Annealing of single lamella nanoparticles of polyethylene

Abstract

We study the change of the size and structure of freely suspended single
lamella nanoparticles of polyethylene during thermal annealing in aqueous
solutions. Using small-angle x-ray scattering and cryogenic transmission
electron microscopy, it is shown that a doubling of the crystalline lamella
sandwiched between two amorphous polymer layers is obtained by annealing the
nanoparticles at 125 C. This thickening of the crystalline lamella can be
understood in terms of an unlooping of polymer chains within a single
nanoparticle. In addition a variation of the annealing temperature from 90 C to
115 C demonstrates that the inverse of the crystalline lamellar thickness
increases linearly with the annealing temperatures leading to a
recrystallization line in a Gibbs-Thomson graph. Since the nanoparticles
consist of about only eight polymer chains, they can be considered as a ideal
candidates for the experimental realization of equilibrium polymer crystals.

Full text

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arXiv:1107.0799v1 [cond-mat.soft] 5 Jul 2011
Annealing of single lamella nanoparticles of polyethylene
Christophe N. Rochette1, Sabine Rosenfeldt1, Katja Henzler2,
Frank Polzer2, Matthias Ballauff2,∗, Qiong Tong4, Stefan Mecking4,
Markus Drechsler5, Theyencheri Narayanan6Ludger Harnau3,∗
1Physikalische Chemie I, University of Bayreuth, 95440 Bayreuth, Germany
2Soft Matter and Functional Materials,
Helmholtz-Zentrum Berlin, 14109 Berlin, Germany
3Max-Planck-Institut f¨ ur Intelligente Systeme,
Heisenbergstr. 3, 70569 Stuttgart, Germany,
and Institut f¨ ur Theoretische und Angewandte Physik,
Universit¨ at Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
4Department of Chemistry, University of Konstanz,
Universit¨ atsstrasse 10, 78457 Konstanz, Germany
5Makromolekulare Chemie II, University of Bayreuth, 95440 Bayreuth, Germany
6ESRF, B.P. 220, 38043 Grenoble Cedex, France
(Dated: July 6, 2011)
Abstract
We study the change of the size and structure of freely suspended single lamella nanoparticles of
polyethylene during thermal annealing in aqueous solutions. Using small-angle x-ray scattering and
cryogenic transmission electron microscopy, it is shown that a doubling of the crystalline lamella
sandwiched between two amorphous polymer layers is obtained by annealing the nanoparticles at
125◦C. This thickening of the crystalline lamella can be understood in terms of an unlooping of
polymer chains within a single nanoparticle. In addition a variation of the annealing temperature
from 90◦C to 115◦C demonstrates that the inverse of the crystalline lamellar thickness increases
linearly with the annealing temperatures leading to a recrystallization line in a Gibbs-Thomson
graph. Since the nanoparticles consist of about only eight polymer chains, they can be considered
as a ideal candidates for the experimental realization of equilibrium polymer crystals.
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I. INTRODUCTION
The crystallization of high molecular weight polymer chains such as polyethylene (PE)
differs qualitatively from the crystallization of simple fluids in a number of important aspects
such as entanglements. At a microscopic scale, entanglements arise from the fact that linear
polymer chains are one-dimensionally connected objects which cannot cross each other. The
resulting topological interaction strongly affects the crystallization process since it imposes
constraints on the motion of the polymer chains. A perfect parallel alignment of all polymer
chains in a crystalline state cannot be obtained starting from a melt because it would take
too long time to disentangle them. By now it is well-established that the crystallization
process leads to a two-phase structure consisting of platelike crystallites which are separated
by amorphous regions. Within a crystallite, the polymer chains are aligned parallel to the
main axis of the platelet, while the remaining entanglements are located in the amorphous
regions. A given polymer chain may fold back into the same crystallite after a transit into
the adjacent amorphous region because the chain length is considerably larger than the
height of the platelike cystallites.
However, it is still under debate whether polymer crystallization is kinetically or thermo-
dynamically controlled (see, e.g., refs1–10and further references therein). The fact that the
melting temperature and the crystallization temperature are different points to a decisive
role of kinetics in polymer crystallization. However, computer simulations and theory4,5,7,11
have demonstrated the existence of equilibrium polymer crystals in the case that only a few
polymer chains form a crystal. Polymer nanocrystals with small amorphous regions are ideal
candidates for reaching thermodynamic equilibrium due the high mobility of the polymer
repeat units.
