Article
pubs.acs.org/Macromolecules
Molecular Modeling Approach to Prediction of Thermo-Mechanical
Behavior of Thermoset Polymer Networks
Natalia B. Shenogina,*,† Mesfin Tsige,*,‡ Soumya S. Patnaik,§ and Sharmila M. Mukhopadhyay†
†
Department of Mechanical and Materials Engineering, Wright State University, Dayton, Ohio, United States
Department of Polymer Science, University of Akron, Akron, Ohio, United States
§
Propulsion Directorate, Air Force Research Laboratory, Dayton, Ohio, United States
‡
ABSTRACT: Molecular dynamics and molecular mechanics simulations have been used to study thermo-mechanical response
of highly cross-linked polymers composed of epoxy resin DGEBA and hardener DETDA. The effective cross-linking approach
used in this work allowed construction of a set of stress-free molecular models with high conversion degree containing up to
35000 atoms. The generated structures were used to investigate the influence of model size, length of epoxy strands, and degree
of cure on thermo-mechanical properties. The calculated densities, coefficients of thermal expansion, and glass transition
temperatures of the systems are found to be in good agreement with experimental data. The computationally efficient static
deformation approach we used to calculate elastic constants of the systems successfully compensated for the large scattering of
the mechanical properties data due to nanoscopically small volume of simulation cells and allowed comparison of properties of
similar polymeric networks having minor differences in structure or chemistry. However, some of the elastic constants obtained
using this approach were found to be higher than in real macroscopic samples. This can be attributed to both finite-size effect and
to the limitations of the static deformation approach to account for dynamic effects. The observed dependence of properties on
system size, in this work, can be used to estimate the contribution of large-scale defects and relaxation events into macroscopic
properties of the thermosetting materials.
1. INTRODUCTION
Chemically cross-linked thermosetting polymer networks are
useful in many applications such as coatings, adhesives and
matrix components in high performance composites.1 Epoxy
resins cross-linked with amine curing agents are popular
materials due to their outstanding thermo-mechanical properties such as high-temperature performance, high stiffness and
fracture strength. Optimizing of the processing conditions,
designing of the new structures with the required properties as
well as better understanding of the physical phenomena in
polymers imply extensive trial-and-error experimental studies
which can be both expensive and time-consuming.
In order to reduce the experimental efforts in synthesis and
optimization of material properties, computer simulations of
the systems of interest can be very useful. Such simulations
allow systematic variation of structural or physical parameters
of the materials and can significantly lower experimental costs
in predicting the properties of new materials. These approaches
© 2012 American Chemical Society
may eventually allow for screening of a greater breadth of
potential resin chemistries than those that can be tried by
experimental testing alone.
Among various computational approaches, finite element
simulations2,3 are successfully employed to study large-scale
events in polymeric materials. However, such methods are
often limited by the lack of realistic parameters for inputs to
constitutive relations. Two other popular computational
approaches used to simulate polymeric systems are Monte
Carlo simulations4−10 and molecular dynamics (MD) simulations using a bead−spring model, where each bead represents
one or several polymer groups.11−17 While these methods can
predict some general trends in the behavior of polymer
networks and generate useful insight into the dependence of
Received: April 13, 2012
Revised: May 25, 2012
Published: June 7, 2012
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DETDA). While some authors23,26−28 have recently reported
simulations showing the influence of the conversion degree on
the properties of thermosetting polymers, there is no previous
report in the literature that systematically investigated the effect
of resin chain length on the thermo-mechanical properties of
epoxy networks to the best of our knowledge.
The paper is organized as follows. In the next section, we
briefly review our simulation methodology, including the
systems of interest and the approach we used in building the
polymer networks. We present and discuss our results for
physical properties of the system in section 3 and for
mechanical properties including the effect of resin chain length
in section 4. In the last section, we summarize our results as
well as the advantages and limitations of the methods used in
this study.
the physical properties on the cross-link networks, these studies
are not capable of providing specific correlation with the
chemical structure of the resin system. However thermomechanical properties of thermosetting networks are known to
be significantly dependent on the molecular-level details of the
structure.
