K, 300K, 320K and 340K, respectively.
108
We have also examined the solid-liquid interface using a segment analysis. There are four types
of segments: fold (loop), tie (bridge), tail and xseg. A fold (loop) is a segment in the amorphous
region with two ends attached to the same crystal nucleus. A tie (bridge) is a segment in the
amorphous region with two ends attached to two different crystal nuclei. A tail is a segment in
the amorphous region that has only one end attached to a crystal nucleus. An xseg is a segment
in the crystal phase with two ends on the solid-liquid interface. Given any crystal nucleus, there
is a simple relation between the numbers of these four types of segments:
2 nfold+ n,ie+ naii = 2 nxseg (5.1)
Shown in Fig.5.8 is a snapshot of a simulation box at the later stage of crystallization. We
performed the segment analysis to the bigger nucleus and found that it has 60 folds, 31 ties, 30
tails and 105 xsegs. Therefore the ratio between folds and ties on the solid-liquid interface of
this particular nucleus is about 2:1.
109
FIG 5.8 A snapshot of a simulation box with one period image on each side. (Blue) the
bigger crystal nucleus, (Red) the smaller crystal nucleus, (Cyan) chain segments in the
amorphous region.
110
Chapter 6 Conclusion and recommendations for future work
Using molecular simulation method, the homogeneous crystal nucleation in the melt
for three n-alkanes, n-octane (C8), n-eicosane (C20) and C150. We have found that
united atom force field was capable of reproducing the physical quantities
crystallization, including the equilibrium melting temperature, heat of fusion and
interfacial free energies.
was studied
the realistic
related to
solid-liquid
Both the dynamics method (MFPT) and the thermodynamic method (free energy analysis) were
applied to interpret the simulation data. In order to properly apply these methods, we have
examined the classical nucleation theory using a 3D Ising model simulation. By studying the
nucleation free energy we proved that the size of the largest nucleus in the system, nmax, is the
controlling reaction coordinate in the nucleation process.
We have found that the critical nucleus has a cylindrical shape and contains around 100 united
atoms. The crystal-liquid interfacial free energies for C8 and C20 were estimated to be o ~ 4
mJ/m2 for the end surface and o - 10 mJ/m 2 for the side surface. Since the size and shape of the
critical nucleus in C150 system are similar to C8 and C20, the interfacial free energies should
also be close. Our measured o is much smaller than the estimations for bundle-like interface
(-280 mJ/m2) and chain-folding interface (-170 mJ/m2)[114]; however, considering that the111
interfacial free energy is dependent on the nucleus size, we expect oe to grow as the lateral
dimension of the crystal increases.
There were very few experimental measurements of interfacial free energy for PE. Cormia et al.
[47] studied the crystal nucleation of PE, and they found the critical nucleus of rod shape with
length of 19.2 nm and radius of 1.1 nm at 56K under-cooling. Such a big n* corresponds to very
high interfacial free energies (q, = 9.6 mJ/m2, o =168 mJ/m2). It is very hard to imagine such a
high oa for a crystal nucleus that is only 1.1 nm wide in cross section. Their oC value is close to
the values quoted by Kraack et al. [115] The oe values quoted there were not from homogeneous
nucleation experiment, but from measuring the melting point of existing crystal lamellae. Again,
the dimension matters, and the comparison of ae from different measurements require careful
examination of the details of the measurements.
We have observed chain folding during the crystal nucleation in the C150 melt. The
configuration of the folds supports the random switchboard reentry model over the adjacent
reentry model. The fact that chain folding starts at a fairly early stage suggests that chain folding
is probably more a result of the configuration of chain segments in the under-cooled melt rather
than being driven by the extra strain due to the accumulation of bundle-like free tails, the
argument that was used against the fringe micelle model, and we found both chain folds and
bundle-like tails and ties on the solid-liquid interface. It will be very interesting to see if the
configuration of one chain changes significantly before and after it enters a crystallite.
The chain length dependence of nucleation process was studied. For all three chain lengths, at
around 20% of under-cooling, AT/Tm, the critical nucleus contains around 70-160 united atoms,
112
and less than 1 nm in diameter. The small n* suggests that homogeneous nucleation is a strictly
local event. Within the range of chain length in our study, we have found that at the same degree
of under-cooling, n* slightly decreases as the chain length increase. Since the heat of fusion per
unit volume is relatively constant for different chain length, the interfacial free energies are
therefore smaller for C150 than for C20. It is particularly meaningful to compare C20 and C150.
C20 has bundle-like nucleus, i.e., chains contribute to the crystal do not fold; C150 has nucleus
with both chain fold and bundle-like free tails. It was suggested [114] that chain fold is more
energetically preferable than dangling free ends in the amorphous region, i.e., the bundle-like
structure, and our observation is consistent with this argument.
We have also studied the temperature dependence of the nucleation process. We have found that
in C8 and C20, the critical nucleus size n* and critical free energy AG* both decrease as the
under-cooling increases, as predicted by classical nucleation theory. However, in C150 melt, the
critical nucleus size n* increases as the under-cooling increases. This abnormal behavior is one
of the remaining questions from this study. It could potentially be resolved in two directions: (1)
the deviation of the dynamics approach from the thermodynamics approach; and (2) the high
viscosity of the melts for long n-alkanes, due to a diffusion free energy barrier.
Another remaining question is the trend of slight decrease of n* as the chain length increases.
We have argued above that it might be associated with the onset of chain folding. Therefore one
practical proposal is to perform a simulation on a system containing only one very long chain,
e.g., a C9000 chain. In this way the total number of united atom is the same as the C150 system
we studied, at the same time there could be no bundle-like free tails.
113
One more interesting project for the future would be to study the interface between the crystal
and amorphous regions, particularly the surface that crystal segments intersect. The constituent
of the interface, i.e., folds, ties, tails can significantly change the physical properties of the
semicrystalline product. The temperature and the concentration of chain molecules will
determine the relative probability of seeing these different types of segments. We could measure
the interfacial free energies directly [60, 135, 136] since so far there is no such measurement
available from experiments.
In our study we applied a step quench, corresponding to a delta function cooling rate. However,
the cooling rate is almost always finite in real experiment. It has been pointed out in Chapter 4
that the cooling rate can significantly change the kinetic pre-factor, I0, of the nucleation rate. It is
worth to study how the cooling rate affects the nucleation process.
Polymer crystallization is a complicated process and there are still many aspects of this field
need to be understood, as pointed out in the beginning of Chapter 2. We believe that our work
has laid a foundation toward a complete understanding of the polymer crystallization on a
microscopic level.
114
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