Up to now, the overwhelming majority of studies have been done using bulk samples
of PE. If the molecular weight is high enough, entanglements will play an important role
for crystallization in these bulk samples. Working with single crystals of the respective
polymer has been a way around this problem and single crystals of PE have been studied
since many years.12Thus, crystals of PE with a thickness of about 10 - 15 nm and lateral
dimensions of the order of 100 nm to a few µm can be generated by crystallization from a
highly dilute solution.13–15These crystals can be used to study the lamellar thickening of the
semi-crystalline PE upon annealing. The morphological changes involved in these processes
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can be easily studied using atomic force microscopy.15,16Thus, Tian and Loos observed that
simultaneous thickening of single crystals of PE was accompanied by the formation of cavities
within the crystal.16It is evident that research on single crystals allows us to monitor subtle
morphological changes that take place upon annealing. However, in most investigations the
PE crystals are lying on a solid substrate which may exert a decisive influence on the shape
transformation (see the discussion of this point in ref15).
Recently, the progress of catalytic polymerization has led to the generation of single
crystals of PE with dimensions in the nanometric range.The catalytic polymerization
technique yields stable aqueous suspensions of well-defined PE nanoparticles that can be
studied by a wide variety of techniques in-situ. Thus, small platelike crystallites of PE
have been prepared in this way.17,18Using a combination of cryogenic transmission electron
microscopy (cryo-TEM) and small-angle x-ray scattering (SAXS), it has been demonstrated
that these freely suspended PE platelets consist of a single crystalline lamella with a thickness
of 6.3 nm. The overall thickness of 9 nm pointed to a small amorphous layer that could
also be inferred precisely from the analysis of the SAXS-data. The facets of these crystals
were clearly visible in micrographs taken by cryogenic transmission electron microscopy
(cryo-TEM). Hence, the new method to create thin nanometric PE platelets provides the
opportunity for new experiments on polymer crystallization using single PE crystals.18
Here we present the first study of the thickening of single lamellar PE nanoplatelets by
thermal annealing. Earlier studies of this process involved either PE crystals supported
by a solid substrate or bulk PE (see, e.g., refs16,19–26and references given therein). In
particular, Tong et al.27presented the first systematic study of the lamellar thickening of
PE nanocrystals on a solid surface by AFM. They showed that thermal annealing leads
to considerable thickening of the crystals. Since the PE nanocrystals are freely suspended
in an inert medium, any influence from solid substrates can be ruled out. Moreover, the
small size allows us to monitor the overall shape and internal structure by a combination of
cryo-TEM and SAXS. The present data can thus be compared to results obtained on solid
substrates16,27and on bulk samples.
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II. EXPERIMENTAL DETAILS AND ANALYSIS OF SAXS-DATA
A. Experimental
The systems have been prepared by catalytic polymerization in aqueous solution as dis-
cussed in refs28–30. Our samples contain 1.6 wt % PE of molecular weight 3.5 × 105g/mol
and the surfactant sodium dodecyl sulfate (SDS) to stabilize the PE particles against co-
agulation. The weight fractions of SDS are 0.87 wt % and 0.36 wt % for the two samples
S87 and S36, respectively. For sample S87 the amount of SDS used during the synthesis of
the PE particles has been increased as compared to our previous study17in order to ensure
colloidal stability even at rather high temperatures during annealing processes. Surface ten-
sion measurements have indicated that virtually all surfactant molecules are adsorbed onto
the strongly hydrophobic PE particles.
Specimens for cryo-TEM have been prepared by vitrification of a thin liquid film of a PE
dispersion supported by a copper grid in liquid ethane at its freezing point. Examinations
were carried out at temperatures around −183◦C. Moreover, no staining agent has been used
to enhance the contrast between the PE particles and the surrounding medium. All images
have been recorded digitally by a bottom-mounted CCD camera system and processed with
a digital imaging processing system.17
The SAXS measurements have been performed using either a custom-built Kratky com-
pact camera or synchrotron radiation using the beamline ID02 at the ESRF. The scattering
intensities of both the empty capillary and the solvent have been subtracted from the scat-
tering intensities presented in the next section. As in our previous study a contrast variation
between the PE particles and the solvent has been used by adding various amounts of glucose
to the solutions. In this way the individual contributions of the amorphous and crystalline
regions of the PE particles to the scattering intensity become available. This allows for
consistency checks of the theoretical modelling.