Detailed microscopic information on the physical properties
of thermosetting polymers can be obtained from atomistic
simulations, which may lead to predictions in quantitative
agreement with experiments. Several groups have developed
atomistic level methods of constructing models of highly crosslinked polymer networks, and used them to predict their
properties.18−28 More notably, Yarovsky and Evans19 proposed
a cross-linking technique in which all cross-linking reactions
were carried out in one step (so-called static approach). The
structure obtained using this method had conversion degree
much lower than in real curing reactions. Alternatively, Wu and
Xu21 used cross-linking procedure that allows the formation of
one cross-link per step and calculated elastic moduli of the
resulting structures. However, this improved approach could
become computationally inefficient with increasing the system
size. Heine et al.20 used a dynamic cross-linking approach based
on a cutoff distance criterion and relaxation procedure of the
cross-linked structure with a modified potential. Komarov et
al.23 employed four-step reverse mapping procedure to perform
cross-linking at coarse-grained level and study properties of the
resulting structures on the fully atomistic level as a function of
the conversion degree. Varshney et al.24 combined the dynamic
cross-linking approach proposed by Heine et al.20 and the
relaxation procedure consisting of cycles of energy minimization and molecular dynamics equilibration stages that was
proposed by Wu and Xu,21 used new method of multistep bond
formation and calculated some thermodynamic and structural
properties. Li and Strachan26,27 proposed the cross-linking
procedure with charge evolution in the course of chemical
reaction and obtained thermo-mechanical properties using their
cross-linked models. Whereas these earlier studies provided a
significant progress in atomistic computer simulations of highly
cross-linked polymer networks, the systems studied were
relatively small in size (less than 15000 atoms). This is a
limitation since it is known that the size of the system can
substantially influence the properties obtained by molecular
dynamics simulations.
In the present work we have addressed this issue by
constructing a significantly larger set of all-atom models,
containing up to 35000 atoms. We have used a method
developed by Accelrys, Inc.30 which uses a cross-linking
procedure that combines several approaches that allow
construction of thermosetting networks having structural
characteristics close to those in real systems. These include
capture sphere growth approach29 and effective relaxation
technique21 in combination with monitoring of local stresses in
the resulting networks on-the-fly. This combination of
approaches yields systems with high conversion degrees and
also free of stresses and geometrical distortions. The resulting
polymer networks have been used in the present study to
investigate the influence of model size, extent of curing and
length of epoxy strands on the thermo-mechanical properties of
epoxy-based polymer networks.
The focus has been on a resin-hardener system that is widely
used in engineering applications: the resin selected is DGEBA
(diglycidylether of bisphenol A) and the aromatic amine
hardener selected is EPI-Cure-W (diethylenetoluenediamine,
2. METHODOLOGY
2.1. Systems of Interest. The molecular structures of the
epoxy monomer and curing agent molecule are shown in Figure
1, parts a and b. To simulate the cross-linking reaction, the
Figure 1. (a) Epoxy resin: DGEBA (diglycidyl ether of bisphenol A)
with activated reactive sites (yellow). (b) Aromatic amine hardener:
DETDA (diethylene toluene diamine). Reactive sites (amine groups)
are highlighted in yellow. (c) The reactive sites in the epoxy resin are
activated by opening the epoxy rings at the ends of the molecule. (d)
Amine groups in curing agent react with opened epoxy rings at the
ends of resin molecules.
reactive sites in the epoxy resin are activated by opening the
epoxy rings at the ends of the molecule as shown in Figure 1c.
Each of the amine groups in curing agent can react with two
resin molecules as terminal carbon atoms of a resin molecule
react with the nitrogen atoms of the amine hardener (Figure
1d). In the present work, we simulate stoichiometrically
balanced compositions of the epoxy resin and hardener
molecules; i.e., in all our systems, the number of the resin
molecules is two times that of the hardener ones, which allows
a theoretical conversion of 100%.
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between reactive atoms whenever a valid atom falls within the
sphere of another one. In the case of densely cross-linked
polymers, the network potential energy has a significant
influence on the properties of the material. Hence, to create
thermosetting polymer with realistic material properties,
additional measures were taken to ensure undistorted geometry
of the molecules with low internal stresses. In order to
minimize the presence of internal stresses and geometric
distortions during cross-linking, energetic and structural
information are analyzed on-the-f ly after each cross-linking
cycle. Unrealistically high stretching energy of any of the
individual bonds in the system can be considered as a sensitive
measure of the distortion. It usually happens when a defect is
introduced in the course of the cross-linking reaction. In the
present work, when a dramatic increase in the maximum bond
stretching energy is detected at any stage of the cross-linking
procedure, the structure is rejected in favor of others with low
internal stresses.
