Characterization of Cure Behavior in Epoxy Using Molecular Dynamics Simulation Compared with Dielectric Analysis and DSC

polymers
Article
Characterization of Cure Behavior in Epoxy Using Molecular
Dynamics Simulation Compared with Dielectric Analysis
and DSC
Shuang Yan * , Wolfgang Verestek , Harald Zeizinger and Siegfried Schmauder

 Institute for Materials Testing, Materials Science and Strength of Materials, University of Stuttgart,
 Pfaffenwaldring 32, 70569 Stuttgart, Germany; wolfgang.verestek@imwf.uni-stuttgart.de (W.V.);
 zeizinger@arcor.de (H.Z.); siegfried.schmauder@imwf.uni-stuttgart.de (S.S.)
 * Correspondence: shuang.yan@imwf.uni-stuttgart.de

 Abstract: The curing behavior of a thermosetting material that influences the properties of the mate-
 rial is a key issue for predicting the changes in material properties during processing. An empirical
 equation can describe the reaction kinetics of the curing behavior of an investigated material, which
 is usually estimated using experimental methods. In this study, the curing process of an epoxy resin,
 the polymer matrix in an epoxy molding compound, is computed concerning thermal influence
 using molecular dynamics. Furthermore, the accelerated reaction kinetics, which are influenced by
 an increased reaction cutoff distance, are investigated. As a result, the simulated crosslink density
 with various cutoff distances increases to plateau at a crosslink density of approx. 90% for the
 investigated temperatures during curing time. The reaction kinetics are derived according to the
 


 numerical results and compared with the results using experimental methods (dielectric analysis
 

 and differential scanning calorimetry), whereby the comparison shows a good agreement between
Citation: Yan, S.; Verestek, W.;
 experiment and simulation.
Zeizinger, H.; Schmauder, S.
Characterization of Cure Behavior in
 Keywords: epoxy; curing; reaction kinetics; molecular dynamics; dielectric analysis (DEA); differen-
Epoxy Using Molecular Dynamics
 tial scanning calorimetry (DSC)
Simulation Compared with Dielectric
Analysis and DSC. Polymers 2021, 13,
3085. https://doi.org/10.3390/
polym13183085
 1. Introduction
Academic Editor: Keon-Soo Jang Thermoset materials are widely used in the industrial sector because of their excellent
 mechanical properties at high temperatures and good chemical resistance. The non-cured
Received: 8 August 2021 thermosetting resin contains many monomers, which crosslink from a specific temperature
Accepted: 10 September 2021 and build three-dimensional macromolecules during processing. Since it causes changes in
Published: 13 September 2021 dynamical viscosity and temperature in the material, it is essential to know how the cure
 behavior changes during processing, e.g., the injection molding process. Several experi-
Publisher’s Note: MDPI stays neutral mental methods are available for monitoring and investigating the thermoset material’s
with regard to jurisdictional claims in
 cure behavior: differential scanning calorimetry (DSC), dielectric analysis (DEA), Fourier
published maps and institutional affil-
 transform infrared spectroscopy (FTIR), near-infrared spectroscopy (NIRs), and ultra-sonic
iations.
 techniques. However, using the measurement technology to monitor curing involves a
 high cost and effort and, in terms of the manufacturing process, the sensors cannot be
 adopted at every position where the curing state in the material needs to be measured. The
 molecular changes in morphological structure caused by the curing, which influence the
Copyright: © 2021 by the authors. material properties, are more easily predicted and analyzed with the help of molecular
Licensee MDPI, Basel, Switzerland. dynamics (MD) simulation.
This article is an open access article
 Molecular dynamics simulation is commonly used in thermoset polymers for ma-
distributed under the terms and
 terial design and to predict material properties. The previous studies [1–15] focused on
conditions of the Creative Commons
 predicting the thermomechanical properties (e.g., glass transition temperature, Young’s
Attribution (CC BY) license (https://
 modulus, thermal conductivity, and compressibility) in already cured molecular struc-
creativecommons.org/licenses/by/
 tures of the epoxy polymers using MD simulation. The molecular structures (Li et al. [4])
4.0/).

Polymers 2021, 13, 3085. https://doi.org/10.3390/polym13183085 https://www.mdpi.com/journal/polymers

Polymers 2021, 13, 3085 2 of 16

 and the crosslink density (Schichtel et al. [10]) strongly influence the thermomechanical
 properties. Okabe et al. [9] investigated the curing behavior of diglycidyl ether bisphenol
 A (DGEBA), tetra glycidyl diamino diphenyl methane (TGDDM), and eight polymers of
 diglycidyl ether bisphenol A (DGEBA8) as base resins, with 40 4-diaminodiphenylsulphone
 (44DDS) and 30 3-diaminodiphenylsulphone (33DDS) used as curing agents. The simu-
 lations showed an acceptable comparability of curing curves in different resin mixtures
 computed using MD simulation with experimental DSC measurements. Although the
 curing behavior was simulated in this study, it focused more on the resulting mechanical
 properties caused by different functional groups. In the study from Li et al. [16], the model
 was extended to a hybrid molecular dynamics/Monte Carlo (MD/MC) approach with
 reaction probabilities based on reaction path energies. Reactions of phenol resin with
 formaldehyde are investigated, and several thermal-mechanical properties agree with the
 experimental values.
 Unger et al. [17] attempted to validate the molecular modeling of a pure epoxy
 resin (base resins: bisphenol A diglycidyl ether, 1,4-Butanediol diglycidyl ether, Alkyl
 (C12-C14) glycidyl ether; hardener: α,ω-poly(oxypropylene)diamine, 3-(aminomethyl)-
 3,5,5-trimethyl-cyclohexanamine, 2,4,6-tris[(dimethylamino)methyl]phenol) according to
 the spectroscopic changes in correlation with crosslink density by means of in situ NIR,
 which showed a good agreement between numerical and experimental results.
 The above-mentioned studies have shown that MD can simulate the curing of epoxy
 networks. However, the cure behavior (reaction kinetics) for different material systems was
 not investigated in detail because the curing was considered simply as a modeling step,
 and only the thermomechanical properties were checked. Either that or united atom force
 fields, e.g., the DREIDING force field, were used, which has been shown by Li et al. [18] to
 result in different morphologies than all-atom force fields.
 Until now, a validation of molecular modeling for curing using experimental methods
 is still challenging due to the difference in time scale between the molecular dynamics
 simulation in the nanoseconds range and the experimental study in the seconds range.
 This work focuses on analyzing the cure behavior of an epoxy resin (base resin: bisphenol
 A diglycidyl ether (BADGE) and hardener: ethylenediamine (EN)) with consideration
 of the thermal influence of using MD simulation. In addition, the accelerated reaction
 kinetics influenced by an increased reaction cutoff distance are investigated. Finally, the
 numerical results are compared with experimental results derived by DSC and DEA,
 whereby DSC is only applicable in a laboratory environment and DEA is suitable for in
 situ cure monitoring. A commercial epoxy composite applied in industrial application
 is measured in our experimental investigations, whose polymer matrix (base resin and
 hardener) is modeled in the MD simulation. Reaction models as part of reaction kinetics
 for describing the cure mechanism are estimated simultaneously using MD, DSC and DEA
 methods for evaluating their comparability.

 2. Materials and Methods
 2.1. Material
 The resin system, also referred to as EP in the following text, consists of epoxy resin
 bisphenol A diglycidyl ether (BADGE) and a hardener, ethylenediamine (EN), that crosslink
 via a polymerization reaction (see Table 1). In this molecular modeling, three possibilities
 for the crosslinking reaction baetween the functional groups are considered:
 (a) Reaction between the epoxide group (R1 -CCO) and the primary amine group (R2 -NH2 );
 (b) Reaction between the epoxide group (R1 -CCO) and the secondary amine group (R3 -
 NH-R4 );
 (c) Reaction between the epoxide group (R1 -CCO) and the hydroxyl group (R5 -OH).

