Material Property Prediction of Thermoset Polymers by Molecular Dynamics Simulations Chunyu Li Purdue University, Birck Nanotechnology Center, Lichunyu@Purdue.Edu

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Birck and NCN Publications Birck Nanotechnology Center

4-2014 Material property prediction of thermoset polymers by molecular dynamics simulations Chunyu Li Purdue University, Birck Nanotechnology Center, [email protected]

Eric Coons Purdue University, Birck Nanotechnology Center, [email protected]

Alejandro Strachan Purdue University, Birck Nanotechnology Center, [email protected]

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Li, Chunyu; Coons, Eric; and Strachan, Alejandro, "Material property prediction of thermoset polymers by molecular dynamics simulations" (2014). Birck and NCN Publications. Paper 1586. http://dx.doi.org/10.1007/s00707-013-1064-2

This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. Acta Mech 225, 1187–1196 (2014) DOI 10.1007/s00707-013-1064-2

Chunyu Li · Eric Coons · Alejandro Strachan Material property prediction of thermoset polymers by molecular dynamics simulations

Received: 22 August 2013 / Revised: 22 October 2013 / Published online: 18 February 2014 © Springer-Verlag Wien 2014

Abstract Molecular dynamics simulations are conducted to predict thermal and mechanical properties of a family of thermoset polymers. We focus on the effect of cross-linkers on density, glass transition temperature, elastic constants, and strength. The polymers are composed of the epoxy resin DGEBA (EPON825) and a series of cross-linkers with different number of active sites and rigidity 33DDS, 44DDS, APB133, TREN, and TAPA. Our simulations quantify effects of cross-linkers on thermal and mechanical properties.

1 Introduction

With the fast increase in computer power and advances in physics-based models and efficient algorithms, computational simulation has become one of three pillars in modern science and engineering. Simulation-Based Engineering Science (SBES) has been classified as a discipline indispensable in science and engineering [1]. Recognizing that predictive computational simulations in materials science and engineering have significant room to grow compared with the huge success of computational simulations in other field such as struc- tural engineering and mechanical engineering, Integrated Computational Materials Engineering (ICME) was recently identified as a transformational discipline for improved competitiveness and national security in a report issued by the US National Academy of Engineering [2]. More recently, the US government unveiled Materials Genome Initiative for Global Competitiveness (MGI) [3]. The goal is to significantly shorten the time elapsed from discovery to deployment of new materials. Reaching this objective will require significant effort in developing advanced computational tools for predictive modeling, simulation, design, and explora- tion of materials. The ability to simulate materials processing and fabrication as well as their performance on computers can bring in deep understanding of fundamental processes, such as chemical reactions and material failure mechanisms. In addition, such simulations can provide design guidelines and thus significantly reduce the number of experiments required for optimization of material formulation and manufacturing conditions. Aligned with these general objectives, our group has devoted significant efforts to predictive simulations of polymers, especially thermoset polymers, in the last few years [4–11]. Thermoset polymers are the matrix of choice for fiber-reinforced composites used in a wide variety of applications, especially in aerospace industry. Compared to thermoplastic polymers, the advantages of thermoset polymers are in their better material properties, such as higher stiffness, higher strength as well as creep and thermal resistance. These properties depend on various factors, including the chemistry and molecular structures of resin and cross-linker, polymerization (conversion) degree, as well as thermal history. Detailed knowledge of the effect of these factors is of great interest to the applications of thermoset polymers.

In honor of Professor George Weng, the 2013 Prager Medalist. C. Li (B) · E. Coons · A. Strachan School of Materials Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47906, USA E-mail: [email protected] 1188 C. Li et al.