In actual chemical reactions, the reactive groups diffuse
through the mixture and bond as they approach to each other
at the appropriate distance. Equilibrating the mixture at high
temperature enhances the movement of the reactive groups
toward each other. However, in the case of a dramatic increase
in molecular weight that usually occurs in the course of the
cross-linking reaction, the diffusion takes much longer time and
is currently beyond the reach of all-atom molecular dynamics
approach. To circumvent this limitation and to relieve network
stresses within a reasonable computational time, the equilibration can be supplemented with energy minimization of the
structure that allows quick changes in geometry which
otherwise would take long time in dynamic simulations.
Thus, to mimic the diffusion in actual cross-linking reactions
our model systems are subjected cascades of both energy
minimization and constant volume and temperature equilibration after each cross-linking cycle.21
Cross-Linking Procedure. In the present study, the initial
amorphous structures of the uncross-linked DGEBA/DETDA
mixtures were built using Accelrys amorphous builder which
uses a Monte Carlo packing algorithm based on the rotational
isomeric states model.33 The DGEBA and DETDA molecules
in stoichiometrically perfect proportions were added to a cubic
periodic simulation box by growing the molecules segment by
segment, taking into account both energy of interaction with all
atoms in the box and chain conformations. As both epoxy and
amine hardener molecules contain aromatic rings, a special care
was taken to check ring spearing during the construction of the
amorphous mixture.
The formation of thermosetting polymer network starts with
the equilibration of reactant mixture. In order to enhance
molecular mobility in the course of chemical reaction and
hence accelerate the network formation the curing is carried out
at elevated temperature of 480 K. The cross-linking cycle
begins with determination of the available reactive sites, i.e.,
terminal carbons of the activated epoxy molecules and nitrogen
atoms of the amine hardener. During the simulation, distances
between these reactive sites are calculated and those reactive
sites falling within the current reaction radius are identified.
The initial cutoff of chemical reaction is set to 5 Å and increase
the cutoff radius by a small value (0.5 Å) after each crosslinking cycle. New bonds between the identified reactive atoms
are created and surplus hydrogen atoms are removed from both
epoxy and amine group reactive sites. The partial charges and
atom types of the atoms that participated in the chemical
To study the effect of system size on the thermo-mechanical
properties of the thermosetting polymers, we constructed seven
different system sizes ranging from (32, 16) to (512, 256),
where the numbers in the parentheses represent the number of
the epoxy resin and hardener molecules respectively, which
corresponds to approximately 2200 to 35100 atoms in the
simulation cells.
Furthermore, to investigate the effect of resin chain length on
the physical and mechanical properties of this class of materials,
we also built the structures using short resin oligomers with
various degree of polymerization, consisting of one, two and
four monomer epoxy molecules (mono-, di-, and tetramers,
respectively).
In addition, in order to examine the role of the degree of
curing on the properties of the system, structures with degrees
of conversion ranging from 50% to 100% are selected during
the curing process described below. We would like to also add
that for better prediction of the mechanical properties of the
thermosetting material under investigations using molecular
dynamics (MD) simulation, we generated up to 70
topologically independent structures for each configuration
with a given system size, degree of conversion and resin chain
length. The details of this approach will be discussed in the
mechanical properties section.
2.2. Simulation Details. The initial mixtures of reactants
were built by packing activated epoxy resin and curing agent
molecules into a cubic simulation cell followed by a geometry
optimization using the Amorphous Cell module of the
Materials Studio commercial package.30 All subsequent MD
simulations were performed using the Discover module of the
Materials Studio software.
In all our simulations, 3D periodic boundary conditions were
imposed to a cubic simulation cell in order to avoid introducing
artificial surface effects. Interatomic interactions are described
using the second-generation COMPASS force field31 due to its
high accuracy in predicting the properties of polymeric
materials. Wu and Xu21 validated COMPASS force field for
predicting the elastic moduli of highly cross-linked polymer
networks. These properties were found to be in better
agreement with experiment than previous calculations of
these properties using DREIDING force field. In addition,
Tack and Ford 32 showed that density prediction of
thermosetting polymer using COMPASS is better than that
using the cff91 force field.