B. Scattering intensity
SAXS determines the scattering intensity I(q,ρ) as a function of the magnitude of the
scattering vector q and the PE particle number density ρ. For a system consisting of monodis-
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perse PE particles the scattering intensity can be written as
I(q,ρ) = ρS(q,ρ)I0(q) + ρIf(q) (1)
where I0(q) describes how the scattering intensity is modulated by interference effects be-
tween radiation scattered by different parts of the same PE particle. The structure factor
S(q,ρ) is related to mutual interactions between different PE particles. Therefore it is
dependent on the degree of order of the PE particles in the samples. For noninteracting
particles the structure factor is unity. Moreover, the contribution to the scattering intensity
due to concentrations fluctuations of PE chains in the amorphous regions If(q) becomes
only important for large scattering vectors.
The scattering intensity I0(q) of a single PE particle is described in terms of
I0(q) =
1
?
0
dαF2(q,α) (2)
with
F(q,α) =
2πRJ1(qR√1 − α2)
q√1 − α2
+ ∆basin(qα(La+ Lc)/2)
×
?
∆bSDSsin(qαL/2)
qα/2
qα/2
+ ∆bcsin(qαLc/2)
qα/2
?
(3)
Here R is the radius of a circular platelet consisting of a crystalline layer of height Lc
sandwiched between layers of amorphous PE and SDS as is shown in Figure 1. As already
discussed in previous work, the faceted nanocrystals can be treated in good approximation as
circular platelets for the SAXS analysis.17The height of the amorphous PE regions including
the hydrocarbon SDS tails is given by La/2, while L denotes the total height of the platelet
(see Figure 1). The electron contrasts are defined as ∆bSDS= bSDS− bs, ∆ba= ba− bSDS,
and ∆bc= bc− ba, where ba= 302 nm−3, bc= 339 nm−3, bSDS= 396 nm−3, and bsare the
electron densities of the amorphous PE layers including the hydrocarbon SDS tails, the
crystalline PE layer, the SDS headgroup layer, and the solvent, respectively. The value of
the electron density of the solvent bsis determined by the concentrations of water and added
contrast agent. Moreover, J1(x) in eq 3 denotes the cylindrical Bessel function of first order.
The effect of size polydispersity of the PE particles is taken into account by an appropriate
average using a distribution function characterizing the degree of polydispersity.
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0.5 La
L
LC
R
FIG. 1. Side view of a platelet of pseudo radius R and total height L. The platelet consists of a
crystalline lamella of height Lcsandwiched between two amorphous sheets. These sheets include
the hydrocarbon SDS tails with thicknesses La/2 and two layers of SDS head groups (marked in
red).
We have used an integral equation theory31in order to calculate structure factors which
characterize intermolecular correlations between different PE particles.This theoreti-
cal approach has been successfully applied to various suspensions consisting of platelike
particles.32–34
Finally, the contribution of concentration fluctuations of the PE chains in the amorphous
phase reads
If(q) =
I(0)
0
1 + (qξ)2
(4)
where ξ is the correlation length and I(0)
0
determines the contribution at vanishing scattering
vector.
III. RESULTS AND DISCUSSION
A.Increase of lamellar thickness
We first discuss the modification of the PE particles of sample S87 due to thermal an-
nealing. A 5 ml glass bottle has been filled with about 2 ml of the original PE suspension.
Thereafter, it has been kept in a metal vessel located on a heating plate at 125◦C for 20
minutes. Finally, the sample has been cooled down to 25◦C. Evaporation of water could
not be avoided completely leading to 1.74 wt % PE and 0.90 wt % SDS after annealing as
compared to 1.70 wt % PE and 0.87 wt % SDS before annealing. Figures 2 (a) and (b) show
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typical cryo-TEM micrographs of the suspension before and after annealing, respectively.