There have been extensive experimental studies to characterize the effect of conversion degree on material properties of thermoset polymers. Among the properties studied, glass transition temperature is the most extensively characterized one. A general trend of all these experimental studies is that Tg increases with increasing conversion in a nonlinear fashion with an increasing dependency for higher conversion degree. Atomistic simulations [5,8,12,13] have also conﬁrmed the general trends observed experimentally. The role of the conversion degree for the elastic properties of thermoset polymers is not as deeply understood, and contradictory results are still in debate. For example, the studies of Theriault et al. [14] showed an increase in the Young’s modulus with increasing conversion degree. Li and Strachan further reported an almost linear increase in tensile modulus with conversion degree [5]. But Morel et al. [15] concluded that the effects of cross-linking density and glass transition temperature on the elastic properties of the polymers in the glassy state are insignificant. Marks and Snelgrove [16] even reported a uniform trend of decreasing tensile modulus with increasing conversion for various amine-cured epoxy thermosets. The material properties of thermoset polymers are also strongly dependent on the detailed molecular structures of the constituent resin monomers and cross-linker molecules. Experiments have shown that the Tg of an epoxy network can be shifted by more than 140 K by just using different cross-linkers [17]. Atomistic simulations have also found that Tg values decrease with increasing chain length of the cross-linkers [18]. The studies of Marks and Snelgrove [16] indicated that the fracture toughness of epoxy thermosets often exhibit a maximum value for conversion degrees between 65 and 95 %, depending on the rigidity of the cross-linker. However, the understanding of the effect of the cross-linker is much less developed than that of the effect of the conversion degree. The objective of this paper was to systematically investigate the effect of cross-linkers on thermal and mechanical properties of epoxy thermosets. We use a procedure developed in our previous research to build network structures of thermoset polymers and characterize the thermomechanical response of the resulting structures using molecular dynamics (MD). In particular, we focus on the effect of cross-linker length and chemical functionality.

2 Molecular dynamics simulations

Since molecular dynamics (MD) was first proposed in 1959 [19] for studying hard-sphere systems followed by soft-sphere systems [20], there have been numerous MD simulations conducted in various fields such as computational physics, computational chemistry, materials science, pharmaceuticals, and biochemistry. There are three basic stages in a classical MD simulation: building a molecular model with atomistic detail to be used as initial conditions, solving classical equations of motion to obtain the time evolution of the atomic positions and velocities subject to appropriate boundary conditions, and finally collecting desired properties from the trajectories following basic rules of statistical mechanics. MD simulations for polymers dates back to the 1970s with most studies focused on thermoplastics [21,22] and significantly less work on thermosets. The first fully atomistic MD simulation on 3D polymer networks was carried out by Hamerton et al. [23] for a 200-atom system, and the elastic modulus and glass transition temperatures were predicted in reasonable agreement with experimental data. Doherty et al. [24] first performed MD simulations that allow progressive polymerization reactions. Yarovsky and Evans [25] developed a computational procedure for constructing molecular models of cross-linked polymer networks and applied it to low molecular weight water-soluble epoxy resins cured with different cross-linking agents. Heine et al. [26] simulated the structure and elastic moduli of end-cross- linked poly(dimethylsiloxane) networks using a united atom force field. Wu and Xu [27] developed a method to construct polymer networks for epoxy resin system based on DGEBA (diglycidyl ether bisphenol A) and IPD (isophorone diamine). Komarov et al. [28] reported a computational method where the polymer network is polymerized at a coarse-grained level and then mapped into a fully atomistic model. Varshney et al. [29] studied molecular modeling of thermosetting polymers with special emphasis on cross-linking procedure. Lin and Khare [30] presented a single-step polymerization method for the creation of atomistic model structures of cross-linked polymers. Bermejo and Ugarte [12] introduced a method for building fully atomistic models of chemically cross-linked poly(vinyl alcohol). A more comprehensive review on MD simulations of thermoset polymers can be seen in the Journal of Applied Polymer Science [31]. The core of a MD simulation is the second stage in which a large number of atoms in the simulation system are allowed to interact with each other based on a well-defined molecular mechanics force field. The trajectories of atoms are deterministically generated by numerically integrating the Newton’s equations of motion with very small time intervals (order of femtosecond) using Verlet’s algorithm [32] or Gear’s predictor–corrector Material property prediction of thermoset polymers by molecular dynamics simulations 1189 algorithm [33]. There have been several molecular force fields specifically developed for polymers, including CVFF (the consistent valence force field) [34], CHARMM (Chemistry at HARvard Macromolecular Mechan- ics) [35], DREIDING [36], PCFF (polymer consistent force field) [37], and COMPASS (condensed-phase optimized molecular potentials for atomistic simulation studies) [38]. Based on our previous experience on the MD simulations of thermoplastic and thermoset polymers, the general-purpose DREIDING force field [36] with harmonic form of covalent potentials is employed in all our simulations. All our MD simulations are conducted using our in-house code based on the open source package LAMMPS [39]. Another important consideration for MD simulations of polymers is atomic partial charge changes during the dynamic process. We obtain the atomic charges based on the self-consistent calculations using the electronegativity equalization method as described in Ref. [4]. This approach to describe atomic interactions, which is generally applicable to any polymer and many composites, has been shown to provide an accurate representation of the thermo-mechanical properties of thermosets [4] and other soft materials, see for example Ref. [40].