2.3. Building of Polymer Networks. Earlier attempts to
construct realistic models of highly cross-linked polymer
networks encountered difficulties resulting from high internal
stresses and unrealistic geometry distortions (see, for example,
Yarovsky and Evans19). As a consequence, these models
produced markedly different predictions with properties of
thermosetting polymers that are often far from the
experimental values. In this study we used the methodology
that is currently most reliable for creating models of free of
stresses and geometrical distortions and yet reaching the levels
of degree of conversion consistent with those typical in real
systems.
To build polymer networks with high degree of conversion,
we used Accelrys software.30 This software employs the capture
sphere growth approach that was originally elaborated by
Eichinger29 and was successfully used for generating the
topology of lightly cross-linked elastomers. In this approach,
a cross-linked system is developed by growing the radius of a
sphere around each cross-link site and bonds are formed
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αG and αR correspond to the volumetric coefficients of thermal
expansion (CTE) in glassy and rubbery regions, respectively.
Glass transition temperatures were determined at the point of
the change in the slope (represented by the intersection of the
solid lines in the Figure 2). To validate the temperature ranges
in which we perform linear fits, we varied the boundaries of
these ranges and found that linear fits obtained are within the
error bars of the volume-temperature plot and bring negligible
changes in glass transition temperature.
Furthermore, for mechanical characterization of the material
both in glassy and rubbery states the systems should be
equilibrated to the temperatures of interest. In this work we
have chosen two temperatures, 298 and 480 K, which are above
and below the glass transition region of a given structure,
respectively. First, we determined densities at these two
temperatures from volume−temperature plots that were
generated by averaging data from five randomly picked
structures for each extent of the reaction. Then all the
structures were equilibrated to these two predefined temperatures and corresponding densities for mechanical properties
characterization of the material under investigation.
reaction process are also modified after the reaction. The cycle
recurs until the specified maximum cutoff radius is achieved or
all available sites are reacted. Unreated epoxy rings remain
open.
After each cross-linking cycle a relaxation procedure is
applied as described in the above section and structural and
energetic characteristics of the obtained configuration are
evaluated. If the maximum bond stretching energy detected
after relaxation remains unusually high, the reaction stops.
Otherwise, the cross-linking cycle continues.
2.4. Determining Thermal Properties of the Network.
As described above, to enhance molecular motions and reduce
stresses in the models, a curing reaction was conducted at an
elevated temperature of 480 K that is above the experimental
glass transition region for this material. To study thermal
response of the constructed networks, we applied stepwise
constant pressure cooling procedure performing a sequence of
constant pressure and temperature molecular dynamics
simulation runs at a set of temperatures. Here we implemented
Andersen thermostat and Berendsen barostat to control
temperature and pressure in our simulations. Cooling of the
model systems was carried out in steps of 10 K with 100 ps
equilibration run at each temperature within the temperature
range of 623 to 223 K. The volume at each temperature was
then computed by averaging results from at least five randomly
picked epoxy structures and is used to construct volume−
temperature plots from which a set of physical properties are
calculated as described below.
Figure 2 shows a representative specific volume−temperature
plot of the cooling process. As can be seen in Figure 2, the
3. PHYSICAL PROPERTIES
3.1. Results. A. Glass Transition Temperature. The results
of our glass transition temperature investigation are presented
in Figure 3a where the glass transition temperature is plotted as
a function of the extent of the reaction for the seven different
system sizes ranging from (32, 16) to (512, 256) DGEBA/
DETDA molecules represented by different colors. As
expected34 we observe an increase in glass transition temperature with increase in degree of curing. In general, the
dependence of Tg on the degree of curing shows better defined
trends with increasing the system size giving about 50 K
divergence in values for the smallest and the largest system at
high conversion degrees. However, even for the largest system
size, small sampling size is prone to larger error bars resulting in
higher standard deviations. These range from 3 K at low curing
degrees to 21 K at 95% of curing. As the confidence interval for
high extents of the reaction is wide, it makes it very hard to
draw any definite conclusion about the shape of the curve in
Figure 3a at high degrees of curing. The experimental values of
the glass transition temperature for this material reported by
different groups under different conditions35−38 also have wide
variation of about 35K (441 to 476 K) and are shown in Figure
3a as open symbols. We see that our simulation results for
extent of reactions higher than 90% are within the range of the
reported experimental values.