Polymers
Polymers 2021,
 2021, 13,13, x FOR
 x FOR PEER
 PEER REVIEW
 REVIEW 3 of
 3 of 17 17

 The
 The detailed
 detailed sequence
 sequence of of
 thethe crosslinking
 crosslinking reaction
 reaction with
 with molecular
 molecular structures
 structures as as
 ex-ex-
Polymers 2021, 13, 3085 amples,
 amples, is shown
 is shown in in Figure
 Figure A1A1in in Appendix
 Appendix A.A. 3 of 16

 Table
 Table 1. Overview
 1. Overview of of
 thethe resin
 resin system.
 system.
 Table 1. Overview of the resin system. Epoxy
 Epoxy Resin
 Resin Hardener
 Hardener
 bisphenol
 bisphenol A diglycidyl
 A diglycidyl ether ether
 Chemical
 Chemical formula
 formula Epoxy Resin ethylenediamine
 Hardener(EN)
 ethylenediamine (EN)
 (BADGE)
 (BADGE)
 Chemical formula
 Molar mass bisphenol A diglycidyl
 340.419 ether (BADGE)
 g/mol ethylenediamine
 60.1 g/mol (EN)
 Molar
 Molar mass
 mass 340.419 g/mol
 340.419 g/mol 60.1 g/mol
 60.1 g/mol
 Molecular
 Molecular
 Molecular formula
 formula
 formula 21C
 C21
 C H HO
 H2124
 24 O2444O4 C2HC
 C228H
 N882N
 H N22

 Chemical
 Chemical
 Chemical structure
 structure
 structure

 2.2.2.2. Molecular
 Molecular
 The detailed
 Modeling
 Modeling
 sequence of the crosslinking reaction with molecular structures as exam-
 ples,ForisFor computing
 computing
 shown thethe
 in Figure chemical
 chemical
 A1 reactions
 reactions
 in Appendix A.(bond(bond breaking,
 breaking, bond bond formation,
 formation, and and reactiv-
 reactiv-
 ity)ity)
 andand predicting
 predicting thethe material
 material properties,
 properties, there
 there arearetwo two typestypes of of force
 force fields:
 fields: fixed
 fixed forceforce
 fields
 2.2.
 fields (also
 Molecular
 (also called
 calledModelingclassical
 classical forceforce fields)
 fields) and and reactive
 reactive force
 force fields.
 fields. ForFor classical
 classical force
 force fields
 fields
 (e.g.,
 (e.g., ForCVFF,
 CVFF, PCFF,
 PCFF,
 computing DREIDING,
 DREIDING,
 the chemical COMPASS),
 COMPASS),
 reactions (bond bonds
 bonds areare defined
 defined
 breaking, bond explicitly,
 explicitly,
 formation, whilewhile maintain-
 andmaintain-
 reactivity)
 ing ing a low
 anda predicting computing
 low computing time
 time and
 the material and good
 good accuracy
 properties, accuracy
 there are [1,3,11,12,19–22].
 [1,3,11,12,19–22].
 two types of force In studies
 In studies
 fields: [8,23], [8,23],
 fixed force a reactive
 a reactive
 fields
 force
 force
 (alsofield field
 called (ReaxFF)
 (ReaxFF)
 classical is
 is force used
 used to to simulate
 simulate
 fields) the chemical
 the chemical
 and reactive reaction.
 forcereaction.
 fields. For The
 Theclassicaladvantage
 advantage force of
 offieldsReaxFF
 ReaxFF is is
 (e.g.,
 thatthat
 CVFF, thethe bond
 PCFF, bond formation
 formation
 DREIDING, is defined
 is COMPASS),
 defined implicitly
 implicitly
 bonds and areand happens
 happens
 defined dynamically
 dynamically
 explicitly, while during
 during compu-
 compu-
 maintaining a
 ting,
 ting,
 low which
 which
 computing needs
 needs time aand
 a very very high
 goodhigh computing
 computing
 accuracy capacity
 capacity
 [1,3,11,12,19–22].and and aInhigh
 a high complexity
 complexity
 studies [8,23], of of parameteri-
 parameteri-
 a reactive force
 zation.
 field
 zation. In In
 (ReaxFF)
 this thisisstudy,
 used
 study, the the
 to crosslinking
 simulate
 crosslinking reactions
 thereactions
 chemical between
 reaction.
 between The
 the the functional
 advantage
 functional groups
 of
 groups ReaxFFareare isimple-
 imple- that
 mented
 the bond
 mented in molecular
 formation modeling
 in molecular modeling
 is definedusing using
 implicitly a
 a fixed fixed
 and force
 forcehappens field (PCFF).
 dynamically during computing,
 field (PCFF).
 which The The
 needs software
 software a veryEMC EMC
 high (Enhanced
 computing
 (Enhanced Monte
 capacity
 Monte Carlo), developed
 and developed
 Carlo), a high byby
 complexity Pieter
 Pieter of J. J. in 't Veld
 parameterization.
 in 't Veld [24],[24],
 Inand
 and this LAMMPS
 study, the
 LAMMPS Simulator
 crosslinking
 Simulator (Large-scale
 (Large-scale Atomic/Molecular
 reactionsAtomic/Molecular
 between the functional Massively
 Massively groups Parallel,
 are
 Parallel, implemented Sandia
 Sandia Na- Na-
 in
 tional
 molecular
 tional Laboratories,
 modelingUSA)
 Laboratories, USA)
 using a[25–27]
 [25–27]fixedwere were
 force applied
 field
 applied to to
 (PCFF). prepare
 prepare thethe initial
 initial molecular
 molecular structures
 structures
 and and molecular
 The
 molecularsoftware modeling.
 EMC (Enhanced Monte Carlo), developed by Pieter J. in ’t Veld [24], and
 modeling.
 LAMMPS Simulator (Large-scale Atomic/Molecular Massively Parallel, resin Sandia National
 • • Step Step 1. Generation
 1. Generation of ofthethe initial
 initial molecular
 molecular network
 network of of thethe uncured
 uncured resin system.
 system.
 Laboratories, USA) [25–27] were applied to prepare the initial molecular structures and
 The The first
 first step
 step was was to to generate
 generate thethe initial
 initial stochastic
 stochastic mixture
 mixture of of
 thethe uncured
 uncured resin
 resin sys- sys-
 molecular modeling.
 tem using EMC software. EMC followed
 tem using EMC software. EMC followed a combined Monte Carlo/MD approach. The a combined Monte Carlo/MD approach. The
 • fixed Step
 force1. Generation
 field PCFF of
 (Sun the et initial
 al. [28])molecular
 was used network
 to of
 parameterize
 fixed force field PCFF (Sun et al. [28]) was used to parameterize intra- and inter-molecular the uncured
 intra- resin
 and system.
 inter-molecular
 interactions
 interactions
 The first between
 between
 step was thetothe atoms.
 atoms.
 generate TheThe
 theratioratio
 initialof of epoxy
 epoxy
 stochastic resin
 resin BADGE
 mixture BADGE theto
 of to the the hardener
 hardener
 uncured resin EN EN was
 system was
 2:1.2:1.
 usingTheThe epoxy
 EMC epoxy resin
 resin
 software. BADGEBADGE
 EMC amount
 amount
 followed isa Nis NBADGE
 combined
 BADGE = Monte
 = 1817,1817, which
 which included
 included
 Carlo/MD thethe
 approach. amount
 amount of
 Theoffixedthethe
 epoxide
 epoxide
 force field group
 group PCFF N(Sun
 N-CCO =et3634.
 = 3634.
 -CCO al.The The
 [28]) amount
 amount
 was of of
 used the the
 to hardener
 hardener
 parameterize ENEN NEN
 Nintra-
 EN = and
 = 908, 908, and and includes
 includes
 inter-molecular thethe
 primary
 primary
 interactions amine
 amine betweengroup
 group N the N
 -NH2 = 1818.
 = 1818.The
 atoms.
 -NH2 The The initial
 initial
 ratio density
 densityresin
 of epoxy after
 afterBADGE creation
 creationtoby by
 theEMC EMC
 hardener was
 was 1EN 1 g/cm³.
 g/cm³. was
 2:1. AsTheAsan an
 epoxy initial
 initialresin equilibration,
 equilibration,
 BADGE amount a short
 a short simulation
 issimulation
 NBADGE sequence
 sequence
 = 1817, which in microcanonical
 inincluded
 microcanonical the amount ensemble
 ensembleof the
 NVE NVE
 epoxide (with
 (with group limited
 limitedN-CCO maximum
 maximum= 3634. The displacement
 amount ofper
 displacement per
 the step),
 hardener
 step), canonical
 EN NEN
 canonical ensemble
 = 908, and
 ensemble NVT, NVT,
 includes
 and and iso-
 the
 iso-
 thermal-isobaric
 primary
 thermal-isobaricamine group ensemble
 ensemble N-NH2 NPT NPT
 = 1818.
 was was The performed in in
 initial density
 performed LAMMPS
 LAMMPS after creation
 withwith abytime
 a time EMCstep step of of
 was 10.25fs,
 g/cm
 0.25fs, at3at
 .
 a target
 As antemperature
 initial of 150
 equilibration, °C, and
 a
 a target temperature of 150 °C, and for a total time of 25ps. for
 short a total
 simulation time of 25ps.
 sequence in microcanonical ensem-
 •ble• NVE
 StepStep (with
 2. limited maximum
 2. Thermodynamical
 Thermodynamical displacement
 equilibration
 equilibration of of per
 thethe step),
 initial
 initial canonical
 uncured
 uncured ensemble
 molecular
 molecular NVT, and
 network.
 network.
 isothermal-isobaric ensemble NPT was performed in LAMMPS with a time step of 0.25 fs,
 The The initial
 initial uncured
 uncured molecular
 molecular mixture
 mixture was was equilibrated
 equilibrated forfor150ps150ps under under periodic
 periodic
 at a target temperature of 150 ◦ C, and for a total time of 25 ps.
 boundary conditions using NPT to achieve
 boundary conditions using NPT to achieve thermodynamical equilibrium. A time step thermodynamical equilibrium. A time step
 • size Step
 of 2.
 1.5fs Thermodynamical
 was applied for equilibration
 these and all of the
 following
 size of 1.5fs was applied for these and all following simulations by utilizing the rRESPA initial uncured
 simulations molecular
 by utilizing network.
 the rRESPA
 algorithm
 algorithm (Plimpton
 The (Plimpton
 initial et al.
 et al.
 uncured [26])[26])
 molecular implemented
 implemented
 mixtureinwas in LAMMPS.
 LAMMPS.equilibrated The The conditions
 conditions
 for 150ps(pressure (pressure
 under P: P:
 periodic423423
 atm;atm;
 boundary temperature
 temperatureconditionsT: T:
 150 150°C,°C,
 using 170
 NPT 170
 °C, °C,
 toand and 190190
 achieve °C),°C), which
 which
 thermodynamical werewere applied
 applied as as
 equilibrium. process
 process A time parameters
 parameters
 step size
 inofin
 DEA DEA measurements
 measurements in in
 an an actual
 actual injection
 injection mold,
 mold, werewere
 1.5 fs was applied for these and all following simulations by utilizing the rRESPA algo- implemented
 implemented for for
 the the equilibration
 equilibration
 procedure.
 procedure.
 rithm (Plimpton Table
 Table et 2 lists
 2 lists
 al. thethe
 [26]) density
 density
 implemented of of
 thethe uncured
 inuncured
 LAMMPS. molecular
 molecular
 The conditions system
 system in in a simulation
 a simulation
 (pressure P: 423 box box
 atm;
 (including
 (including
 temperature its dimension
 its dimension ◦
 T: 150 C, 170 and
 and C,◦ volume)
 volume)
 and 190 after
 ◦
 after equilibration
 C),equilibration
 which were at at three temperatures.
 three temperatures.
 applied as process parameters in
 DEA measurements in an actual injection mold, were implemented for the equilibration
 procedure. Table 2 lists the density of the uncured molecular system in a simulation box
 (including its dimension and volume) after equilibration at three temperatures.