3 Simulations of thermoset curing process

3.1 Model systems

We focus on a typical commercial epoxy resin diglycidyl ether of bisphenol-A (DGEBA) cured with different cross-linkers in this paper. Figure 1 shows the molecular structure of DGEBA. We start simulations with the “activated” monomer shown in Fig. 1b. Figure 2 shows the molecular structures of several cross-linkers. The choice of cross-linkers represents difunctionals, trifunctionals, and different lengths. The reactive sites on the epoxy resin monomer and cross-linker molecules are highlighted in red. The model systems consist of 1024 or 1152 DGEBA monomers with different number of cross-linkers (512 for difunctionals or 384 for trifunctionals) to keep the stoichiometric ratio to be 1. The initial total numbers of atoms in these model systems are in the range of 69,120–76,416.

3.2 MD-based polymerization simulator (MDPoS)

Here, we brieﬂy introduced the simulation procedure for mimicking the curing process of thermoset polymers. The procedure used to create polymer network structure was described in detail in Ref. [4] and consists of threemainstages(seeﬂowchartshowninFig.3):

3.2.1 Pre-cross-linking

A mixture of the epoxy and curing agent with the desired stoichiometry is packed into a simulation cell with 3D periodic boundary conditions at low density (0.5g/cm3). The system is initially energy-minimized using the conjugate gradients method and then equilibrated using an isothermal and isochoric (NVT ensemble) MD simulation for 50 ps at 600 K followed by an isothermal and isobaric (NPT) MD simulation for 400 ps at atmospheric pressure.

Fig. 1 Molecular structures of a DGEBA, b activated DGEBA 1190 C. Li et al.

Fig. 2 Molecular structures of cross-linkers: a 3,3-diaminodiphenyl sulfone (33DDS); b 4,4-diaminodiphenyl sulfone (44DDS); c 1,3-bis (3-aminophenoxy) benzene (APB133); d Tris(2-aminoethyl)amine (TREN); e tris(4-aminophenyl)amine (TAPA)

Fig. 3 Flowchart of curing process simulation

3.2.2 Cross-linking

During the curing stage, chemical reactions are simulated in a stepwise manner using a distance-based criterion. Bonds are created between reactive atoms within a cut-off distance taken as four times the equilibrium N-C bond length (1.41 Å). New bonds are turned on slowly using a 50 ps long multi-step relaxation procedure to avoid large atomic forces. After the new bonds are fully relaxed, an NPT simulation for an additional 50 ps is performed before the new set of bond creations is attempted. This procedure is carried out at 600 K.

3.2.3 Cooling down

The cross-linked systems are built at higher temperature (e.g., 600 K), which is taken to be larger than the expected glass transition temperature. An annealing process is then needed to cool down the systems to room temperature or other desired temperatures. This annealing process can also be used to predict the glass transition temperature and the coefﬁcient of thermal expansion. Material property prediction of thermoset polymers by molecular dynamics simulations 1191

4 Simulation results and discussions

In our MD simulations, the conversion degree is defined as the ratio between the number of bonds created and the maximum possible new bonds, which depend on the functionality and the total number of cross-linkers in the model system. The conversion limit in our simulations is predefined to be 85 % for all five cases (DGEBA/33DDS, DGEBA/44DDS, DGEBA/APB133, DGEBA/TAPA, and DGEBA/TREN) in order to finish the cross-linking process within an affordable simulation time (∼3.0 ns). The simulation time for the cooling down process is about 2.0 ns, and the simulation time for the testing process is about 1.0 ns. Roughly this 6.0 ns simulation for each case takes about 450 CPU hours running on the Purdue University Hansen clusters, which consist of Dell compute nodes with four 12-core AMD Opteron 6176 processors. The simulation results are obtained by running the LAMMPS version released on July 4, 2012.

4.1 Glass transition temperature

Glass transition temperature (Tg) is an important property of polymers. When a polymer is cooled below this temperature, it transforms into a glass (a solid frozen in a nonequilibrium state) becoming hard and brittle. Thermoset polymers for composites are used below their glass transition temperatures. A direct and efficient method to compute Tg of a polymer sample in MD simulations is to examine how the density changes with temperature over a wide range of values. After the polymerization and cross-linking procedure described above produce equilibrated structures at 600 K, we cool the final cross-linked systems down to room temperature using NPT simulations under atmospheric pressure. This is performed in steps of 10 K, and a 60 ps long simulation is performed for each temperature. Figure 4 shows density-temperature plots for DGEBA resin with difunctional and trifunctional cross-linkers. As expected, we observe an increase in density with the decreasing temperature for all cases. A change in the slope of a density-temperature curve marks Tg. To obtain Tg accurately, we perform linear fits to the density-temperature data below and above Tg in various temperature ranges. This procedure leads to a wide range of values for Tg: from ∼470 K for DGEBA/APB133 to 540 K for DGEBA/44DDS, see Table 1. Interestingly,