B. Volumetric Coefficient of Thermal Expansion. Figure 3b
shows the results of the coefficients of thermal expansion
(CTE) calculations for the glassy and rubbery regions at
different degrees of reaction for the seven different system sizes
mentioned above. In both the glassy and rubbery regions, the
CTE monotonically decreases with increase in extent of curing.
In addition, CTE shows well observable dependence on system
size and, on average, it increases with increase in number of
atoms in the model. Experimental values for fully cured
structure at glassy and rubbery states38 are also shown in Figure
3b as open symbols. The CTE values obtained from our
simulations for high degrees of curing are slightly below but in a
reasonable agreement with experiment and will be discussed
later.
C. Density. In Figure 3c, the dependence of density on the
extent of curing reaction at temperatures below and above Tg
Figure 2. Representative specific volume−temperature plot for epoxy
system containing (512, 256) DGEBA/DETDA molecules cured to
90%. The small error bars are evidence of small variation of the specific
volume between the 5 structures used to generate the data.
volume continuously decreases with decrease in temperature
and a linear analysis was then used to predict the behavior
observed at high and low temperature regions. To determine
the properties of interest we perform linear fits of these two
regions using:
VG = VG° + αGT (223−323 K)
(1)
VR = VR ° + αR T (473−623 K)
(2)
where T indicates temperature, VG and VR are specific volumes
in glassy and rubbery regions, while the slopes of these regions,
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increase shows noticeable dependence on temperature. In
addition, at a given extent of reaction the density, on average,
decreases with increase in system size. The density at the
highest extent of the reaction is in excellent agreement with
experimental data (less than 2% difference).39
D. Properties of the Structures with Different Chain
Length of the Epoxy Strands. The dependencies of glass
transition temperature, coefficients of thermal expansion, and
density on the extent of the reaction for different chain length
of the epoxy strands are shown in Figure 4, parts a−c. In this
work we compared three kinds of systems which were
constructed using 512 monomers, 256 dimers and 128
tetramers of the epoxy resin DGEBA of respectively (512,
256), (256, 128) and (128, 64) systems. All properties under
consideration show observable dependence on the resin chain
length. The slopes of the conversion dependencies are
maximum ones for the structures cured with epoxy monomers.
Besides, better correspondence of densities and coefficients of
thermal expansion with experimental values is observed for
longer epoxy strands.
3.2. Discussion. A. Cooling Rate Effect. The transition
between liquid and glass is not a transition between two states
that are in thermodynamic equilibrium. It is a dynamic
transition from an ergodic to a nonergodic state. Glass is a
kinetically locked state of the material and its properties
(density, CTE, Tg) are strongly dependent on its thermal
history. The properties presented above strongly depend on the
rate of cooling which reflects the fact that highly cross-linked
polymer structures do not immediately respond to changes in
temperature. In particular, exceptionally high cooling rates used
in MD simulations usually result in densities that are lower than
experimentally observed values39 as the structure freezes before
its molecules get into more compact equilibrium configuration.
For the same reason, the CTE values from simulation are
usually lower than those estimated by experiment.39
It was shown in many DSC experimental studies (see, for
example, review by Wunderlich40) that glass transition
temperature depends not only on the rate of temperature
change but also on the sign of its change. In other words, glass
transition occurs at different temperatures at cooling and
heating of the same sample giving lower Tg values at cooling
and higher ones at heating. Moreover, it was shown by
experiments that hysteresis increases with increasing the rate of
temperature change. Thus, we could expect that cooling cycles
done using molecular dynamics simulations, as in the present
work, may also yield lower glass transition temperature than
that of heating cycles.
B. System Size Effect. Glass transition temperatures obtained
in our simulations (Figure 3a) show strong dependence on
system size at high extents of the reaction. Indeed, the smaller
the volume, the lesser structural rearrangements it can
accommodate. So glass transition temperatures of smaller
model structures (with periodic boundaries, i.e. without free
surfaces) are higher than that of larger ones. The same reason is
valid for the size dependence of the coefficients of thermal
expansion. An increase in the coefficients of thermal expansion
is observed when the simulation cell size is increased. Similarly,
as larger volumes in atomic scale take more time to equilibrate,
the densities of larger systems obtained from our simulations
are slightly lower.
