Polymer 54 (2013) 3370e3376. Contents lists available at SciVerse ScienceDirect. Polymer. journal homepage:

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1 Polymer 54 (2013) 3370e3376 Contents lists available at SciVerse ScienceDirect Polymer journal homepage: Molecular modeling of elastic properties of thermosetting polymers using a dynamic deformation approach Natalia B. Shenogina a, Mesfin Tsige b, *, Soumya S. Patnaik c, Sharmila M. Mukhopadhyay a a Department of Mechanical and Materials Engineering, Wright State University, Dayton, OH, USA b Department of Polymer Science, University of Akron, Akron, OH, USA c Aerospace Systems Directorate, Wright-Patterson Air Force Base, Dayton, OH, USA article info abstract Article history: Received 14 February 2013 Received in revised form 13 April 2013 Accepted 15 April 2013 Available online 23 April 2013 Keywords: Dynamic deformation simulations Molecular dynamics Elastic properties of epoxy This paper employs fully atomistic molecular dynamics simulations to characterize relationships between structural and elastic properties of thermosetting polymers both in glassy and rubbery state. The polymer system investigated consists of epoxy resin DGEBA and hardener DETDA. An effective crosslinking procedure that enables generation of thermoset structures containing up to atoms with realistic structural characteristics was used. A dynamic deformation approach has been used that takes into consideration both potential energy and thermal motions in the structure. Small uniaxial, volumetric and shear deformations were applied to the systems to obtain elastic moduli. A method to independently determine Poisson s ratio was proposed that reduces statistical errors and circumvents the time scale limitations of molecular dynamics simulations. The influence of variables such as extent of curing and length of epoxy strands on elastic response at various temperatures was explored. Expected trends in the dependence of the elastic constants on these practical process parameters were shown. The relationship between the four independently calculated elastic constants was seen to comply with those predicted by the classical theory of linear elasticity in an isotropic medium, which provides confidence in the validity of these simulations. Moreover, the elastic properties obtained are also in good agreement with experimental data reported in the literature. Close agreements between predicted elastic constants and experimentally measured values underscore the ability of the approaches used in this study to provide realistic predictions of the mechanical response of thermosetting polymers, both in glassy and rubbery states. These results show significant improvement over earlier studies based on a static approach which takes into account the potential energy contribution to the elastic response but ignores temperature effect. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Thermosetting polymers are important materials in a variety of applications due to their high thermal and structural stability compared to thermoplastic polymers. Properties of thermosets depend to a large extent on chemical structure and cross-linking density, as well as on the processing conditions such as temperature and pressure. Computer simulations for property prediction reduce experimental costs and help with accelerating the development and applicability of these materials. Recently, molecular dynamics simulations have proven to be powerful in understanding the behavior of state-of-the-art thermosetting polymers. * Corresponding author. addresses: Natalia.Shenogina@wright.edu (N.B. Shenogina), mtsige@ uakron.edu (M. Tsige). Atomistic molecular dynamics simulations reported in the literature have provided great insight into the elastic response of highly cross-linked polymer networks. Wu and Xu [1] calculated elastic moduli for diglycidyl ether of bisphenol A (DGEBA) cured with isophorone diamine (IPD) using the static deformation approach. It was seen that use of the DREIDING force field resulted in unrealistically high elastic constants whereas the COMPASS force field yielded more reasonable but still high values compared to experimental measurements. Heine et al. [2] calculated the elastic modulus of the united atom model of poly(dimethylsiloxane) (PDMS) networks as a function of strand length and found that to be in qualitative agreement with experimental data. Fan and Yuen [3] used a PCFF forcefield for EPON862/TETA (triethylenetetramine) structures and calculated Young s modulus higher than the experimental value. Tack and Ford [4] used fully atomistic MD simulations of EPON862/DETDA structures of oligomeric mixtures using CFF91 and COMPASS force fields, and their calculated value of bulk /$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved.