Polymers 2021, 13, 3085 4 of 16

 Table 2. The density of the initial molecular mixture in a simulation cell equilibrated at 150 ◦ C,
 170 ◦ C, and 190 ◦ C.

 T Initial Density Initial Dimension Initial Volume
 [◦ C] [g/cm3 ] [nm3 ] [nm3 ]
 150 0.988 10.4 × 10.4 × 10.4 1131.3
 170 0.977 10.4 × 10.4 × 10.4 1144.2
 190 0.962 10.5 × 10.5 × 10.5 1162.1

 • Step 3. Identification of the reactive bonding and definition of criteria for the beginning
 of crosslinking.
 For crosslinking resin and hardener, the REACTER protocol [29,30] as implemented in
 LAMMPS was used. The cutoff distance Dcut , which correlated to the bond energy, was
 defined as a criterion for bond breaking and creation. Bond dissociation energy described
 the amount of energy that was required to split the bond homolytically. Numerically, the
 reactivity of the crosslinking reaction was changed through varying the cutoff distance
 Dcut . By increasing cutoff distance, the reactivity increased, and the duration until the
 material was fully crosslinked reduced. For example, during curing between epoxide and
 amine groups, if the distance between functional groups satisfied the boundary condition
 (≤cutoff distance Dcut ), the bond C-O in the epoxide group will break up and link up to the
 bond N-H, which also breaks simultaneously.
 As a reference, the bond dissociation length of the newly created bonds (O-H, C-N,
 and C-O) was calculated using the class2 force field (see Table 3). The bond potentials
 Vbond between the atom i and j were generated with Equation (1) using class2 bond style
 presented by Sun [31]:
 
 
2 
3 
4
 Vbond,ij rij = K2 · rij − rij.0 + K3 · rij − rij.0 + K4 · rij − rij.0 (1)

 Table 3. The bond dissociation length of the newly created bonds.

 Bond-Type (i– j) Unit O–H C–N C–O
 Bond dissociation energy Vbond,ij [32] [kJ/mol] 463 305 358
 Bond dissociation energy Vbond,ij [32] [kcal/mol] 110.7 72.9 85.56
 rij.0 [Å] 0.965 1.452 1.42
 K2 - 532.5062 327.1657 400.3954
 K3 - −1282.905 −547.899 −835.1951
 K4 - 2004.7658 526.5 1313.0142
 Bond dissociation length rij [Å] 1.4858 2.0717 1.9373

 Parameter rij.0 presents the equilibrium bond distance. K2, K3, and K4 are bond coefficients.
 This molecular modeling of crosslinking performed using the classic force field
 showed several advantages. It needed less computing capacity and performed more
 efficiently. The temperature influence and the effect of cutoff distance Dcut on the cure ki-
 netics were analyzed. The curing process was simulated at temperatures of 150 ◦ C, 170 ◦ C,
 and 190 ◦ C while varying the cutoff distance Dcut of the O-H bond from 1.5 Å→2.5 Å.