C. Effect of Chain Length of the Resin Strands. The
dependence of Tg on resin chain length is not as clear as the
trends seen for CTE and density. The observed dependence of
Figure 3. (a) Glass transition temperature, (b) volumetric coefficients
of thermal expansion in glassy state (squares) and rubbery state
(circles), and (c) density in glassy state (squares) and rubbery state
(circles) as a function of the extent of the reaction for the atomic
structures ranging from (32, 16) to (512, 256) DGEBA/DETDA
molecules. In all cases the open symbols represent experimental values
and are taken from the following: (a) square, Jansen et al.;35 triangle,
Ratna et al.;36 circle, Shen et al.;37 rhomb, Liu et al.;38 (b) in glassy
(square) and rubbery (circle) states; 38 (c) in glassy state (square).39
The color codes are as follows: orange (32, 16); cyan (64, 32); black
(96, 48); red (128, 64); green, (256, 128); blue (432, 216); magenta
(512, 256).
for seven different system sizes is shown. As expected the
density increases in the course of the reaction and the rate of
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Note that the structures built using epoxy monomers contain
about 35000 atoms, while dimer structures have 27500 atoms
and tetramer structures have 24000 atoms. As these systems are
not of the same size, quantitative comparison of absolute values
of the properties is not advisible since they are influenced by
both chain length and size of the simulation cell. However,
comparison of the individual slopes is still informative, and
worth noting.
4. MECHANICAL PROPERTIES OF THE HIGHLY
CROSS-LINKED POLYMERIC NETWORKS
4.1. Approach Used in the Mechanical Properties
Calculations. Any small volume element of an amorphous
material in atomic scale can be characterized by a unique
distribution of matter within it and consequently displays
unique properties that sometimes can be pretty far from the
macroscopic properties of the sample. It means that a
macroscopically homogeneous amorphous material can be
viewed as heterogeneous at the nanoscale. We can thus
partition the macroscopic sample into many small elements and
take an average of the properties of the individual elements to
predict the macroscopic properties of the material. In this work
we estimate elastic constants of the thermosetting polymer by
calculating elastic constants for a number of nanoscopically
small simulation cells with subsequent averaging of the
obtained properties.
Simple averaging of the stiffness and compliances matrices
gives so cold Voigt and Reuss bounds representing upper and
lower bounds of the elastic constants of the material. However,
these bounds are often pretty broad for polymer networks while
the aim of this study is to be able to distinguish between
properties of very similar materials, e.g., materials with extent of
reaction differing by 5%. For this purpose, in the present work
we used two different bound estimation techniques to study
statistical variation of elastic constants. The simplest, but not
necessarily the best, is rough estimation technique giving Voigt
and Reuss bounds mentioned above. The second technique
proposed by Hill and Walpole41−44 is more sophisticated but
gives significantly narrower bounds.
To calculate elastic constants, we used Accelrys software30
which implements the static approach.45 First, three tensile and
three shear deformations of a small magnitude were applied to
the systems in three directions with subsequent energy
minimization. The obtained stress tensor is then used to
calculate stiffness and compliance matrices Cij and Sij of the
simulation cells followed by estimation of Voigt−Reuss and
Hill−Walpole bounds. Finally, assuming isotropic symmetry of
the model, these stiffness and compliances matrices are used to
calculate two Lamé elastic constants from which Young’s, shear,
and bulk moduli and Poisson’s ratio are calculated.44
Because of high computational efficiency, this methodology
allows one to analyze the elastic constants of a large number of
nanoscopically small volume elements giving narrow bounds of
the material properties. Such statistical treatment allows
mimicking the effect of nanoscopic heterogeneities that are
always present in real macroscopic samples and partially
overreach the size limitation of the molecular dynamic
simulation method. For this purpose we generated batches of
up to 70 topologically distinct thermosets for each extent of the
reaction. To the best of our knowledge, we are not aware of any
all-atom molecular modeling that treated such a great number
of statistical data (while using large simulation cells of up to
35000 atoms) to calculate elastic constants.