2 N.B. Shenogina et al. / Polymer 54 (2013) 3370e modulus was also found to be higher than the experimental value of a similar material (DGEBA/IPD). Clancy et al. [5] simulated curing of DGEBA/DETDA systems up to 86%. While their calculated Young s and shear moduli variations with temperature demonstrated monotonic decrease as expected, the dependence of these moduli on degree of cure was less consistent, particularly for high degrees of cure. Li and Strachan [6,7] studied EPON-862/DETDA systems containing up to 16,000 atoms using DREIDING force field with atomic charges obtained using electronegativity equalization method. An increase in Young s modulus with conversion degree was seen, but the trend of Poisson s ratio was not clear, since the predicted values were extremely scattered, ranging from 0.2 to 0.5. Bandyopadhyay et al. [8,9] modeled EPON862/DETDA systems containing up to 25,000 atoms and cured up to 76%. They applied different modes of deformation to the structures to directly obtain two elastic constants and then used linear elastic formulae assuming isotropic materials to calculate the other two. While the Young s and shear moduli were seen to increase, bulk modulus was found to slightly decrease with conversion degree. Although all of the above studies have made significant progress in predicting the mechanical response of thermosetting polymer networks, many questions still remain unanswered. For example, the influence of degree of conversion and length of strands between cross-links on mechanical properties is still not fully understood, mainly due to difficulties in creating realistic systems with conversion degree consistent with that typical in experiments. Moreover, small system sizes used in simulations lead to substantial amounts of scattering in mechanical properties. Besides, the choice of force field is found to be critical in obtaining reasonable values of elastic constants. It must also be noted that a complete range of mechanical studies are generally not performed using atomistic simulations. Typically, one or two modes of deformation are simulated with the assumption that elastic constants obtained from those can fully characterize the elastic response of the particular amorphous polymeric material. However, the long simulation times needed to reproduce macroscopic mechanical behavior of thermosets has led to studies using larger deformations (beyond the elastic limit) and with limited statistical sampling, causing inaccurate prediction of individual elastic constants. To answer some of the above-mentioned questions and to also provide a systematic study of available modeling approaches, we have recently investigated the dependence of thermo-mechanical properties of thermosetting polymer DGEBA/DETDA on the extent of curing reaction, length of resin strands and size of simulation cell at two different temperatures [10]. We used the effective crosslinking procedure developed by Accelrys [11] that allows construction of highly cross-linked polymer networks having structural characteristics close to those in real systems. The resulting systems are characterized by high conversion degrees and are free of internal stresses and geometrical distortions. In our recent paper [10] we found that while properties such as density, coefficient of thermal expansion and glass transition temperature were found to be in good agreement with experimental data available in the literature [12e17], the values of elastic constants, calculated using static deformation approach, showed notable deviation from values reported in experiments [16,18e23]. In the present work we are employing a dynamic deformation approach, which takes into account both the potential energy contribution and the influence of thermal motions in the structure on its mechanical behavior. We explore the influence of temperature, extent of curing and length of epoxy strands on elastic properties of thermosetting materials. To verify that the acquired elastic properties meet assumptions of linear elasticity, we performed uniaxial, volumetric and shear deformation of the simulation cells and determined all four elastic constants independently using small deformations. We also proposed a novel methodology of Poisson s ratio determination that reduces statistical errors and avoid time scale limitations of molecular dynamics simulations. 