51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<BR>18th 1-15 April 010, Orlando, Florida AIAA 010-811 Atomistic Modeling of Cross-linked Epoxy Polymer Ananyo Bandyopadhyay 1, Benjamin D. Jensen Michigan Technological University, Mechanical Engineering-Engineering Mechanics, Houghton, MI, 49931 Pavan K. Valavala 3 Johns Hopkins University, Department of Materials Science and Engineering, Baltimore, MD, 118 and Gregory M. Odegard 4 Michigan Technological University, Mechanical Engineering - Engineering Mechanics, Houghton, MI, 49931 Abstract Molecular Dynamics simulations are used to study cross-linking of an epoxy polymer. OPLS force field parameters are used for modeling a :1 stoichiometric mixture of epoxy resin and the cross-linking agent. The model has 17,98 united atoms and a static cross-linking method is used along with molecular minimization and molecular dynamics techniques to achieve two different cross-link densities. The crosslinked models can be used for understanding various phenomenon occurring in cross-linked epoxy resins at the atomic scale. Glass-transition temperature ranges of two differently cross-linked samples have been predicted using the models. These models will be used for studying aging behavior at the atomic level in epoxy materials and understanding the influence of aging on mechanical properties. E I. Introduction poxy Resins are prime constituents in adhesives, sealants, and aircraft composite structural components. A wide range of studies have focused on epoxy-based materials to establish physical and mechanical properties. 1-3 The excellent specific-stiffness and specific-strength properties of epoxy-based composite materials are due to the complex microstructure of their constituent materials. There is significant interest in understanding the aging response of these material systems due to their wide-spread use in commercial aircraft. A. Computational Studies on Epoxy Polymers Epoxy resins are formed when epoxy monomers react with compounds known as cross-linking or curing agents with active hydrogens such as amines and anhydrides. 4 A trial-and-error approach to experimentally optimize the processing conditions of epoxy materials can become time-consuming and expensive. With the advancement of computational technology, computational modeling has provided an efficient route to study these polymer resins. 5-14,4,15 Molecular dynamics (MD) simulations based on the bead-spring model 10,11 and Monte-Carlo simulations based on the bond-fluctuation model 16,8,9 have been used in the last two decades for studying epoxy materials. The beadspring models did not take into account the details of the molecular structures and thus cannot predict the influence of specific groups of atoms on the physical properties. In the last few years, MD at the atomic scale has been quite successful in exploring different phenomena occurring at pico- to nano-second time scales in epoxy resins. 14 Many researchers have studied the formation of cross-linked epoxy resins using different approaches of simulated cross-linking. Doherty et al. 5 modeled PMA networks using lattice-based simulations in a polymerization MD scheme. Yarovsky and Evans 15 discussed a cross-linking technique which they used to crosslink low molecularweight, water-soluble, phosphate-modified epoxy resins (CYMEL 1158). The cross-linking reactions were carried out simultaneously (static cross-linking process). Dynamic cross-linking of epoxy resins was performed by Xu et al. 4 1 PhD Candidate PhD Candidate 3 Post-Doctoral Researcher, Member, AIAA 4 Associate Professor, Associate Fellow, AIAA 1 Copyright 010 by Gregory M. Odegard. Published by the, Inc., with permission.