The PE particles are platelets with rather narrow size distribution. The hexagonal facetting
of the nanocrystals is clearly visible.17Evidently, the platelets do not form aggregates due
to the added SDS molecules. Those PE particles that appear to be very close to each other
are located along the optical path (perpendicular to the plane of the figures) but at different
depths inside the suspension. Moreover, it is worthwhile to mention that the different gray
scales for different platelets are also related to different angles between the normal of the
platelets and the direction of the electron beam. The length of the optical path through
a platelet with its normal oriented perpendicular to the electron beam is larger than that
through a platelet with its normal oriented parallel to the electron beam.
From the cryo-TEM images we have determined platelet radii of 14 ± 4 nm and 9 ± 2 nm
before and after annealing, respectively. The corresponding heights of the platelets have been
estimated to be 7 ± 1 nm and 13 ± 2 nm before and after annealing. These values have been
derived from the image analysis of 20 particles with their normal oriented perpendicular to
the electron beam. On the basis of this analysis one may conclude that a pronounced change
of the shape of the PE nanoplatelets occurred during the annealing process (see also the
insets of Figure 2). However, one has to take into account that it is not possible to detect
amorphous PE with the help of the cryo-TEM micrographs shown in Figures 2 (a) and (b)
because the electron density of amorphous PE is very similar to the one of the surrounding
low density amorphous ice.17However, the images shown in Figure 2 demonstrate clearly
that the process of annealing did not lead to formation of cavities, disintegration, or any
other gross morphological change of the platelets. In order to elucidate the shape and the
structure of the PE particles in more detail, the systems have been investigated using SAXS.
Figures 3 (a) and (b) display SAXS intensities of the PE nanoplatelets before and af-
ter annealing, respectively. These scattering intensities have been measured at different
contrasts starting from a stock solution of the nanoplatelets dispersed in pure water. The
different contrasts have been adjusted by adding different amounts of glucose. The vol-
ume fraction of added glucose increases from 0 (lower symbols in Figures 3 (a) and (b)) to
0.14 (upper symbols in Figures 3 (a) and (b)) while the corresponding volume fractions of
the nanoparticles decrease upon increasing the amount of added glucose. For clarity, the
scattering intensities related to different contrast have been shifted vertically in Figure 3.
Figure 3 demonstrates that annealing the sample leads to marked differences in the scat-
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FIG. 2. Cryo-TEM micrographs of polyethylene nanoplatelets S87 in aqueous solution. The gray
background is the low-density amorphous ice. In panel (a) the original sample was maintained at
25◦C after catalytic polymerization at 15◦C, while in panel (b) the same sample was subject to
an annealing process at 125◦C for about 20 minutes. Thereafter, the sample was cooled down to
25◦C. The insets displays single particles in stronger magnification. Here the platelets are oriented
with their normal parallel to the plane of the figure.
tering intensities. Moreover, contrast variation furthermore enhances the difference between
these states. Hence, the change of shape and internal structure of the platelets can be stud-
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ied with high precision. The lines show the results obtained from eqs 1 - 4 for noninteracting
particles (dashed lines), i.e., S(q,ρ) = 1, and interacting particles (solid lines). For small
magnitudes of the scattering vector q the calculated scattering intensities for noninteracting
particles (dashed lines) on the one hand, and the integral equation results for interacting
particles (solid lines) as well as the experimental data (symbols) on the other hand deviate
due to strong repulsive electrostatic interactions between the particles brought about by the
adsorbed SDS molecules. The correlation length associated with the peak of the scattering
intensities at q = 0.14 nm−1is given by d = 2π/q = 45 nm and is related to the average
distance between two platelets (see also Figure 2). Both this correlation length and the
isothermal compressibility which is related to I(0,ρ)31are the same before and after anneal-
ing. Hence, the intermolecular pair correlations and the equation of state of the suspension
are not influenced by the annealing process. This corroborates our finding that the number
of platelets before and after annealing is equal (see Figure 2).