2. Methodology 2.1. Systems of interest and simulation details We focused on a widely used resin-hardener system composed of DGEBA (diglycidyl ether of bisphenol A) epoxy oligomers and aromatic amine hardener DETDA (diethylene toluene diamine). The molecular structures of the initial components used for the crosslinking reaction are shown in Fig. 1 and, we examined stochiometrically balanced compositions of reactants permitting a theoretical conversion of 100%. Initially, the reactants were randomly distributed in a simulation box using the Amorphous Cell module of the Materials Studio commercial package [11], and all subsequent molecular dynamics simulations were done with the Discover module of this software. Atomic interactions are based on the Class II force field COMPASS [24], which has been shown to provide accurate predictions of thermo-mechanical properties of thermosetting polymers [1,4,10]. The cross-linking method developed by Accelrys [11] was used to build highly cross-linked polymer networks. With this crosslinking method, it is possible to achieve high extents of reaction typical of real systems, with no internal stresses and no geometrical distortions in the structures. More details about this method are given in Ref. [10]. To study the influence of the extent of curing reaction on the mechanical properties of thermoset networks (DGEBA/DETDA epoxy resin), six conversion degrees ranging from 50% to 95% were selected for each system. These structures were then equilibrated at room temperature and at elevated temperature, as discussed in detail in our earlier paper [10], to determine the mechanical properties of the obtained networks both in glassy and rubbery states. Elevated temperature was chosen to 480 K which is above the range of glass transition temperatures (396e430 K) found for these structures in our previous study [10]. To examine the effect of epoxy chain length on the mechanical properties, we constructed Fig. 1. (a) Epoxy resin: DGEBA (diglycidyl ether of bisphenol A) with activated reactive sites (yellow); (b) aromatic amine hardener: DETDA (diethylene toluene diamine). Reactive sites (amine groups) are highlighted in yellow. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3 3372 N.B. Shenogina et al. / Polymer 54 (2013) 3370e3376 several systems using short epoxy oligomers of one, two or four monomer resin molecules, referred hereafter as mono-, di-, and tetramers, respectively. The system based on epoxy monomers consisted of 512 epoxy monomers and 256 cross-linkers and will be denoted hereafter by (512,256). Similarly, dimer and tetramer based structures are denoted by (256,128) and (128,64), respectively. Note that similar system sizes were used for all cases. For better prediction of the elastic response of polymer networks and to reduce statistical scattering due to nanoscopically small simulation cells, data obtained from five topologically independent structures were averaged for each extent of reaction and epoxy chain length Choice of deformation approach In an earlier paper [10], we used the so-called static deformation approach [25], where uniaxial and shear deformations of a small magnitude are instantly applied to a simulation cell in different directions and energy minimization subsequently performed. This approach was introduced by Theodoru and Suter to study small deformations of polymers at relatively low temperatures. Due to its computational efficiency, this methodology allows the analysis of a large number of nanoscopically small volume elements to compensate for large scattering in mechanical properties data and to partially circumvent size limitations of the molecular dynamics simulations method. However, this approach takes into account only potential energy contribution to the mechanical response of the material, neglecting the contribution of thermal motions. It is known that local motions in polymers occur even at temperatures a few degrees above absolute zero and become significant in the rubbery state. For this reason, deformation in polymers should be considered an intrinsically dynamic event that involves thermal activation of molecular rearrangements and, therefore, the effect of temperature on the mechanical properties cannot be ignored. This assertion was confirmed in the course of our previous study, where we found that static approach did not accurately predict the mechanical response, especially at elevated temperatures. In the present study, we modeled stressestrain behavior using a dynamic approach, taking into account potential energy as well as entropic and vibrational contributions to the elastic response of thermosetting polymers. The deformation of a given structure was applied in stepwise fashion at a rate of 10 8 [1/s], which is typical for MD simulations. At each step the structure was deformed by 0.1% followed by energy minimization and equilibration at constant temperature and volume for 10 ps. Elastic moduli were then calculated as initial slopes of stresse strain curves obtained using appropriate components of stress and strain tensors. More specifically, Young s modulus was obtained by applying tension and compression uniaxial strains individually at each coordinate direction of the simulation cell and calculated as s ii =ε ii, where s ii and are diagonal elements of the stress and strain tensors, respectively. Bulk modulus describes the material response to uniform pressure. In the present study it was obtained by simultaneously applying equal compression or dilatation strains in all three directions and determined as the initial slope of the curve representing the average of stress tensor diagonal components vs. volumetric deformation as: ½1=3ðsÞŠii þ s jj þ s kk B ¼ (1) ε ii þ ε jj þ ε kk Similarly, shear