Their model was used to study the diffusion of water in cross-linked networks. An iterative Molecular Dynamics (MD)/Molecular Minimization (MM) procedure was used to cross-link an epoxy resin (DGEBA), with one crosslink established per iteration. Other computational studies 17,6 involving cross-linking of epoxies have been performed. All of the studies discussed thus far were performed on relatively small model systems (less than 00 atoms). Heine et al. 7 simulated large PDMS networks using a dynamic cross-linking approach and Varshney et al. 14 used Heine s dynamic cross-linking approach and Xu s MD/MM concept 4 to cross-link EPON86 with DETDA. Varshney et al. 14 modeled two different systems having molecules of EPON and DETDA in the ratios of 18:64 (EPON: DETDA) and 56:18. Although dynamic approaches to establishing crosslinks may provide a more realistic physical understanding of the crosslinking process than static crosslinking, it is more difficult to control the ultimate cross-link density of a molecular model using dynamic approaches. Therefore, a multi-step static approach to crosslinking large molecular models of epoxy is necessary to efficiently establish such models for parametric studies involving multiple, pre-defined cross-link densities. B. Objective The objective of this study was to establish a method of statically cross-linking large systems of EPON 86 and DETDA molecules having a molecular ratio of 43:16. In this method, the MD/MM techniques used by Varshney et al. 14 for establishing an EPON-DETDA modeled structure with a :1 molecular ratio was coupled with the static cross-linking method described by Yarovsky and Evans. 15 A description of the modeling procedure for a monomer solution is followed by a description of the cross-linking mechanism. Finally, constant pressure and temperature simulations were run on two models with different cross-link densities for determining the glass-transition regions and the results were found to agree with results reported in the literature. II. Modeling A. Modeling EPON-DETDA structure having 16:8 stoichiometric ratio The initial structure consists of the EPON-86 monomer (Di-glycidyl ether of Bisphenol-F) and the cross-linking agent DETDA (Diethylene Toluene Diamine). EPON-86 is produced by Hexion Chemicals Inc. The molecules of EPON-86 and DETDA are shown in Figure 1. A stoichiometric mixture (:1 ratio) of molecules of EPON-86 and 1 molecule of DETDA was modeled first using the LAMMPS (Large Scale Atomic/Molecular Massively Parallel Simulator) simulation program. 18 The initial atomic coordinates were written in a coordinate file in the native LAMMPS format and the OPLS United Atom force field 19-1 was used for defining the bond, angle, and dihedral parameters. The non-bonded van der Waals interactions were modeled using the 1-6 Lennard-Jones potential. By using this particular OPLS United Atom force field, all CH 3, CH, and CH groups were modeled as single united atoms with their corresponding masses. The Carbon and Hydrogen atoms of the two benzene rings present in one EPON-86 molecule were considered as single atoms only. Similarly for the DETDA molecule, the Carbon and Hydrogen atoms of the benzene ring and the Nitrogen and Hydrogen atoms present in the amine groups were considered as single atoms only. In the DETDA molecule, the alkyl groups attached to the benzene rings were considered as united atoms. Thus in a :1 structure the number of atoms were reduced from 116 atoms to 83 atoms with the use of united atoms. Figure 1. Molecular Structure of EPON-86 resin and DETDA cross-linking molecules
The initial :1 structure was formed in a 10 10 10 Angstrom simulation box with periodic boundary conditions. This structure was subjected to four molecular mechanics (MM) minimizations and three MD simulations in order to minimize internal forces (thus reduce internal residual stresses) resulting from the construction of bonds, bond angles, and bond dihedrals. After reaching a relatively low energy value, this structure was replicated to form eight more structures within the simulation box so that a 16:8 molecular mixture of EPON and DETDA monomers was established. A slow stress relaxation procedure was performed over a cycle of 0 MM and 10 MD simulations. All MD simulations were conducted in the NVT (constant volume and temperature) ensemble for 100 picoseconds at 600 K. The NVT ensemble made use of the Nose/Hoover thermostat and barostat for temperature and pressure control, respectively. After every cycle of MD and MM, the box size was reduced by a small amount. After all minimization and MD runs, a specific gravity of 1.13 was achieved. The final pressure value of the last minimization was less than 1 atmosphere which indicated that the structure had almost no residual stress. This equilibrated structure was used for the subsequent cross-linking step. B. Force Field The OPLS United Atom force field was developed by Jorgensen and co-workers. 