Figure 4. (a) Glass transition temperature, (b) volumetric coefficient
of thermal expansion, and (c) density as a function of the extent of the
reaction for the atomic structures built using monomers (black),
dimers (red), and tetramers (green) of the epoxy resin. The open
symbols in part b represent experimental values in glassy (square) and
rubbery (circle) states,38 and in part c, the open square represents
experimental value.39
CTE and density on the chain length of the resin strands can be
understood using following considerations. For infinitely long
strands the cross-linking density tends to zero and any of the
properties discussed above should not show any observable
dependence on the degree of conversion. Moreover, it is known
that small amount of resin oligomers is always present in
commercial products. Our results for density and coefficients of
thermal expansion for oligomer-based structures match better
with experimental values and clearly show the correct trend.
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Figure 5. Elastic moduli at 298 K(squares) and 480 K (circles) as a function of the extent of the reaction for the atomic structures ranging from (96,
48) to (512, 256) DGEBA/DETDA molecules: (a) Young’s modulus; (b) Poisson’s ratio; (c) bulk modulus; (d) shear modulus. Error bars represent
Hill−Walpole bounds. Color codes are black (96, 48), red (128, 64), green (256, 128), blue (432, 216), and magenta (512, 256).
4.2. Results. The dependence of the elastic moduli on the
extent of reaction at two temperatures for five different system
sizes ranging from (96, 48) to (512, 256) DGEBA/DETDA
molecules are shown in Figures 5a-d. As expected Hill-Walpole
bounds are significantly narrower than Viogt and Reuss ones so
in all our elastic moduli plots we show only Hill-Walpole
bounds, where the size of the error bars reflect the width of the
bounds. The elastic moduli for small systems containing (32,
16) and (64, 32) molecules are not shown here due to
considerable scattering and extremely wide bounds. As
expected, the results from our simulation demonstrate the
pronounced increase in Young’s, bulk and shear moduli with
extent of curing at both temperatures. However, the values of
the Young’s modulus determined from our simulation are
above the experimental value (2.71 GPa) at high extents of the
cross-linking reaction.46 The Young’s, bulk and shear moduli
values are lower at the elevated temperature, which is evidence
of the reduced stiffness of the polymer with increase in
temperature. Besides, these elastic constants demonstrate
systematic dependence on the system size giving more accurate
shapes of the curves and narrowing the bounds significantly
with increase in system size. We do not detect any dependence
of the Poisson’s ratio on the extent of the reaction or system
size. From our simulations we determined an average Poisson’s
ratio value of 0.31 at ambient temperature for all extensions and
sizes of the simulation cell, which is in good agreement with
experimental values for epoxy materials.47
The dependence of mechanical properties on the length of
the epoxy strands in the material are shown in Figure 6, parts
a−d. All elastic constants show a pronounced dependence on
the length of the epoxy strands. The dependence of the
Young’s, bulk and shear moduli on the degree of conversion
shows different slopes with the maximum slope corresponding
to the structures with the shortest epoxy strands. For the
structures with the shortest epoxy strands these moduli have
higher value at high extent of the reactions. As expected,
Poisson’s ratio increases for the structures with longer epoxy
strands showing the correct trend.
4.3. Discussion. A. Size Effect. In this study, we managed to
successfully build models that are free of stresses and distorted
geometry (bonds, angles etc.). However, these defects are not
the only ones present in real polymers. For example, big voids
and large-scale cooperative motions in macroscopic samples
can lower the measured stiffness of the material. This
phenomenon was observed experimentally e.g. by Shen et
al.37 and Possart et al.48 The authors measured micrometerscale Young’s modulus and compared it with macroscopic
values obtained from tensile tests. In both studies Young’s
modulus measured at micrometer scales using nanoindentation
test is higher than integral values measured at macroscale by
tensile test. At nanoscale, the deviation from integral values is
expected to be even more pronounced. Therefore, it is not
surprising that simulation predictions at nanoscale are higher
than those measured experimentally (at micro and macro
scales). All the models we constructed for our MD simulations
are nanoscopically small and nearly perfect. As a consequence,
such structures cannot accommodate either big defects or large5313
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Figure 6. Elastic moduli at 298 (squares) and 480 K (circles) as a function of the extent of the reaction for the atomic structures built using
monomers (black), dimers (red) and tetramers(green) of the epoxy resin: (a) Young’s modulus; (b) Poisson’s ratio; (c) bulk modulus; (d) shear
modulus. Error bars represent Hill−Walpole bounds.
properties dependence on this parameter. The shorter the
strands the more sensitive the material property to changes in
the conversion degree and the higher the slope of the
dependence of the mechanical property on extent of reaction.