 2.3. Dielectric Analysis
 The dielectric analysis is a measuring procedure to study the material’s response to an
 applied electric field. The electrical current Ielectr (f) flowing through an electric capacitor, in
 which the epoxy material as dielectric material was located, was measured as a response
 to an alternating electric field depending on field frequency f. This analyzed the changes
 in dielectric properties, e.g., the relative permittivity εr of the polymer, depending on the
 changes in molecular and morphological structures in the material. Therefore, the the
 dialectric behavior of thermosets can be used to monitor the curing process which causes
 changes in dielectric properties. Permittivity ε, determined by the material polarization in

Polymers 2021, 13, 3085 5 of 16

 response to the electrical field, is a physical quantity describing the interaction between the
 electric field and the dielectric material. Relative permittivity εr is defined as the ratio of
 dielectric permittivity ε to vacuum permittivity ε0 .
 The electric field polarized the dipoles and freed the charges in the investigated
 material, leading to a change in relative permittivity related to the curing. The relative
 permittivity was calculated according to the measured electrical current and electrical
 voltage. The electrical current Ielectr resulted from dipole interaction in the capacitor’s
 electric field and the mobility of ionized molecules. The relative permittivity εr of the
 investigated material was a complex quantity with angular frequency ω = 2πf, shown
 in Equation (2).
 ε r (ω ) = ε0 (ω ) − i ·ε00 (ω ) (2)
 The term ε’ is the real part of relative permittivity that represents stored electric energy
 in the material. The imaginary part ε” of the relative permittivity (dielectric loss) describes
 the lost energy of the applied external electric field.
 In this study, dielectric measurements were performed using a sensor TMC 16/3
 (NETZSCH-Gerätebau GmbH, Selb, Germany) with 1 kHz at a pressure of 423 atm and
 temperatures of 150 ◦ C, 170 ◦ C, and 190 ◦ C. Figure 1 shows an example of the evaluation
 procedure of a DEA measuring result obtained at the temperature of 150 ◦ C. The change of
 relative permittivity εr during heating is shown in Figure 1a. Firstly the relative permittivity
 εr increases until t0 , which indicates the melting of the materials and the increased amount
 of available charge carriers in the material. After the onset of curing, the polymer system
 transits from a highly viscous to solid-state, and the content of free movable charges
 reduces, leading to a decrease in the relative permittivity εr and an increase in crosslink
 density, meaning that εr ~ 1/α. The maximal relative permittivity εr,0 at t0 means the curing
 begins (α ≈ 0%). After the material is almost fully cured, the relative permittivity εr reaches
 a plateau. Termination of curing (α ≈ 100%) is determined at time t1 when the relative
 permittivity’s change rate is less than 1%. Thereby, the value of relative permittivity εr,1
 is determined at time t1 . Based on the relative permittivity’s change in correlation with
 crosslink density α during curing, the crosslink density is estimated as a function of heating
 time according to Equation (3), shown in Figure 1b. In the further step, the reaction rate
 dα/dt in dependency on the crosslink density α, shown as dots in Figure 1c, can be derived
 regarding the results of the last step to determine the unknown parameter in reaction
 model f(α). Whereby the black curve of f(α) is the fitted line regarding the values of dα/dt
 vs. α.
 α DEA = (ε r − ε r,∆ )/ε r,∆
 (3)
 ε r,∆ = ε r,1 − ε r,0
 αDEA is the crosslink density determined using DEA. The term εr,0 means the maximal
 relative permittivity occurring at t0 ; εr,1 is the relative permittivity determined at time t1 ,
 when the reaction is terminated, and εr,∆ is the value of the difference between εr,0 and εr,1 .

 2.4. Differential Scanning Calorimetry
 DSC is a classic method for analyzing the correlation between the curing amount,
 temperature, and time, based on the amount of heat released from crosslinking reactions.
 The equipment DSC Q100 (TA Instruments–Waters LLC, New Castle, DE, USA) was used
 for studying the reaction kinetics. Non-isothermal measurements were performed with
 heating rates of 2 ◦ C/min, 5 ◦ C/min, 10 ◦ C/min, 15 ◦ C/min, and 20 ◦ C/min with nitrogen
 N2 as the purge gas.

lymers 2021, 13, x FOR PEER REVIEW 6 o
 Polymers 2021, 13, 3085 6 of 16

 Figure 1. Figure 1. Evaluation
 Evaluation procedure
 procedure of DEA
 of DEA result:
 result: (a)(a)maximum
 maximum andandminimum
 minimum of relative permittivity,
 of relative (b) crosslink
 permittivity, (b) density
 crosslink density
 based on DEA result, and (c) reaction rate in dependency of crosslink
 based on DEA result, and (c) reaction rate in dependency of crosslink density. density.

 The thermally stimulated depolarization currents method [33,34] and DSC are applica-
 2.4. Differential Scanning determination
 ble for an experimental Calorimetry of the activation energy, while the DSC method has
 been
 DSCemployed in our
 is a classic study. Concerning
 method the results
 for analyzing DSC measurements,
 the ofcorrelation between thethe
 activation
 curing amou
 energy EA and the pre-exponential A could be determined using the Kissinger method [35].
 temperature, andthe
 For predicting time, basedparameter
 unknown on the amount ofmodel
 in reaction heat released from crosslinking
 f(α), the following procedure is reactio
 The equipment DSC the
 performed. During Q100 (TAprocess,
 curing Instruments–Waters
 the change of heatLLC,
 flow inNew Castle,ofDE,
 dependency timeUSA)
 and was u
 for studying
 temperaturethe reactionbykinetics.
 is detected DSC shownNon-isothermal measurements
 in Figure 2a. According were
 to the resultant performed
 change of w
 heat amount, temperature- and time-dependent crosslink density is a ratio
 heating rates of 2 °C/min, 5 °C/min, 10 °C/min, 15 °C/min, and 20 °C/min with nitro of the released
 heat amount Hrelease to the maximum heat amount Hmax in a fully crosslinked state, using
 N2 as the purge gas.
 Equation (4) (see Figure 2b). Figure 2c shows the reaction rate in dependency on the
 The thermally
 crosslinking degree stimulated depolarization
 to estimate the currents
 unknown parameter method
 in the reaction[33,34] and DSC are ap
 model f(α):
 cable for an experimental determination of the activation energy, while the DSC meth
 has been employed in our study.α DSC = Hrelease /H
 Concerning the
 maxresults of DSC measurements, (4) the a
 vation energy
 The termEAαDSC
 andrepresents
 the pre-exponential A could
 the crosslink density be determined
 determined using DSC. using the Kissin
 The Hrelease
 method
 is the[35]. For heat
 released predicting
 amount the unknown
 by the parameter
 material in in cured
 the partially reaction
 state,model f(α),
 and Hmax is the
 the follow
 maximum heat amount released by the material in the fully cured state.
 procedure is performed. During the curing process, the change of heat flow in depe
 ency of time and temperature is detected by DSC shown in Figure 2a. According to
 resultant change of heat amount, temperature- and time-dependent crosslink density
 ratio of the released heat amount Hrelease to the maximum heat amount Hmax in a fully cro
 linked state, using Equation (4) (see Figure 2b). Figure 2c shows the reaction rate in
 pendency on the crosslinking degree to estimate the unknown parameter in the react
 model f(α):