3,19,1 In this force field, the total energy of a molecular system is a sum of all the individual energies associated with bond, angle, dihedral, and 1-6 Lennard-Jones interactions. The equilibrium spacing parameter of the Lennard-Jones potential was taken to be the arithmetic mean of the individual parameters of the respective atom types while the well depth parameter was taken to be the geometric mean of the values of the respective atom types. The bond energy is given as E K r r bonds r ( o ) (1) bonds where K r is a force constant, r is the distance between the two atoms considered, and r 0 is the equilibrium bond distance. The energy associated with bond-angle bending is E K ( ) angle 0 (0) angles where K is a force constant, is the bond angle, and 0 is the equilibrium bond angle. The dihedral potential is given by V V V V 1 1 3 1 3 1 4 (0) 1 4 Edihedral cos cos cos cos where V 1, V, V 3, and V 4 are coefficients in the Fourier series 19,0 and is the dihedral angle. C. Cross-Linking Procedure The equilibrated structure of the 16:8 model was statically cross-linked based on the root mean square (RMS) distance between the Nitrogen atoms of DETDA and CH groups of the EPON molecules. 14 Simultaneous breaking of CH -O bonds in the epoxide ends of the EPON molecules and N-H bonds of the DETDA molecules made the activated CH ends capable of forming cross-links with activated N atoms of the DETDA molecules. A particular activated N could form a cross-link with the activated CH of any adjacent EPON molecule within a cutoff distance. Figures and 3 demonstrate the cross-linking process. Three assumptions were made for the cross-linking process: 1) Both primary amines in DETDA were assumed to have the same reactivity ) The CH -O and N-H bonds were broken simultaneously (Figure ) 3) A Nitrogen atom was partially activated when it had only one activated CH within a defined cutoff distance. The starting point of the cross-linking reaction is shown in Figure where the lone pair of electrons from the N atom of NH end of one DETDA molecule attacks the carbon atoms close to it and oxygen attains a negative charge by breaking the C-O bond. The N forms a bond with the C and attains a positive charge and by this way, the neutrality of the EPON-DETDA system is maintained. Cross-links were formed by computing all RMS distances 3
between each N atom and the CH united atoms within the defined cutoff distance. The cutoff distance was defined as the maximum RMS distance that was chosen to find all possible CH N pairs. The CH radicals located outside the cut-off distance of a particular NH group were not cross-linked to that particular group. In the next step, the H + ions were formed by breaking NH bonds and were reacted with the O - atoms of the broken epoxide ends. This bond formation was also performed based on the closest RMS distances between the O - and H + atoms. The second step of the cross-linking reaction is shown in Figure 3. Two different cross-linked structures were formed for a range of cutoff distances. Figure. 1 st step of Cross-Linking reaction: The lone pair of electrons of the N atom attacks the Carbon atom next to the epoxide Oxygen, giving a negative charge on the oxygen and a positive charge on the nitrogen. (The wavy lines represent the remaining parts of the EPON and DETDA molecules in the respective structures) Figure 3. nd and Final steps of Cross-Linking reaction: (A) The oxygen s extra pair of electrons takes the nearest hydrogen from the ammonium nitrogen, making an alcohol group and an amine group and the cross-linking is complete. (B) The same cross-linked N reacts with another epoxide end of EPON in the same way and forms two cross-links. 4
D. Modeling EPON-DETDA structure having 43:16 stoichiometric ratio After cross-linking, new bond, angle, and dihedral parameters were defined in the structure. The cross-linked 16:8 models were equilibrated by performing one cycle of two minimizations and one MD run alternately to remove the residual stresses generated during the formation of the cross-links. The MD runs were NVT simulations for 100 picoseconds at 500K. The equilibrated, cross-linked 16:8 models were oriented and translated into 7 more structures and these structures formed large systems that were a 3 3 3 array of 16:8 structures. The large systems had 43 molecules of EPON and 16 molecules of DETDA. For two different defined cutoff distances (thus two different cross-link densities), the 16:8 models had differences in the number of bonds, angles, and dihedrals. In one 16:8 structure, 3 possible cross-linking sites exist. The crosslink density of the polymer was defined as the total number of these sites that were crosslinked. For example, a specimen having 16 out of 3 cross-links was defined as having a 50% cross-link density. Each of these two samples had 17,98 united atoms while the number of modeled individual atoms in each chemical structure was 5,7. At this point it is important to note the advantage of modeling united atoms instead of individual atoms. Simulations can be performed more efficiently when fewer atoms are modeled. Figure 4. 