modulus was obtained by applying shear deformation in each direction and calculated as s ij =ε ij, where s ij and ε ij are off-diagonal elements of the stress and strain tensors. Poisson s ratio was obtained in uniaxial deformation mode. The details of its determination are discussed in the next subsection. Keeping in mind that nanoscopically small structures investigated using MD simulations are not perfectly isotropic and homogeneous, we performed uniaxial and shear deformations of simulation cells in the three different directions and subsequently averaged the acquired data. It is also important to keep in mind that one of the characteristic features of amorphous polymers is shallow energy landscape [26]. As a consequence of the ability to rearrange at the molecular level, even at low deformations, linear mechanical response of these materials can be observed only for very small deformations, typically below 1%. At such low deformation, the statistical scattering of the stresses computed from MD simulations can be significant. Nevertheless, the slope of the stressestrain curve at infinitesimal deformation could be estimated by taking into account that it smoothly changes at small deformations from compression to tension deformation mode. Hence, to a first approximation, the slope in the vicinity of zero deformation can be calculated as an average over the slopes of tension and compression stress-strain curves measured within a few percent of deformation. In this work, we deformed simulation cells up to 1.5% in uniaxial, volumetric and shear deformation modes, and ε ii elastic moduli were determined as an average over the slopes in tension and compression modes Poisson s ratio calculation Poisson s ratio can be measured as the ratio of lateral to longitudinal strain in uniaxial tests. However, due to the viscoelastic behavior of polymers in the course of uniaxial deformation, lateral stresses approach equilibrium zero values over a finite period of time and lateral contractions are strain rate dependent. Keeping in mind that experimental and MD simulations strain rates differ by 10e12 orders of magnitude one can expect simulated Poisson s ratio values to be lower than experimental values. To address this issue and to obtain realistic values of Poisson s ratio, we employed the following procedure to calculate this elastic constant. We applied stepwise uniaxial tension and simultaneous compression deformation in the transverse directions, corresponding to certain values of Poisson s ratio. Several Poisson s ratios ranging from 0.0 to 0.5 were probed for each structure both at room and at elevated temperature and lateral stresses were monitored in the course of deformation. At constant lateral conditions, when transverse shrinkage does not occur during uniaxial tension (n ¼ 0.0), positive lateral stresses are developed during tensile deformation yielding positive slope of the lateral stress-axial strain curve s ii =ε kk. Such a behavior in simulated systems is predictable as typical experimental values of thermoset Poisson s ratio fall in the range of 0.33e0.40 at room temperature and rise to about 0.5 in the rubbery state [19e22]. In another limiting case of incompressible material (n ¼ 0.5), lateral stresses approach equilibrium zero values at elevated temperature (rubbery state), giving negligible s ii =ε kk slopes. At room temperature, these slopes are negative in the case of tension, denoting too-large lateral contraction for glassy state. Probing up to six values of Poisson s ratio reveals linear dependence of the slopes of the lateral stress-axial strain curves s ii =ε kk on the probed Poisson s ratios. Fig. 2 represents such a dependence, which characterizes lateral stresses developed in the structure during uniaxial deformation at various probed Poisson s ratios. Such dependences were plotted for each extent of reaction, both at room and at elevated temperature. Each point on the plot represents an average over five structures and all three directions in tension and compression simulations to reduce statistical errors and take into account anisotropic effects. To

4 N.B. Shenogina et al. / Polymer 54 (2013) 3370e level within the range typically employed in MD simulations of amorphous polymers, elastic constant values obtained using two different deformation modes could be different beyond statistical errors [28]. It is thus very important to verify the compatibility of the results obtained using the different deformation modes. There is a set of experimental papers [28e32] in which the authors examined the experimental limitations of deriving elastic constants from properties measured directly. Similar to these studies, we conducted independent simulations of uniaxial, volumetric and shear deformations to acquire all four elastic constants. We then used the theory of linear elasticity to calculate elastic moduli from any two constants obtained by means of direct simulations. Finally, the calculated moduli were compared with the moduli obtained from direct simulations. The similarities