Furthermore, the shortening of the distance between crosslinking sites increases the stiffness of the polymer network
reducing the Poisson’s ratio and raising the values of the
remaining three moduli of the material.
scale motions and thus resulted in higher than experimentally
measured Young’s modulus values.
B. Mechanical Properties at Temperatures above Tg. At
temperatures above glass transition we should expect to observe
high value of Poisson’s ratio that is close to 0.5 as well as a
rubbery plateau modulus that is two or 3 orders of magnitude
lower than the modulus at glassy state. However, the Young’s
moduli determined at 480 K from our simulations are slightly
lower than the corresponding Young’s moduli at room
temperature, while Poisson’s ratio takes distinctly lower values
than that for perfectly incompressible material. Nevertheless,
these results could be interpreted in terms of frequency
dependence of elastic constants in amorphous polymers.
Large-scale cooperative motions of big segments of the
molecules characterize the rubbery state. However, static
method of deformation does not accurately take into account
the dynamic effects that are especially noticeable at high
temperatures. Moreover, typical sizes of the simulation cells
used in atomistic simulations could not accommodate these
kinds of motions and thus resulting in higher Young’s modulus
values. Besides, these kinds of structure rearrangements could
take macroscopic-scale time. So, one could say that elastic
constants simulated in this study correspond to elastic
constants measured at extremely high frequencies where
relaxation motions are frozen.
C. The Role of Epoxy Chain Length. The explanation given
in section 3.1 on the observed dependence of material
properties on the length of the resin strands composing the
network is generally valid in explaining the mechanical
5. SUMMARY AND CONCLUSIONS
In the present study, we were able to generate a set of stressfree thermoset models with high degree of cure containing up
to 35000 atoms. These models were used to predict the
dependence of the thermo-mechanical properties of highly
cross-linked polymers on several parameters: the extent of
curing reaction, temperature, size of the model, and chain
length of the resin strands. It was seen that densities,
coefficients of thermal expansion, and glass transition temperatures of the systems are in good agreement with experimental
data. However, some of the elastic constants from our model
systems were found to be higher than in real macroscopic
samples.
The advantages and limitations of the methods used in this
study are summarized as follows.
A. Cross-Linking Method. The capture sphere growth
approach employed in this study to build polymer networks
allows one to achieve high extents of the reaction close to the
real systems. The effective on-the-fly monitoring of the model
quality and analysis of energetic and structural information
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enables to minimize the presence of internal stresses and
geometric distortions in the produced structure. The relaxation
procedure used to build thermoset structures has successfully
helped to overreach time-scale limitations of the MD method.
B. Static Deformation Approach for Mechanical
Properties Calculation. Large number of samples averaged
in the analysis of our data allows us to compensate for large
scattering in mechanical properties data that is usually caused
by unique distribution of matter in each nanoscopically small
simulation cell. While this deformation method makes it
possible to distinguish between properties of very similar
materials that have minor differences, it may have difficulties in
correctly predicting mechanical properties at high temperatures.
Though simulated structures were equilibrated at predefined
temperatures, this approach does not accurately take into
account dynamic effects that become more important at
elevated temperatures.
C. Size and Time Scale Effects on Thermo-Mechanical
Properties. Since systematic size effect on the material
properties is observed one may extrapolate to predict properties
at macroscopic sizes. However, due to size limitations of
atomistic simulations, simulation cell could not accommodate
large-scale events such as large voids and cooperative motions
that are important in macroscopic-sized polymers. Therefore,
the elastic constants obtained can be considered as properties
for “perfect” highly cross-linked polymer networks with no
large-scale events. Moreover, time scales used in all-atomistic
molecular dynamics simulations of thermosetting polymers do
not allow monitoring macroscopically long structural rearrangements, which are characteristic features of amorphous
polymers. In this sense, the properties obtained in the present
study allow to estimate contributions of dynamic effects as well
as large-scale defects and relaxation events into the macroscopic
properties of thermosetting materials.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: (N.B.S.) [email protected]; (M.T.)
[email protected].
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
This work was supported by the Low Density Materials
Program of the Air Force Office of Scientific Research Grant
Number: FA9550-09-1-0358. The authors gratefully acknowledge Dr. Charles Lee (AFOSR) for valuable discussions, and
the Air Force Research Laboratory DoD Supercomputing
Resource Center High Performance Computing for computer
time.
■
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