43:16 model of EPON-DETDA containing 17,98 united atoms (5,7 real atoms) The models having a 43:16 stoichiometric ratio of EPON and DETDA chains were further equilibrated using MD and MM techniques with continuous shrinking of the volume until the models reached densities close to 1. gm/cc. Around 30 minimizations and 1 NVT simulations were required for the equilibration of each individual 43:16 EPON-DETDA model. These models were further cross-linked based on RMS cutoff distances. The 7 structures, each consisting of 16:8 ratio of resin and hardener chains, were cross-linked with one another, thus increasing the cross-link densities further. The 50% cross-linked structure had a 54% cross-link density after this step and the 7% cross-linked structure increased to 76%. After this additional process of cross-linking, the structures were further equilibrated at the same volume with two NVT simulations at 500K and 300K with inbetween minimizations. After equilibration, 1 NPT (constant temperature and constant pressure) simulations were run for 400 picoseconds at temperatures from -3 o C (50K) to 7 o C (500K) at 1.5 degree intervals at a pressure of 1 atm. These simulations were used to study the density changes with respect to temperature. The results of these simulations are discussed in the next section. III. Results Using the results of the NPT simulation, density versus temperature curves were plotted as shown in Fig. 5 for the 54% cross-linked and 76% cross-linked systems. Data from the final 350 picoseconds of each NPT simulation were 5
used to establish the data points in the graph to eliminate the effects of molecular relaxation and initial oscillation of the temperature and pressure around the set values. The density-temperature curves showed a characteristic change in slope in the glass-transition temperature (T g ) region. The glass-transition temperature marks the point in which decreases in temperature no longer result in significant decreases in free volume in the polymer structure. Since the transition does not take place suddenly, the glass-transition temperature (T g ) is usually assumed to occur over a finite range of temperatures. For the 54% cross-linked structure, the T g was found to be in the range of 80 o C to 100 o C and intersection of the trendlines suggested T g was roughly around 88 o C. For the 76% cross-linked structure, the T g was found to be in the range of 90 o C to 110 o C and the intersection of the trendlines suggested a value of 97 o C. The results were found to be consistent with those reported by Varshney et al. 14 and Fan et al. 6 Varshney et al. predicted a T g for the same EPON-DETDA system at 105 o C, though the corresponding cross-link density was not given. Fan et al. found the T g for a 100% cross-linked EPON-DETDA system to be 109 o C, but their model contained only 68 atoms. As expected, the T g for the 76% cross-linked structure is larger than that of the 54% cross-linked system. The reason for this increase in T g is likely due to the presence of more number of covalent bonds present in the 76% cross-linked model due to the higher degree of cross-linking. Higher temperatures are needed to increase atomic vibrations and deform the extra covalent bonds, angles, and dihedrals associated with these cross-links. As a result, there is more resistance to increases in free volume as the temperature increases for increased levels of cross-linking. Analysis of the curves also shows that for the 76% cross-linked structure, the density changed from 1.1 gm/cc to 1.03 gm/cc over a temperature range of 50 degrees. While, for the 54% crosslinked system, density changed from 1.14 gm/cc to 0.99 gm/cc over the same temperature range. This is also likely due to the difference in the number of covalent bonds present in the two structures. The 54% cross-linked structure has less covalent bonds and higher number of free chains that are mobile within the polymer structure. The 76% cross-linked structure has chains that are more constrained than the 54% cross-linked structure. Therefore, at high temperatures, the 54% cross-linked model can occupy a higher volume than the 76% cross-linked model. Density Temperature Curves Density (gm/cc) 1.14 54% Cross Linked 1.13 76% Cross Linked 1.1 1.11 1.10 1.09 1.08 1.07 1.06 1.05 1.04 T g ~88 o C 1.03 T 1.0 g ~ 97 o C 1.01 1.00 0.99 0.98 40.0 0.0 0.0 0.0 40.0 60.0 80.0 100.0 10.0 140.0 160.0 180.0 00.0 0.0 40.0 Temperature (degrees C) Figure 5. The dependence of the crosslink density on cross-linking cutoff distance 6