and differences between moduli will be discussed at the end of the next section. To the best of our knowledge, this is a first reported comprehensive study of the mechanical response of thermosetting polymers which compares elastic constants acquired by direct simulations with the corresponding material functions computed using the theory of linear elasticity. Fig. 2. The ratio of lateral stress s ii to axial strain ε kk, developed in the course of uniaxial deformation of monomer-based structure at room temperature and cured to 95%, at various probed Poisson s ratios. s ii and ε kk are diagonal elements of the stress and strain tensors, respectively. Indexes denote transverse (i) and longitudinal (k) directions. The red line represents a linear fit of the data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) estimate the Poisson s ratio at which lateral stresses are not developed during deformation of a given structure, a linear fit to the data was used (Fig. 2) to interpolate the Poisson s ratio value at the intersection of the fit with the horizontal axis. This Poisson s ratio value is identified as true Poisson s ratio of the given structure. This method was applied for each extent of reaction to determine the Poisson s ratios in the glassy state (at 298 K) and rubbery state (at 480 K). For a given structure at a given temperature Poisson s ratio found using this approach was then used in the direct measurement of Young s modulus, i.e., by keeping the Poisson ratio of the structure to this value during uniaxial deformation Comparing simulation results with the predictions of linear elasticity theory The theory of linear elasticity [27] has been successfully used to describe the mechanical response of materials to infinitesimal deformations. The most common experimental means of polymer characterization are uniaxial tension and shear deformations, while bulk (volumetric) response and Poisson s ratio experimental data are limited due to difficulties associated with making precise measurements of very small deformations. Nevertheless, in experiments, having any two elastic constants available from direct measurements of an amorphous polymer at small deformation is considered sufficient to fully characterize the mechanical properties of the system, since the remaining constants are often obtained by using the theory of linear elasticity with the assumption that structures are homogeneous and isotropic. However, nanoscopically small simulation cells of thermosetting polymers used in molecular dynamics simulations are less homogeneous and less isotropic than macroscopic samples used in experiments, as each simulation cell is characterized by a unique distribution of matter and therefore generates unique mechanical properties, causing significant scattering in the data. Furthermore, different deformation modes such as uniaxial, volumetric and shear strains may result in significantly different molecular rearrangements in the structure. Therefore, starting with some deformation 3. Results and discussions 3.1. The role of deformation approach Fig. 3 shows the elastic constants of monomer-based structures obtained from direct deformation simulations using both static (reported in Ref. [10]) and dynamic approaches. Young s, bulk and shear moduli show monotonic increase with the degree of cure both in static and dynamic simulations, reflecting the expected increase in material stiffness. The slopes of dynamic curves, however, were found to be notably lower than that of static curves, as can be seen in the figure. The values of all three elastic moduli at both temperatures, obtained using dynamic simulations, show excellent improvement compared to those found using static simulations, a clear justification that one must fully account for dynamic effects. The Young s modulus of the dynamic simulations is in very good agreement with experimental data at room temperature [18]. Though no experimental data exist for the other moduli of the DGEBA/DETDA thermoset, the values obtained using dynamic simulations are within the range for common thermosetting polymers [19e23]. The simulation results for elastic moduli can be understood if we recall that epoxy polymers demonstrate nonlinear behavior even at very low deformations. In Fig. 4, a representative stresse strain curve obtained from tensile simulations is shown. In this seemingly stepwise curve, regions of increasing of internal stresses alternate with regions of stress relaxation. The relaxation in the stress is due to thermally activated molecular relaxations and results in decreasing Young s modulus values. In contrast to this, the deformation level in static simulations was less than 0.1%, where molecular relaxations are less intensive, causing high stress derivative and unrealistically large elastic moduli values. Similar behavior can be observed at the initial 0.3% deformation portion of the stress-strain curve. Moreover, extremely high deformation rates used in molecular dynamics simulations result in less intensive molecular relaxations in polymeric systems and may increase the elastic moduli values. Poisson s ratios (Fig. 3d) obtained using dynamic simulations show different behavior from previously obtained static simulation results (also shown in the figure) both qualitatively and quantitatively. While the results of the static simulations show no dependence of Poisson s ratio on degree of cure, the dynamic results show a monotonic decrease in Poisson s ratio with degree of cure. The static approach gives unrealistically low values of Poisson s ratio