(a) 1.20 Difunctional crosslinkers Cooling rate: 10K/60ps

) 1.16 3

1.12

1.08 Density (g/cm 1.04 33DDS 44DDS APB133 1.00 300 350 400 450 500 550 600 650 Temperature (K)

(b) 1.12 Trifunctional crosslinkers 1.10 Cooling rate: 10K/60ps )

3 1.08 1.06 1.04 1.02

Density (g/cm 1.00 TAPA 0.98 TREN 0.96 300 350 400 450 500 550 600 650 Temperature (K) Fig. 4 Density as a function of temperature: a difunctional cross-linkers; b trifunctional cross-linkers 1192 C. Li et al.

Table 1 Thermal and mechanical properties of thermoset polymers

Epoxy resin DGEBA

Cross-linker 33DDS 44DDS APB133 TREN TAPA Density (g/cm3, 300 K) 1.161 1.156 1.132 1.112 1.109 Tg (K) 519 536 471 509 530 CTE (glassy, 10−6/K) 196 200 219 289 230 CTE (rubbery, 10−6/K) 418 450 400 536 485 Young’s modulus (GPa) 2.81 2.45 3.15 2.89 3.16 Yield strength (MPa) 178 175 192 179 189 Poisson’s ratio 0.34 0.32 0.36 0.39 0.35 thermosets with trifunctional cross-linkers appear to show a possible secondary relaxation below 400 K; this deserves further investigation and is beyond the scope of this report. Comparing these predictions with experimental values requires accounting for several factors that affect Tg both in the experiments and simulations. One such factor is the conversion degree. A wide range of conversion degrees has been reported for epoxy resins, mostly in the range 60–100 % because of the conversion degree dependence on chemistry, stoichiometry, and processing conditions and the challenges faced by quantifying the conversion degree in experiments [16]. Typically, conversion degrees in well-controlled curing processes are in the range 80–95 % [8]; this is similar to our conversion degree, so direct comparison can be done. Another factor is cooling rate, and typical experimental values (10 K/min) are many orders of magnitude lower than that in our MD simulations. Because Tg is a kinetic quantity and strongly depends on the cooling or heating rate, extensive research have been devoted to understanding the effect of cooling rate on Tg. It is generally accepted that an increase of about 3–5 K is expected per order of magnitude increase in cooling rate [8,41]. The Tg predicted from our MD simulations (cooling rate 12 orders of magnitude higher) should be expected to be about 36–60 K higher than the values obtained under normal testing conditions. Experimental values of Tg for DGEBA/33DDS were reported to be 413 K [42] and 417–432 K [43]. Taking into account the role of the cooling rate, our simulation results for Tg are only slightly higher than the available experimental results. We note that these represent true predictions, and no parameter is adjusted to reproduce experiments. Prior MD simulation value for the Tg of DGEBA/APB133 was reported to be 437 K, slightly lower than our value [44].

4.2 Coefﬁcient of thermal expansion

Thermal expansion is another important physical property for polymers that affects their processing and applications. The volumetric thermal expansion α is defined as follows: ∂ α = 1 V , ∂ (1) V0 T P where V is the volume at temperature T ,andV0 is the volume at a reference temperature (300 K in this paper). Figure 5 shows the change of normalized volume with temperature. The coefficient of volumetric thermal expansion α can be determined from the slope of the linear fitting of the MD data. As expected, the CTE is observed very different in the glass state and the rubbery state. The effect of cross- linkers is also obvious for CTE at both states. The thermosets with trifunctional cross-linkers exhibit larger thermal expansion. The experimental results for the CTE of DGEBA/33DDS were reported to be 207×10−6/K −6 below Tg and 528 × 10 /K above Tg [8]. Again, the simulations are in good quantitative agreement with experiments.