Polymers 2021, 13, x FOR PEER REVIEW 7 of 17
 Polymers 2021, 13, 3085 7 of 16

 Figure
 Figure2.2.Evaluation
 Evaluation procedure ofDSC
 procedure of DSCresult
 result measured
 measured under
 under thethe non-isothermal
 non-isothermal condition:
 condition: (a) change
 (a) change of heatofflow
 heatduring
 flow dur-
 ing the curing process with heating ◦
 rate 10 °C/min, (b) crosslink density, and (c) reaction rate in dependency
 the curing process with heating rate 10 C/min, (b) crosslink density, and (c) reaction rate in dependency of curing amount.of curing
 amount.
 3. Results and Discussion
 3. Results
 3.1. Curing and Discussion
 Behavior
 3.1. Curing Behavior
 Once the initial uncured molecular mixtures were finished to equilibrate with NPT,
 the crosslinking simulations
 Once the initial uncured were performed
 molecular with varied
 mixtures were cutoff
 finisheddistances for O-H with
 to equilibrate bondsNPT,
 from 1.48 Å up to 2.5 Å at 150 ◦ C, 170 ◦ C, and 190 ◦ C (shown in Figure 3a–c). The number
 the crosslinking simulations were performed with varied cutoff distances for O-H bonds
 of completed
 from 1.48 Å upreactions
 to 2.5 Å isat counted
 150 °C, 170 during simulation.
 °C, and Since the
 190 °C (shown epoxide
 in Figure group
 3a–c). The-CCO
 number
 participates in each of the modeled crosslinking reactions, Equation
 of completed reactions is counted during simulation. Since the epoxide group -CCO (5) is derived for par-
 estimating the crosslink density as the ratio of the number of already utilized epoxide
 ticipates in each of the modeled crosslinking reactions, Equation (5) is derived for estimat-
 groups Nu,CCO to the total number of epoxide groups Ntotal,CCO .
 ing the crosslink density as the ratio of the number of already utilized epoxide groups
 The computing time and effort, with a calculated bond dissociation length (Dcut = 1.48 Å),
 Nu,CCO to the total number of epoxide groups Ntotal,CCO.
 are enormous until the material is fully cured as obtained by the MD simulation. Then,
 The
 further computingare
 simulations time and out
 carried effort,
 withwith
 the aincreasing
 calculated bonddistance
 cutoff dissociation
 Dcut oflength
 1.48 Å(D cut = 1.48
 up to
 Å), are enormous until the material is fully cured as obtained
 2.5 Å, which increases the reactivity of the chemical reactions, respectively. by the MD simulation. Then,
 further simulations are carried out with the increasing cutoff distance D
 Based on computing results, the cutoff distance has a significant influence on theup to cut of 1.48 Å
 2.5 Å, which
 curing increases
 reaction’s activity.the
 Atreactivity of the chemical
 three temperatures, reactions,
 the molecules respectively.
 crosslink very slowly with
 a lowBased
 cutoffon computing
 distance of 1.5 results,
 Å, and itthe cutoff
 is to distancethat
 be expected hasa asignificant
 significant influence
 change on the cur-
 in crosslink
 ing reaction’s
 density is only activity. At athree
 visible over very temperatures,
 long simulationthe molecules
 time. However, crosslink
 since thevery
 cutoff slowly with a
 distance
 is setcutoff
 low to larger than 1.6ofÅ,1.5
 distance theÅ,crosslinking
 and it is toprocess becomes
 be expected significantly
 that faster.
 a significant Additionally,
 change in crosslink
 density is only visible over a very long simulation time. However, since the cutoff distance

Polymers 2021, 13, x FOR PEER REVIEW 8 of 17

Polymers 2021, 13, 3085 8 of 16

 is set to larger than 1.6 Å, the crosslinking process becomes significantly faster. Addition-
 ally, the maximal achieved crosslinking amount also increases. A similar effect of cutoff
 the maximal
 distance achieved
 on the crosslinking
 crosslinking behavioramount
 is also also increases.
 observed in theAreport
 similar
 byeffect
 Unger of et
 cutoff distance
 al. [17].
 on the crosslinking behavior is also observed in the report by Unger et al. [17].

 (a) (b) (c)
 Figure 3. Curves of crosslink density depending on curing
 Figure curing time
 time concerning
 concerning the
 the influences
 influences of
 of cutoff
 cutoff distances
 distances (1.48
 (1.48 Å→2.5
 Å→2.5 Å)
 Å) at (a) 150
 ◦ °C, (b) 170
 ◦ °C, and (c)
 at (a) 150 C, (b) 170 C, and (c) 190 C.190
 ◦ °C.

 Generally,
 Generally,aahigher
 higherenvironment
 environmenttemperature
 temperature(e.g., (e.g.,190 190°C)◦ C)excites
 excitesstronger
 strongermolecu-
 molecular
 lar oscillations,
 oscillations, whichcause
 which causeaahigher
 higher molecular
 molecular diffusion
 diffusionininthe thesimulation
 simulation system.
 system. Thus,
 Thus,
 while
 whilethethemolecular
 molecularmovement
 movementbecomes becomes more
 moreactive
 active at ata higher
 a higher temperature
 temperature of 190 °C ◦ C
 of 190
 than
 thanatataalower
 lowertemperature
 temperatureofof150 150°C,◦ C,
 thetheepoxy
 epoxyresin,
 resin,over time,
 over time,hashas
 a higher
 a higherprobability
 probability
 for meeting a hardener and linking up. Respectively, an achievement
 for meeting a hardener and linking up. Respectively, an achievement of a higher of a higher crosslink
 crosslink
 density
 densityatataahigher
 highertemperature
 temperatureisisexpected,expected,whichwhich is isdemonstrated
 demonstrated in inourour
 simulation
 simulation
 study
 studywith
 withthe thecutoff
 cutoffdistance
 distanceofof 1.61.6
 Å.Å.However,
 However, thethe
 influence
 influence of the rising
 of the environmental
 rising environmental
 temperature
 temperatureononthe crosslink
 the crosslink density
 density is noismore significantly
 no more significantly recognized in respect
 recognized of nu- of
 in respect
 merical
 numericalresults computed
 results computed withwith
 a cutoff distance
 a cutoff of ≥ 1.7
 distance of ≥ Å.1.7 Å.
 As
 Asdescribed
 describedininstep step2 2ininSection
 Section 2.2, thethe
 2.2, molecular
 molecular mixture
 mixture is previously
 is previously equilibrated
 equilibrated
 at the given temperature, and all molecules distribute homogeneously
 at the given temperature, and all molecules distribute homogeneously in space. in space. Since theSince
 diffusion of molecules is a long-term process which only
 the diffusion of molecules is a long-term process which only occurs at a lower occurs at a lower cutoff distance
 cutoff
 (e.g., 1.6 Å),
 distance (e.g.,indicating a longeracuring
 1.6 Å), indicating longer time
 curinguntil
 timethe untilmaximum
 the maximum crosslink density
 crosslink is
 density
 achieved, the influence of the increasing temperature (150
 is achieved, the influence of the increasing temperature (150 C→190 C) on diffusion °C→190 ◦ °C) on ◦diffusion and
 the
 andcorresponding
 the corresponding crosslink density
 crosslink is observed
 density in Figure
 is observed 3. By increasing
 in Figure the cutoff
 3. By increasing thedis-
 cutoff
 tance (≥ 1.7
 distance (≥Å), 1.7the
 Å), molecules
 the molecules havehavea higher probability
 a higher probability of reacting
 of reacting with the the
 with neighboring
 neighboring
 reaction
 reactionpartners
 partnersinina homogenized
 a homogenized polymer
 polymer mixture,
 mixture, while the simulated
 while the simulated curing time is
 curing time
 reduced, therefore preventing molecular diffusion. As a result,
 is reduced, therefore preventing molecular diffusion. As a result, excited intermolecular excited intermolecular in-
 teractions
 interactions areare
 notnot
 significantly
 significantly observable
 observable duedue
 to the increasing
 to the increasing temperature,
 temperature, which leads
 which leads
 to a similar crosslink density value simulated with the equal
 to a similar crosslink density value simulated with the equal cutoff distance at varying cutoff distance at varying
 temperatures
 temperatures(150 (150°C→190
 ◦ C→190 °C).
 ◦ C).
 Additionally,
 Additionally, the molecularsimulation
 the molecular simulation demonstrates
 demonstrates thatthata fully
 a fullycured
 curedmolecular
 molecular
 network (α ≈ 100%) is hardly realizable. During the curing process,
 network (α ≈ 100%) is hardly realizable. During the curing process, more crosslinked more crosslinked mo-
 lecular networks are created and grow continuously to a larger
 molecular networks are created and grow continuously to a larger size. The large molecules size. The large molecules
 restrict
 restrictthe
 theremaining
 remaining epoxy
 epoxy resin andand
 resin hardener fromfrom
 hardener free movement
 free movement and finding the re-the
 and finding
 action
 reactionpartner.
 partner.
 , Nu,CCO
 α MD= = (5) (5)
 N.total.CCO
 Theterm
 The termααMD MD represents
 represents thethe crosslink
 crosslink density
 density determined
 determined usingusing molecular
 molecular dynamics
 dynamics
 simulation.NN
 simulation. u,CCO is
 u,CCO the
 is number
 the number of epoxide
 of epoxide groups
 groupsalready
 already utilized
 utilizedduring curing,
 during andand
 curing,
 NNtotal,CCO is
 total,CCO the
 is total
 the number
 total numberof epoxide
 of groups.
 epoxide groups.
 Table
 Table44 lists
 lists the
 the densities
 densities of the molecular mixtures at at the
 the end
 endof ofthe
 thecuring
 curingsimulations
 simula-
 tions for cutoff
 for cutoff distances
 distances Å→2.5
 (1.48(1.48 Å→2.5 at ◦150
 Å) atÅ)150 ◦ C, and
 °C, 170
 C, 170 °C, and ◦ C. °C.
 190 190 Referenced
 Referenced to the
 to the initial
 densities of the uncured molecular system in Table 2, the molecular mixture’s density at
 the end of simulation increases when the achieved crosslink density grows by increasing
 the cutoff distances shown in Figure 3. Additionally, the density decreases during the