5 3374 N.B. Shenogina et al. / Polymer 54 (2013) 3370e3376 Fig. 3. Elastic moduli at 298 K (blue closed symbols) and 480 K (black open symbols) as a function of the extent of the reaction obtained from static (squares) and dynamic (circles) simulations: (a) Young s modulus; (b) bulk modulus; (c) shear modulus; (d) Poisson s ratio. Red lines represent experimental values at room temperature: (a) Young s modulus for GDEBA/DETDA structure [18]; (c) shear modulus for similar epoxy structure [23]; (d) Poisson s ratio for similar epoxy structures (lower border of shaded area e Ref. [20]; upper border of shaded area e Refs. [21,22]). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) and will not be discussed further. The values of Poisson s ratio at high temperature are higher than that at room temperature, displaying the behavior typical in rubbery and glassy states. The Poisson s ratio value at high degree of cure at room temperature is found to be within the range of experimental values for similar materials [20e22] Relationships between elastic constants Fig. 5 shows a comparison of the elastic constants obtained from direct simulations and by applying linear elasticity theory both at Fig. 4. Stressestrain curve obtained at uniaxial tension dynamic deformation of monomeric structure at 10 8 [1/s] deformation rate. room and elevated temperatures. Almost all properties are found in excellent mutual agreement. However, both Poisson s ratio and bulk modulus calculated using direct Young s and shear simulation results, represented as yðe; GÞ and BðE; GÞ, respectively, are characterized by significant scattering of the data (Fig. 5b, d). A close examination of the following equations: EG B E; G ¼ 3ð3G EÞ yðe; GÞ ¼ (2) E 1 (3) 2G used to calculate bulk modulus and Poisson s ratio, respectively, reveals that these calculated values are sensitive to the ratio of Young s modulus to shear modulus. In Eq. (2), when the Young s modulus is about three times higher than shear modulus, the uncertainty in the calculated bulk modulus value grows dramatically and may grow to several orders of magnitude. In addition, since G is small, especially at elevated temperature, the ratio of E and G values in the calculation of Poisson s ratio using Eq. (3) causes significant scattering of the data. Note that a good agreement between all four elastic constants determined by different deformation modes is expected only under certain conditions, since molecular mechanisms contributing to different deformation modes are not the same. For instance, uniform compression involves local motions of molecular segments, while shear deformation involves both local and extensive molecular rearrangements. Such diversity in mechanical responses can cause significant variation in the values of elastic constants obtained by different deformation modes if large strains are imposed. Mutual agreement between all four elastic constants in our simulations confirms that slopes of the stressestrain curves at

6 N.B. Shenogina et al. / Polymer 54 (2013) 3370e Fig. 5. Elastic moduli at 298 K (closed symbols) and 480 K (open symbols) as a function of the extent of the reaction calculated using direct dynamic simulations (squares) and using linear elasticity theory. (a) Young s modulus: E(B,G) e red circles; E(B,n) e green triangles; E(G,n) e blue stars. (b) Bulk modulus: B(E,G) e red circles; B(E,n) e green triangles; B(G,n) e blue stars. (c) Shear modulus: G(B,E) e red circles; G(B,n) e green triangles; G(E,n) e blue stars. (d) Poisson s ratio: n(b,e) e red circles; n(b,g) e green triangles; n(e,g) e blue stars. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 6. Elastic moduli at 298 K (closed symbols) and 480 K (open symbols) as a function of the extent of the reaction for the atomic structures built using monomers (black squares), dimers (red circles) and tetramers (green triangles) of the epoxy resin: (a) Young s modulus; (b) bulk modulus; (c) shear modulus; (d) Poisson s ratio. Solid lines represent linear fits to the data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