The overall dimensions following from the theoretical description of the scattering inten-
sities are the average radius R = 10 ± 3 nm and R = 7.5 ± 3 nm as well as the thickness
of the crystalline layer Lc= 6.5 ± 1 nm and Lc= 13 ± 1 nm before and after annealing, re-
spectively. The thickness of the amorphous layer slightly increases from La= 3.1 ± 0.8 nm
to La= 3.8 ± 1.0 nm during the annealing process. The change of the shape of the platelets
can be directly seen from the shift of the location of the side maxima of the scattering in-
tensities to lower q values after annealing. The model parameters characterizing the size of
the platelets can be expressed in terms of dimensionless scaling variables qR, qL, q(La+Lc),
and qLcaccording to equation 3. Hence a variation of the location of the q values of the
side maxima of the scattering intensity implies a change of these model parameters. The
pronounced suppression of the scattering intensities for the samples with 4.5 vol. % added
glucose is due to the fact that the electron density of the solvent, i.e., water and added
glucose, is similar to that of the crystalline layer. Therefore, there is only a minor con-
tribution of the crystalline layer to the scattering intensities in this case. From the model
parameters and by taking into account the densities of the amorphous and crystalline phase
reported in the literature35, we have calculated that about 2 × 105CH2groups are forming
each nanoparticle. Given the molecular weight 3.5×105g/mol this corresponds to approxi-
mately 8 polyethylene chains by particle. Furthermore, we emphasize that we haven’t found
alternative models which lead to agreement with the experimental data shown in Figure 3.
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10
-1
10
0
10
-2
10
0
10
2
10
4
(a)
I(q, ρ) [nm-3 ]
q [nm-1]
10
-1
10
0
10
-2
10
0
10
2
10
4
(b)
I(q, ρ) [nm-3 ]
q [nm-1]
FIG. 3. Measured scattering intensity I(q,ρ) of polyethylene nanoplatelets of sample S87 before
(a) and after (b) annealing at 125◦C as a function of the magnitude of the scattering vector q
(symbols). The volume fraction of added glucose acting as contrast agent increases from bottom
to top (0, 4.5, 10.0, 14.3 vol. %) while the volume fraction of the nanoplatelets decreases (2.6,
2.5, 2.3, 2.2 vol. % ). As a result the electron density of the solvent increases from bottom to top
according to bs= 333,342,352,359 nm−3. The three lowermost scattering intensities are shifted
down by a factor of 10, 102, 103, respectively. The dashed lines represent the result of the modeling
of the SAXS data assuming a dispersion of noninteracting polydisperse platelets. The solid lines
represent the scattering intensity calculated for a dispersion of interacting polydisperse platelets.
The differences between the dashed and solid lines reflect the repulsive interaction between the
nanoplatelets.
The analysis of the SAXS data and the TEM micrographs leads to the conclusion that
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annealing the sample S87 at 125◦C for about 20 minutes leads to a doubling of the thickness
of the crystalline lamella from Lc= 6.5 nm to Lc= 13 nm. We have checked that longer
annealing times (up to 60 minutes) do not lead to further increase of Lc. The thickening of
the crystalline lamella during annealing can be understood in terms of an unlooping within
a single lamella19,20as is illustrated in Figure 4. With increasing temperature the chain
mobility increases leading to cooperative motion of repeat units parallel to the main axis
of a crystalline platelet (see, e.g., refs36,37and references therein). In doing so the chains
can partly unfold and reduce the amount of amorphous loops. As a result the height of the
platelets increases while their radius decreases.
?T
FIG. 4. Illustration of the mechanism leading to a doubling of a crystalline lamella after annealing.
The figure shows schematically a side view of platelets consisting of an inner crystalline layer
sandwiched between two amorphous layers. The increase of the height of the crystalline lamella
during annealing is due to an unlooping within a single lamella.19,20
It must be kept in mind that the number of particles inferred from SAXS is the same
before and after annealing. This is in agreement with the cryo-TEM micrographs. Within
the present investigation this is an important observation because the alternative scenario,
namely the stacking of adjacent lamella22,23can be ruled out definitely. In this model the
thickening is due to a fusion of two crystalline layers. Such a model may be invoked for
solution crystallized films involving a semicrystalline polymer gel in which the crystalline
domains are connected by amorphous material. For the present system a stacking of lamellae
would lead to the reduction of the number of particle number by a factor two after annealing
which is not observed.