4.3 Mechanical properties

The mechanical properties of a material describe how it responds under a certain loading condition. The mechanical properties of polymers often determine the application, and the degradation of these properties establishes their lifetime in service. The actual molecular networks of thermoset polymers play a critical role in their mechanical properties. The most common mechanical properties include elastic modulus, Poisson’s Material property prediction of thermoset polymers by molecular dynamics simulations 1193

0.12 TREN 0.10 TAPA 33DDS 0.08 44DDS

300K APB133 0.06 dV/V 0.04

0.02

Cooling rate: 10K/60ps 0.00 300 350 400 450 500 550 600 650 Temperature (K) Fig. 5 Volume of simulation cells normalized with reference volume at T = 300 K

(a) 250 8 -1 Strain rate=5x10 s T=300K 200

150

100

Stress (MPa) Crosslinker 50 APB133 44DDS 33DDS 0 0 5 10 15 20 25 Strain (%)

(b) 250 8 -1 Strain rate=5x10 s T=300K 200

150

100 Stress (MPa) Crosslinker 50 TAPA TREN 0 0 5 10 15 20 25 Strain (%) Fig. 6 Stress–strain relationships under uniaxial tension conditions ratio, yield strength, fracture strength, and toughness. All these properties can, in principle, be predicted using MD simulations if proper molecular force ﬁelds are used. However, toughness involves processes with length scales often in the micrometer regime, beyond what is possible with MD. Here, we only characterize elastic modulus, Poisson’s ratio, and yield strength of thermosets via nonequilibrium MD simulations of uniaxial tension. The uniaxial tension simulations are carried out by increasing the length of the simulation cell along the loading direction at every MD step while maintaining atmospheric pressure in the transverse directions using a barostat. The limit of the longitudinal strain is set to be 25 %, and the strain rate is 5 × 108 s−1,whichis extremely large by experimental standards but is typical for MD simulations due to the small affordable time scale. Uniaxial deformations are carried out along all three Cartesian directions, and then, the stress–strain curves are obtained based on their average. 1194 C. Li et al.

0.00 APB133 33DDS -0.02 44DDS

(%) -0.04 y ε

-0.06 Strain

-0.08 T=300K

-0.10 0.00 0.05 0.10 0.15 0.20 0.25 Strain εz (%)

-1 TAPA TREN -2 -3 (%) x

ε -4 -5

Strain -6 -7 T=300K -8 0 5 10 15 20 25 Strain εz (%) Fig. 7 Strains in transverse directions under uniaxial tension conditions

Figure 6 shows the stress–strain curves for model systems at 300K under uniaxial tension conditions at a strain rate of 5 × 108 s−1. Figure 7 shows the transverse strain as a function of the tensile strain from the same simulations. We calculate Young’s modulus by linear fitting the stress–strain curves up to 4 % strain and define yield strength as the maximum stress attained in stress–strain curves. The Poisson’s ratio is obtained by linear fitting the transverse strain and longitudinal strain data up to 4 % tensile strain. Table 1 lists the values of Young’s modulus and Poisson’s ratio as well as yield strength. White et al. [45] reported that the measured Young’s modulus of DGEBA/44DDS is in the range of 2.4– 3.2 GPa. Our group’s prior MD simulations [9] on DGEBA/33DDS (using the same approach and force fields) led to Young’s modulus is about 3.0 GPa and the yield strength is about 198 MPa. These are slightly larger (about 10 % difference) than the present values; this may have resulted from different system size and thermal history but also from the intrinsic molecular-level variability in these systems. This level (10 %) gives a rough idea of the degree to variability intrinsic in our predictions.

5 Conclusions

Material properties of thermoset polymers are strongly dependent on the detailed molecular structures of the constituent resin monomers and cross-linker molecules. This paper focuses on the effect of the cross-linker on the thermo-mechanical response of epoxy thermosets. MD simulations are used to characterize the thermo- mechanical response of the thermoset polymer composed of the epoxy resin DGEBA and various cross-linkers, including 33DDS, 44DDS, APB133, TREN, and TAPA. The results show a significant effect of cross-linking agent on thermo-mechanical properties. In the case of bifunctional cross-linkers, the increased separation between cross-links in the case of APB133 leads to lower density as compared to 33DDS and 44DDS; we also observe a depression in the glass transition temperature. Surprisingly, the stiffness of the APB133 system is higher than that of the other two bifunctional cross-linkers; the origin of this observation is unclear at this point. For the trifunctional cross-linkers, TREN and TAPA, we ﬁnd that the more rigid and large TAPA Material property prediction of thermoset polymers by molecular dynamics simulations 1195 leads to slightly lower density but higher glass transition temperature and stiffness. Such predictions, based on atomistic simulations without tunable parameters, can dramatically reduce the number of experiments and speed up the process of materials development.

Acknowledgments This work was supported by a grant with The Boeing Company and the US National Science Foundation (NSF) under contract CMMI-0826356.

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