initial densities of the uncured molecular system in Table 2, the molecular mixture’s den-
Polymers 2021, 13, 3085 sity at the end of simulation increases when the achieved crosslink density grows 9 ofby
 16 in-
 creasing the cutoff distances shown in Figure 3. Additionally, the density decreases dur-
 ing the increasing temperature of 150 °C→190 °C, which correlates to the thermal expan-
 sion of the molecular system influenced by a higher temperature.
 increasing temperature of 150 ◦ C→190 ◦ C, which correlates to the thermal expansion of
 the molecular
 Table 4. Density system influenced
 of the molecular by a higher
 mixture temperature.
 at the end of the crosslinking simulation for Dcut = 1.48
 Å→2.5 Å at 150 °C, 170 °C, and 190 °C.
 Table 4. Density of the molecular mixture at the end of the crosslinking simulation for Dcut = 1.48 Å→2.5 Å at 150 ◦ C,
 170 ◦ C, and 190 ◦ C. Density [g/cm³]
 T [°C]
 Dcut = 1.48 Å Dcu = 1.5 Å Dcu =3 1.6 Å Dcu = 1.7 Å Dcu = 1.8 Å Dcu = 2.5 Å
 Density [g/cm ]
 T [◦ C] 150 0.984 0.992 1.050 1.077 1.078 1.080
 Dcut = 1.48 Å D = 1.5 Å Dcu = 1.6 Å Dcu = 1.7 Å Dcu = 1.8 Å D = 2.5 Å
 170 cu 0.980 0.982 1.058 1.074 1.075 cu 1.077
 150 0.984 0.992 1.050 1.077 1.078 1.080
 190 0.960 0.963 1.028 1.063 1.064 1.068
 170 0.980 0.982 1.058 1.074 1.075 1.077
 190 0.960 0.963 1.028 1.063 1.064 1.068
 Changes in the molecular structures regarding the result with a cutoff distance of 1.8
 Å at 190 °C are schematically shown as an example in Figure 4. The colors of electrically
 Changes in the molecular structures regarding the result with a cutoff distance of
 charged atoms change according to changes in the atoms’ charges during bond breaking
 1.8 Å at 190 ◦ C are schematically shown as an example in Figure 4. The colors of electrically
 and formation.
 charged atoms change according to changes in the atoms’ charges during bond breaking
 and formation.

 (a)

 (b)
 Figure 4. 4.
 Figure Changes
 Changesininmolecular
 molecularstructures
 structuresduring
 during the
 the curing process
 processwith
 withthe
 thecutoff
 cutoffdistance
 distanceofof
 1.81.8
 ÅÅ at at ◦ C:°C:
 190190 (a) (a) total
 total
 view of of
 view thethe
 simulation
 simulationbox
 boxwith
 withperiodic
 periodic boundary, and(b)
 boundary, and (b)growth
 growthofofthe
 themolecule
 moleculeasas
 anan example.
 example.

 Thereaction
 The reactionspeed
 speed simulated
 simulated by bymolecular
 moleculardynamics
 dynamicsisissignificantly higher
 significantly higherthan thethe
 than
 experimentally determined value. In contrast to the experiment in which a
 experimentally determined value. In contrast to the experiment in which a larger test spec- larger test
 specimen
 imen is considered
 is considered in millimeter
 in millimeter scale,the
 scale, thematerial
 material volume
 volume ininnanometer-scale
 nanometer-scale is taken
 is taken
 into account in MD simulation, shown in Figure 5. An ideal thermodynamic state is set in
 into account in MD simulation, shown in Figure 5. An ideal thermodynamic state is set in
 molecular dynamics, under a constant pressure and temperature with a periodic boundary
 molecular dynamics, under a constant pressure and temperature with a periodic bound-
 condition, meaning that there is no heat transfer within the material and no heat loss
 ary condition, meaning that there is no heat transfer within the material and no heat loss
 occurs between the edge and the environment. In the experimental analysis, heat transfer
 occurs
 withinbetween the edge
 the material and theexchange
 and thermal environment. Inthe
 within theenvironment
 experimentalcauses
 analysis, heat transfer
 the imbalance
 within the material and thermal exchange within the environment causes
 of the temperature distribution across the layer thickness in the polymer matrix from the imbalance
 EP.
 ofAdditionally,
 the temperature distribution
 monomers across the layer
 are homogeneously thickness
 distributed in the
 in the polymer
 simulation matrixleading
 volume, from EP.
 to a shorter balancing time of the molecular movements and a faster reaction. Furthermore,
 besides pure epoxy resin and hardener, the commercial EP material investigated in the

Additionally, monomers are homogeneously distributed in the simulation volume, lead
 Polymers 2021, 13, 3085 ing to a shorter balancing time of the molecular movements and a faster reaction. 10 of 16 Furthe

 more, besides pure epoxy resin and hardener, the commercial EP material investigated i
 the experiment contains additives such as the reaction inhibitor, which delays and slow
 experiment
 down contains
 the curing additives such as the reaction inhibitor, which delays and slows down
 reaction.
 the curing reaction.