Acknowledgments This research was funded by NASA Langley Research Center under the Aging Aircraft Program (Grant NNX07AU58A). The authors thank Dr. Kristopher E. Wise, Dr. Thomas C. Clancy, and Dr. Sarah-Jane V. Frankland for their input regarding the cross-linking procedure. References 1 Boinard, P., Banks, W. M. and Pethrick, R. A., "Changes in the dielectric relaxations of water in epoxy resin as a function of the extent of water ingress in carbon fibre composites," Polymer, Vol. 46, 005, pp. 18-9. Liu, H. P., Uhlherr, A. and Bannister, M. K., "Quantitative structure-property relationships for composites: prediction of glass transition temperatures for epoxy resins," Polymer, Vol. 45, 004, pp. 051-060. 3 Ni, Y., Zheng, S. X. and Nie, K. M., "Morphology and thermal properties of inorganic-organic hybrids involving epoxy resin and polyhedral oligomeric silsesquioxanes," Polymer, Vol. 45, 004, pp. 5557-5568. 4 Wu, C. F. and Xu, W. J., "Atomistic molecular modelling of crosslinked epoxy resin," Polymer, Vol. 47, 006, pp. 6004-6009. 5 Doherty, D. C., Holmes, B. N., Leung, P. and Ross, R. B., "Polymerization molecular dynamics simulations. I. Cross-linked atomistic models for poly(methacrylate) networks," Computational and Theoretical Polymer Science, Vol. 8, 1998, pp. 169-178. 6 Fan, H. B. and Yuen, M. M. F., "Material properties of the cross-linked epoxy resin compound predicted by molecular dynamics simulation," Polymer, Vol. 48, 007, pp. 174-178. 7 Heine, D. R., Grest, G. S., Lorenz, C. D., Tsige, M. and Stevens, M. J., "Atomistic simulations of end-linked poly(dimethylsiloxane) networks: Structure and relaxation," Macromolecules, Vol. 37, 004, pp. 3857-3864. 8 Jo, W. H. and Ko, M. B., "Structure Development in Epoxy-Resin Modified with Thermoplastic Polymer - a Monte-Carlo Simulation Approach," Macromolecules, Vol. 6, 1993, pp. 5473-5478. 9 Jo, W. H. and Ko, M. B., "Effect of Reactivity on Cure and Phase-Separation Behavior in Epoxy-Resin Modified with Thermoplastic Polymer - a Monte-Carlo Simulation Approach," Macromolecules, Vol. 7, 1994, pp. 7815-784. 10 Stevens, M. J., "Interfacial fracture between highly cross-linked polymer networks and a solid surface: Effect of interfacial bond density," Macromolecules, Vol. 34, 001, pp. 710-718. 11 Tsige, M. and Stevens, M. J., "Effect of cross-linker functionality on the adhesion of highly cross-linked polymer networks: A molecular dynamics study of epoxies," Macromolecules, Vol. 37, 004, pp. 630-637. 1 Valavala, P. K., Clancy, T. C., Odegard, G. M. and Gates, T. S., "Nonlinear multiscale modeling of polymer materials," International Journal of Solids and Structures, Vol. 44, 007, pp. 1161-1179. 13 Valavala, P. K., Clancy, T. C., Odegard, G. M., Gates, T. S. and Aifantis, E. C., "Multiscale modeling of polymer materials using a statistics-based micromechanics approach," Acta Materialia, Vol. 57, 009, pp. 55-53. 14 Varshney, V., Patnaik, S. S., Roy, A. K. and Farmer, B. L., "A molecular dynamics study of epoxy-based networks: Crosslinking procedure and prediction of molecular and material properties," Macromolecules, Vol. 41, 008, pp. 6837-684. 15 Yarovsky, I. and Evans, E., "Computer simulation of structure and properties of crosslinked polymers: application to epoxy resins," Polymer, Vol. 43, 00, pp. 963-969. 16 Carmesin, I. and Kremer, K., "The Bond Fluctuation Method - a New Effective Algorithm for the Dynamics of Polymers in All Spatial Dimensions," Macromolecules, Vol. 1, 1988, pp. 819-83. 17 Fan, H. B., Chan, E. K. L., Wong, C. K. Y. and Yuen, M. M. F., "Molecular dynamics simulation of thermal cycling test in electronic packaging," Journal of Electronic Packaging, Vol. 19, 007, pp. 35-40. 18 Plimpton, S., "Fast Parallel Algorithms for Short-Range Molecular-Dynamics," Journal of Computational Physics, Vol. 117, 1995, pp. 1-19. 19 Jorgensen, W. L., Maxwell, D. S. and TiradoRives, J., "Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic liquids," Journal of the American Chemical Society, Vol. 118, 1996, pp. 115-1136. 0 Watkins, E. K. and Jorgensen, W. L., "Perfluoroalkanes: Conformational analysis and liquid-state properties from ab initio and Monte Carlo calculations," Journal of Physical Chemistry A, Vol. 105, 001, pp. 4118-415. 1 Weiner, S. J., Kollman, P. A., Case, D. A., Singh, U. C., Ghio, C., Alagona, G., Profeta, S. and Weiner, P., "A New Force- Field for Molecular Mechanical Simulation of Nucleic-Acids and Proteins," Journal of the American Chemical Society, Vol. 106, 1984, pp. 765-784. Hoover, W. G., "Canonical Dynamics - Equilibrium Phase-Space Distributions," Physical Review A, Vol. 31, 1985, pp. 1695-1697. 3 Duffy, E. M., Kowalczyk, P. J. and Jorgensen, W. L., "Do Denaturants Interact with Aromatic-Hydrocarbons in Water," Journal of the American Chemical Society, Vol. 115, 1993, pp. 971-975. 7