T. Okabe et al. / Polymer 54 (2013) 4660e4668
4663
transition state. If so, the energy of the process wherein the atoms
were statically moved from the activated complex toward the
molecular structure of the reactant was calculated. Moreover, using
the same technique, the energy of the process by which the atoms
were moved from the activated complex to the molecular structure
of the reaction product was also calculated. The energy curve, the
activation energy and the heat of formation were obtained from the
above-mentioned calculation process. In addition, electric charge of
the stable molecular structure was calculated.
The curing reaction at a molecule scale was modeled using MD
based on the energy curve obtained from MO. In the initial system
for our simulations, according to the compounding ratio of Table 1,
the monomer of base resin and curing agent was arranged at
random so that the number of epoxy groups was set to 40. The
compression and relaxation calculations were then performed to
achieve a density of 1.0 g/cm3.
In this study, the curing reaction was modeled using the
following steps.
Step 1: Judge whether reactive sites (i.e., each functional group)
approach each other within the distance of reaction judgment.
Step 2: Compare the reaction probability discussed below using
a random number.
Step 3: If it is judged that a reaction occurs, rearrangement of the
force field parameter and structural optimization calculation are
performed for the molecular chains that include the reactive sites.
Structural optimization is performed using the conjugate gradient
method (CG).
Fig. 6. Experiment (solid lines) and simulation (dashed lines) results of the conversion
for samples with a fixed curing agent.
Step 4: Apply heat of formation to the system as atomic velocity
after a reaction.
As stated above, the reaction probability was calculated when an
epoxy group and an amino group approach within a constant dis-
tance. Reaction probability k, which is described in the transition
state theory as the reaction rate, was calculated by the following
samples 5 and 6. However, the change in curing temperature for
each sample with mixed base resins was small, compared with the
three systems with mixed curing agents when the curing agent was
changed.
3. Molecular simulation
formula using activation energy DG, which is determined by MO.
ꢀ
ꢁ
Using atomistic simulation, we investigated the influence of
selection and mixture of base resins and curing agents on the
curing characteristics. In the present study, the curing process of an
epoxy resin was simulated by considering the effect of activation
energy, heat of formation, and polarization of a molecule. In the
appendix, we address why all of them should be considered to
simulate the curing process of epoxy resin.
D
RT
G
k ¼ A exp ꢁ
(2)
Here, A is an acceleration factor to minimize the scale difference
between the actual time and the time of MD. The value of 1 ꢂ 108
was used in this simulation. R is the gas constant, and T is the
approaching local temperature between an epoxy group and an
amino group. After calculating the reaction probability using Eq.
(2), we used the Monte Carlo method to judge whether a reaction
occurred or not. Using uniform random numbers P (0e1), we
assumed that a chemical reaction occurs when P < k, and that no
chemical reaction occurs when P > k. The judgment was performed
in all combinations that satisfied this condition when the distance
3.1. Simulation procedures
In this simulation, the curing reaction was modeled using the
molecular orbital method (MO) and MD. MO was used to calculate
the energy curve when one monomer of base resin and one
monomer of curing agent approached and reacted. Activation en-
ergy and heat of formation in the chemical reaction were obtained
from this curve. We used the semi-empirical MO in this simulation.
This method uses an experiential parameter, so it is less accurate
than a non-experiential MO; however, its calculation cost is low,
and it can be used with a large system. The semi-empirical MO
program is MOPAC2009, and the calculation used the PM6 Hamil-
tonian [14,15]. The input data were created, and the results were
displayed using the free software program Winmostar [16].
ꢀ
between functional groups was less than 6A which is four times the
equilibrium CeN bond length [17]. When it was judged that a re-
action occurred, the combination of atoms that applied interatomic
potential was re-arranged before and after the reaction. Further-
more, the geometry was optimized using CG with the molecule
chain containing the functional group judged as carrying out the
curing reaction. The velocity of each atom of a molecule to which
the reaction functional group belonged was updated using the
energy calculated by Eq. (3).
First, the molecular structures of reactant and reaction product
were created. In the molecular structure of the reactant, each
monomer was arranged so that the distance of the oxygen atom of
Kafter ¼ Kbefore þ H:O:F:
(3)
ꢀ
an epoxy group and the nitrogen atom of an amino group was 3A.
The molecular structure of a transition state (i.e., activated com-
plex) was then calculated from these structures. Next, vibration
analysis was conducted near the structure of an activated complex
to determine whether the activated complex obtained was in a
Here, Kafter is the kinetic energy of the molecule chain after the
reaction, Kbefore is that before the reaction, and H.O.F. is the heat of
formation. The velocity of each atom of the molecular chain was
calculated using the velocity-scaling method. The degree of
Okabe, Tomonaga