 (a) (b)
FigureFigure
 5. Macro- and and
 5. Macro- micro-structure
 micro-structureofofthe
 the epoxy resin:
 epoxy resin: (a)(a) Nano-CT
 Nano-CT ofepoxy
 of the the epoxy resin,
 resin, and (b) and
 view (b) view of simulation
 of simulation cell in cell
 molecular
in molecular dynamicssimulation.
 dynamics simulation.

 3.2.3.2. Reaction Kinetics Using Molecular Dynamics Simulation
 Reaction Kinetics Using Molecular Dynamics Simulation
 Reaction kinetics describe the rate of the chemical reaction, which considers tempera-
 Reaction
 ture, pressurekinetics describe the
 and concentration rate of the
 dependency: chemical
 k(T), f(α), and reaction, which considers
 h(p) (see Equation (6)). tempe
 ature, pressure and concentration dependency: k(T), f(α), and h(p) (see Equation (6)).
 dα/dt = k( T )· f (α)·h( p) (6)
 / = ( ) ∙ ( ) ∙ ℎ( ) (6
 Supposing that the pressure p is kept constant during the chemical reaction process, the
 Supposing
 pressure that the
 dependency pressure
 can be ignored,pand
 is kept constant
 Equation during into
 (6) is simplified the Equations
 chemical(7)
 reaction
 and (8). proces
 the pressure dependency can be ignored, and Equation (6) is simplified into Equations (7
 and (8). dα/dt = k( T )· f (α) (7)

 The temperature dependency, / k(T) ==A ( )
 ·exp(-E∙ ( )
 A /(Rgas ·T)), generally known as the (7
 Arrhenius equation, consists of the material-related activation energy EA and the pre-
 The temperature
 exponential term A: dependency, k(T) = A·exp(-EA

 /(Rgas·T)), generally known as the A
 rhenius equation, consistsdα/dt
 of the
 =A material-related activation
 ·exp − E A / R gas · T · f (α). energy EA and the(8)pre-expo
 nential The
 termreaction
 A: model f(α) describes the reaction mechanisms in dependency on the
 proportion of crosslinked / 
 molecules.
 = There are already
 ∙ (− /( empirical equations available (such
 ∙ )) ∙ ( ). (8
 as nth-order catalytic, Avrami–Erofeev, Prout–Tompkins, and Sestak–Berggren reaction
 model
 The [36,37]),
 reactionwhose
 model unknown factors can
 f(α) describes the be determined
 reaction by regression
 mechanisms methods. Since
 in dependency on the pro
 the empirical model (nth-order catalytic) best fits the numerical results for this
 portion of crosslinked molecules. There are already empirical equations available (such a investigated
 material, only this model is considered further in our study.
 nth-order catalytic, Avrami–Erofeev, Prout–Tompkins, and Sestak–Berggren reactio
 Concerning the difference in the time scale between the molecular dynamics simu-
 model [36,37]), whose unknown
 lation in the nanoseconds range andfactors can be determined
 the experimental study in by theregression methods.
 seconds range, the Sinc
 theevaluation
 empiricalprocedure
 model (nth-order catalytic)
 for determining best model
 the reaction fits the
 f(α)numerical
 regarding theresults for this invest
 experimental
 gated material,
 results can alsoonly this model
 be applied for theiscalculation
 considered further
 based on theinsimulation
 our study. results. As shown
 Concerning the difference in the time scale between the molecular dynamics simula
 tion in the nanoseconds range and the experimental study in the seconds range, the eva

Polymers 2021, 13, x FOR PEER REVIEW 11 of 17

Polymers 2021, 13, 3085 11 of 16

 uation procedure for determining the reaction model f(α) regarding the experimental re-
 sults can also be applied for the calculation based on the simulation results. As shown in
 inFigure
 Figure6a–c,
 6a–c,thethetime
 time dependent
 dependent reaction
 reaction raterate is calculated
 is calculated based based on crosslink
 on crosslink density
 density de-
 depending on curing time, which is simulated using various
 pending on curing time, which is simulated using various cutoff distances and tempera-cutoff distances and tem-
 peratures in Figure
 tures in Figure 3. With
 3. With a large
 a large cutoffcutoff (2.5
 (2.5 Å), Å),
 the the maximal
 maximal reaction reaction rate is achieved
 rate is achieved imme-
 immediately after the beginning of curing. As the cutoff distance
 diately after the beginning of curing. As the cutoff distance decreases, the maximal decreases, the maximal
 reac-
 reaction
 tion raterate decreases
 decreases untiluntil no reaction
 no reaction occurs
 occurs during during the simulation
 the simulation time. time. Theofdata
 The data reac-of
 reaction
 tion raterate and
 and thethe associated
 associated crosslink
 crosslink densityatatthe
 density thesame
 samecuring
 curingtime
 timearearetransferred
 transferredtoto
 Figure
 Figure6d–f
 6d–ftotofurther
 furtherexplain
 explainthe thedetermination
 determination of of the
 the unknown parametersin
 unknown parameters in reaction
 reaction
 modelf(α).
 model f(α).After
 Afterthat,
 that,thethereaction
 reactionraterate curve
 curve is is normalized
 normalized according
 according to to the
 the maximum
 maximum
 reaction
 reactionrate,
 rate,related
 relatedtotothe theassociated
 associatedcutoff
 cutoffdistance
 distanceand andtemperature.
 temperature. SinceSince the tempera-
 temper-
 ture
 aturedependency
 dependency is considered
 is considered by by
 factor k(T)
 factor in in
 k(T) Equation
 Equation (7),(7),
 thethe
 data (normalized
 data (normalized dα/dt
 dα/dtvs.
 simulated
 α),vs. with with
 α), simulated the same cutoff
 the same distance
 cutoff at different
 distance temperatures,
 at different temperatures,are are
 merged
 merged for for
 the
 subsequent regression
 the subsequent procedure
 regression procedure(see Figure 7). 7).
 (see Figure

 (a) (b) (c)

 (d) (e) (f)
 Figure
 Figure 6. (a–c)
 6. (a–c) Curves
 Curves of of reaction
 reaction rate
 rate dependingon
 depending oncuring
 curingtime
 timeand
 and(d–f)
 (d–f)curves
 curves of
 of reaction
 reaction rate depending
 depending on
 oncrosslink
 crosslink
 density
 density concerning
 concerning thethe influence
 influence of of cutoff
 cutoff distance(1.48
 distance (1.48ÅÅ→2.5
 →2.5 Å)Å)atat150
 150◦°C, 170 ◦°C,
 C, 170 and 190
 C, and 190 °C.
 ◦ C.

 Figure7 7shows
 Figure showsthe thenormalized
 normalizedreaction
 reaction rate
 rate in
 in the
 the dependency
 dependency of of crosslink
 crosslinkdensity
 density
 computedusing
 computed usingdifferent
 differentcutoff
 cutoffdistances.
 distances.The The plot
 plot of reaction rate
 rate scatters
 scattersless
 lessby
 byincreas-
 increas-
 ingcutoff
 ing cutoffdistance.
 distance. However,
 However, supposing
 supposing that thatthe
 thecutoff
 cutoffdistance
 distance is is
 setset
 tootoo
 high, such
 high, suchas at
 as
 at1.9
 1.9Å,
 Å,the
 thechemical
 chemicalreaction
 reactiontakes
 takesplace
 place tootoo
 fastfast
 so so
 that thethe
 that maximum
 maximum reaction raterate
 reaction is al-is
 ready reached,
 already reached,even
 evenatata alow
 lowcrosslink
 crosslink density (Figure
 density 7d).7d).
 (Figure The The
 reaction model
 reaction f(α) isf(α)
 model esti-is
 mated based
 estimated basedononthe
 thenumerical
 numericalresults
 results(normalized
 (normalizeddα/dtdα/dtvs.
 vs.α)α)with
 withthethecutoff
 cutoffdistance
 distanceof of
 1.8 Å, whereby the regression result of the reaction model (nth-order catalytic)
 1.8 Å, whereby the regression result of the reaction model (nth-order catalytic) is presented is pre-
 insented
 Figurein7c.
 Figure 7c.