7 3376 N.B. Shenogina et al. / Polymer 54 (2013) 3370e3376 infinitesimal deformations are calculated correctly and the acquired results were within elastic limits The role of epoxy chain length Fig. 6 represents elastic constants of monomer-, dimer- and tetramer-based structures obtained from direct deformation simulations using the dynamic approach. The dynamically obtained Young s, bulk and shear moduli of all oligomer-based structures show monotonic increase with the degree of cure, while Poisson s ratio values demonstrate notable decrease with extent of reaction, particularly at high temperature. It can be seen that elastic properties of oligomer-based structures reveal the same trend: structures with shorter distance between cross-links tend to show more pronounced dependence on degree of cure as can be observed from the slope of the fitted lines. 4. Conclusions This study uses a proven cross-linking procedure on large polymer systems containing up to 35,000 atoms, to generate stressfree thermoset networks with high degree of cure. A dynamic deformation approach has been used to simulate the elastic response of the generated structures and predict their mechanical properties. The dependence of elastic constants on process parameters such as temperature, degree of conversion and length of resin strands has been investigated in these simulations. Young s, shear and bulk moduli, as well as Poisson s ratio, were obtained directly at small deformations from atomistic simulations using uniaxial, volumetric and shear deformation modes. A novel algorithm to calculate the Poisson s ratio was proposed and found to successfully reduce statistical errors and circumvent the time-scale limitations of molecular dynamics simulations. Finally, all simulation results of individual parameters were shown to compare favorably with values calculated using linear elasticity theory. The dynamic deformation approach has provided realistic values for mechanical properties, both in glassy and rubbery states. Values of Young s, shear and bulk moduli and Poisson s ratio at high extents of curing reaction were found to be in very good agreement with experimental data of actual cured polymers and show excellent improvement compared to elastic constants calculated using the static deformation approach. This finding supports that thermal motions have significant influence on the mechanical response of highly cross-linked polymers, both in glassy and rubbery states. Further insight into the role of extent of reaction and length of resin strands on elastic properties of thermosets was attained showing realistic mechanical response. To the best of our knowledge, this is the first reported paper that predicts all four elastic coefficients of a thermosetting polymer by direct simulation and then successfully compares them with the corresponding values computed using the theory of linear elasticity. The approaches used in this study exhibit significant promise in their ability to predict the mechanical behavior of highly crosslinked polymeric materials. Acknowledgment This work was primarily supported by the Low Density Materials Program of the Air Force Office of Scientific Research (gs1) Grant Number: FA The authors gratefully acknowledge Dr. Charles Lee (AFOSR) for valuable discussions, Wright State University for partial salary support, and the Air Force Research Laboratory DoD Supercomputing Resource Center High Performance Computing for computer time. References [1] Wu C, Xu W. Polymer 2006;47:6004. [2] Heine DR, Grest GS, Lorenz CD, Tsige M, Stevens MJ. Macromolecules 2004;37: [3] Fan HB, Yuen MMF. Polymer 2007;48:2174. [4] Tack JL, Ford DMJ. Mol Graphics Modell 2008;26:1269. [5] Clancy TC, Frankland SJV, Hinkley JA, Gates TS. Polymer 2009;50:2736. [6] Li C, Strachan A. Polymer 2010;51:6058. [7] Li C, Strachan A. Polymer 2011;52:2920. [8] Bandyopadhyay A, Valavala PK, Clancy TC, Wise KE, Odegard GM. Polymer 2011;52:2445. [9] Bandyopadhyay A, Odegard GM. Modelling Simul Mater Sci Eng 2012;20: [10] Shenogina NB, Tsige M, Patnaik SS, Mukhopadhyay SM. Macromolecules 2012;45:5307. [11] Accelrys Software Inc. Materials studio. Available from: HYPERLINK " \o " products/materials-studio/" [accessed ]. [12] Gao JG, Li YF, Zhao M, Liu GD. J Appl Polym Sci 2000;78:794. [13] Jansen BJP, Tamminga KY, Meijer HEH, Lemstra PJ. Polymer 1999;40:5601. [14] Ratna D, Manoj NR, Varley R, Raman RKS, Simon GP. Polym Int 2003;52:1403. [15] Ratna D, Varley R, Singh RK, Simon GP. J Mater Sci 2003;38:147. [16] Shen L, Wang L, Liu TX, He C. Macromol Mater Eng 2006;291:1358. [17] Liu W, Varley RJ, Simon GP. Polymer 2006;47:2091. [18] Qi B, Zhang QX, Bannister M, Mai YW. Compos Struct 2006;75:514. [19] Kalantar J, Drzal LTJ. Mater Sci 1990;25:4186. [20] Kalantar J, Drzal LTJ. Mater Sci 1990;25:4194. [21] O Brien DJ, Sottos NR, White SR. Exp Mech 2007;47:237. [22] Tcharkhtchi A, Faivre S, Roy LE, Trotignon JP, Verdu J. J Mater Sci 1996;31: [23] Drzal LT. Mater Sci Eng A 1990;126:289. [24] Sun H. J Phys Chem B 1998;102:7338. [25] Theodorou DN, Suter UW. Macromolecules 1986;19:139. [26] Wales DJ. Energy landscapes. Cambridge: Cambridge University Press; [27] Landau LD, Lifshitz EM. Theory of elasticity. Oxford: Butterworth Heinemann; [28] Sane SB, Knauss WG. Mech Time-Depend Mater 2001;5:325. [29] Sane SB, Knauss WG. Mech Time-depend Mater 2001;5:293. [30] Yee AF, Takemori MT. J Polym Sci 1982;20:205. [31] Lu H, Zhang X, Knauss WG. Polym Eng Sci 1997;37:1053. [32] Deng TH, Knauss WG. Mech Time-Depend Mater 1997;1:33.