For comparison we note that a simple theoretical model5,7for the equilibrium shape of a
cylindrical polymer crystal yields
Lc
R= 2σeff
σc
, (5)
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where σcis the solvent-crystalline layer interfacial tension and σeff is the effective solvent-
amorphous layer interfacial tension that includes entropic and bending contributions40,41of
the PE chains. The temperature dependence of σeff is discussed in Ref.7on the basis of a
statistical model that takes into account loops and tails of polymer chains in the amorphous
regions. Depending on the model parameters σeff decreases or increases upon increasing
the temperature (see Figure 7 in Ref.7). Equation 5 follows from a minimization of the free
energy
F = −πR2Lcǫc+ 2πRLcσc+ 2πR2σeff
(6)
with respect to Lc under the constraint that R2Lc = const. Here ǫc is the energy gain
per volume of the bulk crystalline phase as compared to the bulk amorphous phase. After
annealing, the shape of the PE nanocrystals under consideration is characterized by the ratio
Lc/R ≈ 2 implying that σeff/σc≈ 1 if the nanocrystals are thermodynamically stable. In
view of the fact that no direct experimental values for the interfacial tensions are available,
we note that σeff/σc ≈ 2 for polymer crystal melt interfaces according to a very recent
computer simulation study.42The presence of the SDS molecules and the solvent water
will lead to a ratio σeff/σc < 2, but at the moment it is not possible to decide whether
the PE nanocrystals after annealing exhibit their equilibrium shape or not. Nevertheless,
the size ratio Lc/R ≈ 2 of the PE nanocrystals is remarkably large as compared to the
corresponding ratios of the well known PE crystals with a thickness of about 10 - 15 nm and
lateral dimensions of the order of 100 nm to a few µm. Moreover, the scattering intensities
shown in Figure 3 (b) can be modelled using a size polydispersity with a constant size ratio
Lc/R according to equation 5. Based on these results we see avenues for future research
devoted to the experimental realization of equilibrium polymer nanocrystals.
B. Variation of annealing temperature
The results of the previous subsection show that thermal annealing at the high tempera-
ture 125◦C leads to a strong increase of the height of the crystalline lamella of the order of
two. We now study possible changes of the PE platelets after annealing at lower tempera-
tures using the sample S36 which contains a smaller amount of added SDS as compared to
the sample S87 studied in the last subsection. Figure 5 (a) displays SAXS intensities of the
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10
-1
10
0
10
-2
10
-1
10
0
10
1
10
2
10
3
(a)
I(q, ρ) [nm-3 ]
q [nm-1]
115◦C
105◦C
90◦C
25◦C
0.00 0.05 0.100.15
30
60
90
120
150
180
micron crystals
on substrate
nanocrystals
freely suspendend
(b)
T [°C]
1/Lc
FIG. 5. (a) SAXS intensities of polyethylene nanoplatelets of sample S36 dispersed in pure water
at 25◦C together with the measured scattering intensities after annealing at 90◦C, 105◦C, and
115◦C (from bottom to top). For clarity, the three lowermost scattering intensities are shifted up
by a factor of 10, 102, 103, respectively. The differences between the lower data set and the one
shown in Figure 3 (a) are due to the different amounts of added SDS in samples S36 and S87. The
solid lines display the calculated results for interacting platelets. In (b) it is shown that the inverse
of the crystalline lamellar thickness 1/Lcincreases linearly with the annealing temperatures (upper
three squares) leading to a recrystallization line (solid line). The lower square corresponds to the
original sample studied at T = 25◦C. For comparison the open symbols display the thickness
evolution during annealing of solution-grown micron crystals deposited on a solid substrate.16The
lowermost triangle and circle correspond to the micron crystals originally crystallized at T = 85◦C
and T = 95◦C, respectively. The stars mark the data taken from Figure 4 of ref.27.
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original sample together with the measured scattering intensities after annealing for about
20 minutes at 90◦C, 105◦C, and 115◦C (from bottom to top).