 3.3. Comparison between Simulation and Experimental Results
 3.3.1. Reaction Kinetics Using DEA
 DEA measurements were performed under the pressure of 423 atm and at temperatures
 of 150 ◦ C, 170 ◦ C, and 190 ◦ C. The change in reaction rate depending on the crosslinking
 amount, which is calculated according to the relative permittivity’s changes at different
 temperatures, is plotted in Figure 8. At the beginning of curing (α ≈ 0%), a large number of
 ionized charge carriers in an alternating electrical field lead to the maximal relative permittivity

Polymers 2021, 13, 3085 12 of 16

 in the material. The number of ions reduces, and the ions’ mobility is restricted strongly
 by the increasing crosslink density, reflecting the reduced relative permittivity in a DEA
 measurement. The relative permittivity converges when the material almost finishes curing
 Polymers 2021, 13, x FOR PEER REVIEW 12 of 17
 (α ≈ 100%), even though the small content of reaction agents that cannot find a reaction
 partner is still available. The regression results using Trust-Region algorithms for determining
 the unknown parameters regarding the DEA measurements are listed in Table 5.

 (a) (b)

 (c) (d)
 Figure7.7.Reaction
 Figure Reactionrate
 ratedepending
 dependingon
 on crosslink
 crosslink density
 density computed using
 using varied
 varied cutoff
 cutoffdistances
 distancesofof(a)
 (a)1.6
 1.6Å,Å,(b)
 (b)1.7
 1.7Å,Å,(c)
 (c)1.8
 1.8ÅÅand
 and(d)
 (d)1.9
 1.9Å.
 Å.

 Table 5. Parameters3.3. Comparison
 in the between
 reaction model Simulation
 estimated and Experimental
 by regression Results interval of 95 %.
 with a confidence
 3.3.1. Reaction Kinetics Using DEA
 logA
 Method Expression of Reaction Model
 DEA f(α)
 measurements cα were performed
 nα Kα the pressure of 423 atm
 under R2 and atRMSE
 tempera-
 [1/s]
 tures of 150 °C, 170 °C, and 190 °C. The change in reaction rate depending on the cross-
 MD 0.3218 3.392 28.03 7.57 0.99 0.03
 linking(nOk):
 nth-order catalytic amount, which is calculated according to the relative permittivity’s changes at dif-
 DSC nα 0.03 1.25 150.4 7.44 0.99 0.05
 f (α) = cα · (1 −ferent
 α) ·(1temperatures,
 + Kα · α ) is0.03
 plotted in1.41
 Figure 8. At the beginning
 DEA 145.8 7.45 of curing
 0.97 (α ≈ 0%), a large
 0.03
 number of ionized charge carriers in an alternating electrical field lead to the maximal
 relative permittivity in the material. The number of ions reduces, and the ions’ mobility is
 restricted strongly by the increasing crosslink density, reflecting the reduced relative per-
 mittivity in a DEA measurement. The relative permittivity converges when the material
 almost finishes curing (α ≈ 100%), even though the small content of reaction agents that
 cannot find a reaction partner is still available. The regression results using Trust-Region
 algorithms for determining the unknown parameters regarding the DEA measurements

Polymers 2021, 13, x FOR PEER REVIEW 13 of 17
Polymers 2021, 13, 3085 13 of 16

 Figure
 Figure 8. Comparison of
 8. Comparison of the
 the kinetic
 kinetic model
 model derived
 derived using
 using MD, DEA, and DSC methods.

 Table 5. Parameters in the
 3.3.2. reactionKinetics
 Reaction model estimated by regression with a confidence interval of 95 %.
 Using DSC
 Concerning DSC measurements, the unknown parameters logA in Equation (8) are deter-
 Method Expression of Reaction
 mined in Model f(α) First, the
 two steps. reaction model f(α) 
 is fitted with the Trust-Region 
 algorithm,
 [1/s]
 MD whereby the results are0.3218listed in Table
 3.3925. Figure 28.038 shows the fitted
 7.57 curve regarding
 0.99 0.03 the
 nth-order catalytic
 nth-order(nOk):
 catalytic reaction model to the reaction rate, resulting in the dependency of
 DSC 0.03 1.25 150.4 7.44 0.99 0.05
 ( ) = ∙ (1 −the
 ) crosslinking
 ∙ (1 + ∙ )amount measured with different heating rates β. Second, the Kissinger
 DEA 0.03 1.41 145.8 7.45 0.97 0.03
 method was applied for determining the activation energy EA and the pre-exponential
 term A. Equation (9) can be evaluated according to a linear relationship y = aK · x + bK ,
 3.3.2. Reaction Kinetics Using DSC
 reported by Yan et al. [38]. Then, the activation energy EA and the pre-exponential term A
 Concerningusing
 are determined DSC Equations
 measurements,
 (10) andthe (11):
 unknown parameters in Equation (8) are deter-
 mined in two steps. First, the reaction model f(α) is fitted with the Trust-Region algorithm,
 A· Rthe
    
 whereby the results are listed β in Table 5. E AFigure 8 shows gas fitted
 0 curve regarding the nth-
 ln =− + ln − · f (α P ) (9)
 order catalytic reaction modelTP 2 to the R gasreaction
 · TP rate, resulting
 EA in the dependency of the
 crosslinking amount measured with different heating rates β. Second, the Kissinger
 method was applied for determining E
 aK = − A the => activation
 E A = 77.66 energy
 kJ/mol EA and the pre-exponential (10)
 term A. Equation (9) can be evaluated R gas according to a linear relationship = ∙ + ,
 reported by Yan et al. [38].  Then,A·the
 R gasactivation energy EA−and
 
 10bKthe · E Apre-exponential term A
 are determined using = ln − (10) and
 bK Equations · f 0(α(11):
 P) = > A = 1/s (11)
 EA R gas· f 0(αP )
 ∙ 
 where f 0 (α) = d f (α) /dα, Rgas
 = −is the gas+ constant,
 − ( heating
 β is∙ the ) rate, Tp represents
 (9)
 ∙ 
 the temperature, and αp is the crosslink density with the index p, which means that the
 indicated values correspond to the position of the rate peak maximum.
 =− => = 77.66 / (10)
 3.3.3. Comparison
 The results of the reaction∙ model determined by different ∙ methods (MD, DSC, and
 DEA) are transmitted = − in the∙ ′( 
 together ) 8 =>
 Figure =
 (normalized dα/dt1/ (11)
 vs. α) and Table 5. While
 ∙
 the maximum response rate appears at a low network level (approx. 20%) in the MD
 where ( )the
 simulation, ( )⁄ , reaction
 = maximum Rgas is the gas
 rate constant,
 occurs β is the heating
 at a crosslink density rate, Tp represents
 of 50%, concerningthe
 ex-
 perimental results. According to the MD calculation, the crosslinking reactionthat
 temperature, and α p is the crosslink density with the index p, which means the
 ends indi-
 almost
 cated values
 at a high correspond
 crosslink density to of
 the>85%.
 position
 Theofmobility
 the rateofpeak
 freemaximum.
 monomers/molecules is high at
 a low crosslink density, leading to the increased probability of the crosslinking reaction
 occurring. The barrier that prevents the molecular movements through morphological
 structures by the growing molecules is proportional to the increasing crosslink density. In
 an experimental investigation like DSC, the reaction heat is measured to determine the