These scattering intensities have been obtained from a solution of PE platelets dispersed
in pure water. For clarity, the three lowermost scattering intensities have been shifted
vertically. The increase of the height of the nanoplatelets upon increasing the annealing
temperature can be directly seen from the shift of the side maxima and minima to lower
q-values. Longer annealing times (up to 60 minutes) do not lead to further changes of the
scattering intensities. We note that the differences between the measured scattering inten-
sities of the original sample S36 (lower symbols in Figure 5 (a)) and the original sample
S87 (lower symbols in Figure 3 (a)) are due to the different amounts of added SDS. How-
ever, the amount of added surfactant does not influence the size and structure of the PE
nanoparticles.
The solid lines in Figure 5 (a) show the calculated scattering intensities for inter-
acting platelets using Lc= 6.5 ± 1 nm for the original sample and Lc= 8.5 ± 1.4 nm,
Lc= 10.2 ± 1.4 nm, Lc= 11.8 ± 1.4 nm for the samples which have been annealed at 90◦C,
105◦C, and 115◦C, respectively. Figure 5 (b) displays a plot of the annealing temperature
against the reciprocal thickness Lcas suggested by the Gibbs-Thomson equation
Tc= T∞
c(1 −
2σ
∆hLc) (7)
where ∆h is the heat of fusion, σ the surface free energy of the lamellae, and T∞
c
the
temperature limit referring to fully crystalline samples. The open symbols refer to the
respective data taken from micron-sized PE crystals on a solid substrate. These data have
been taken from the work of Tian and Loos.16Evidently, the temperatures T∞
c
obtained
by extrapolation of PE crystals of different sizes agree within prescribed limits of error.
This is to be expected since the influence of both the SDS molecules in the case of the
freely suspended nanocrystals and the solid substrate in the case of the micron crystals
on the recrystallization process is vanishing in the limit of infinitely long polymer chains.
Moreover, the data obtained by Tong et al.27on the thickening of PE-nanocrystals lying on
a solid substrate (crosses in Figure 5 (b)) fit very well on this line.
It is also interesting to compare the lamella thickening of the PE nanocrystals with
crystallization results obtained on PE bulk samples. Figure 6 displays the recrystallization
line (solid line) of the PE nanocrystals together with the crystallization line (dashed line)
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of PE crystallized from melt.39In the limit Lc → ∞ the recrystallization line and the
crystallization line lead to the same temperature T∞
c
≈ 182◦C. Note that here the initial
thickness of the lamellae has been used for this plot (see the discussion of this point in
ref.43). Together with the extrapolation shown in Figure 5 this plot demonstrates that
T∞
c
is a well-founded number describing the melting point of a fully crystalline PE-sample.
However, in case of bulk sample, the recrystallization process ends at the intersection of the
recrystallization line with the melting line (dash-dotted line in Figure 6) which is given by
T = 141.1 − 259.7/Lcaccording to ref38.
For comparison the supercooling dependence of the crystalline lamellar thickness after
post-crystallization reorganization is shown by the dotted line. After initial crystallization,
the PE chains are known to reorganize toward the equilibrium state leading to an increase of
the crystalline lamella thickness. The PE nanoparticles (lowermost square in Figure 6) went
through post-thickening after initial crystallization. Extrapolation of these data according
to eq. (7) would lead to a considerably smaller value for T∞
c.
IV. CONCLUSION
In conclusion, our findings elucidate the change of the size and the structure of individual
polyethylene nanocrystals during annealing. The nanoparticles have been synthesized and
stabilized by a nickel-catalyzed polymerization in aqueous solution. Hence freely suspended
polymer nanocrystals can be studied, while earlier studies of larger polyethylene crystals
involved either the bulk polymer phase or a supporting solid substrate. The combination of
small-angle x-ray scattering and cryogenic transmission electron microscopy demonstrates
that the thickening of the crystalline lamella can be controlled by varying the annealing
temperature. The resulting recrystallization line defines a linear relationship between the
annealing temperature and the reciprocal of the crystalline lamellar thickness. The recrys-
tallization lines for both the present PE nanocrystals and for micron-sized PE crystals (see
Figure 5) lead to the same temperature T∞
c ≈ 182◦C as the crystallization line for for bulk
samples (see Figure 6). All data obtained herein point to a thermodynamic control of the
thickness of the lamellae. The finite thickness measured after annealing can be rationalized
by a simple theoretical model5,7for the equilibrium shape of a cylindrical polymer crystal.
15