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Polymer 54 (2013) 3370e3376

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Polymer

journal homepage: www.elsevier .com/locate/polymer

Molecular modeling of elastic properties of thermosetting polymersusing a dynamic deformation approach

Natalia B. Shenogina a, Mesfin Tsige b,*, Soumya S. Patnaik c, Sharmila M. Mukhopadhyay a

aDepartment of Mechanical and Materials Engineering, Wright State University, Dayton, OH, USAbDepartment of Polymer Science, University of Akron, Akron, OH, USAcAerospace Systems Directorate, Wright-Patterson Air Force Base, Dayton, OH, USA

a r t i c l e i n f o

Article history:Received 14 February 2013Received in revised form13 April 2013Accepted 15 April 2013Available online 23 April 2013

Keywords:Dynamic deformation simulationsMolecular dynamicsElastic properties of epoxy

* Corresponding author.E-mail addresses: Natalia.Shenogina@wright.edu

uakron.edu (M. Tsige).

0032-3861/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.polymer.2013.04.034

a b s t r a c t

This paper employs fully atomistic molecular dynamics simulations to characterize relationships be-tween structural and elastic properties of thermosetting polymers both in glassy and rubbery state. Thepolymer system investigated consists of epoxy resin DGEBA and hardener DETDA. An effective cross-linking procedure that enables generation of thermoset structures containing up to 35000 atoms withrealistic structural characteristics was used. A dynamic deformation approach has been used that takesinto consideration both potential energy and thermal motions in the structure. Small uniaxial, volumetricand shear deformations were applied to the systems to obtain elastic moduli. A method to independentlydetermine Poisson’s ratio was proposed that reduces statistical errors and circumvents the time scalelimitations of molecular dynamics simulations. The influence of variables such as extent of curing andlength of epoxy strands on elastic response at various temperatures was explored. Expected trends in thedependence of the elastic constants on these practical process parameters were shown. The relationshipbetween the four independently calculated elastic constants was seen to comply with those predicted bythe classical theory of linear elasticity in an isotropic medium, which provides confidence in the validityof these simulations. Moreover, the elastic properties obtained are also in good agreement with exper-imental data reported in the literature. Close agreements between predicted elastic constants andexperimentally measured values underscore the ability of the approaches used in this study to providerealistic predictions of the mechanical response of thermosetting polymers, both in glassy and rubberystates. These results show significant improvement over earlier studies based on a static approach whichtakes into account the potential energy contribution to the elastic response but ignores temperatureeffect.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Thermosetting polymers are important materials in a variety ofapplications due to their high thermal and structural stabilitycompared to thermoplastic polymers. Properties of thermosetsdepend to a large extent on chemical structure and cross-linkingdensity, as well as on the processing conditions such as tempera-ture and pressure. Computer simulations for property predictionreduce experimental costs and help with accelerating the devel-opment and applicability of these materials. Recently, moleculardynamics simulations have proven to be powerful in understandingthe behavior of state-of-the-art thermosetting polymers.

(N.B. Shenogina), mtsige@

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Atomistic molecular dynamics simulations reported in theliterature have provided great insight into the elastic response ofhighly cross-linked polymer networks. Wu and Xu [1] calculatedelastic moduli for diglycidyl ether of bisphenol A (DGEBA) curedwith isophorone diamine (IPD) using the static deformationapproach. It was seen that use of the DREIDING force field resultedin unrealistically high elastic constants whereas the COMPASS forcefield yielded more reasonable but still high values compared toexperimental measurements. Heine et al. [2] calculated the elasticmodulus of the united atom model of poly(dimethylsiloxane)(PDMS) networks as a function of strand length and found that tobe 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 exper-imental value. Tack and Ford [4] used fully atomistic MD simula-tions of EPON862/DETDA structures of oligomeric mixtures usingCFF91 and COMPASS force fields, and their calculated value of bulk

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modulus was also found to be higher than the experimental valueof a similar material (DGEBA/IPD). Clancy et al. [5] simulated curingof DGEBA/DETDA systems up to 86%.While their calculated Young’sand shear moduli variations with temperature demonstratedmonotonic decrease as expected, the dependence of these modulion degree of cure was less consistent, particularly for high degreesof cure. Li and Strachan [6,7] studied EPON-862/DETDA systemscontaining up to 16,000 atoms using DREIDING force field withatomic charges obtained using electronegativity equalizationmethod. An increase in Young’s modulus with conversion degreewas seen, but the trend of Poisson’s ratio was not clear, since thepredicted values were extremely scattered, ranging from 0.2 to 0.5.Bandyopadhyay et al. [8,9] modeled EPON862/DETDA systemscontaining up to 25,000 atoms and cured up to 76%. They applieddifferent modes of deformation to the structures to directly obtaintwo elastic constants and then used linear elastic formulaeassuming isotropic materials to calculate the other two. While theYoung’s and shear moduli were seen to increase, bulk modulus wasfound to slightly decrease with conversion degree.

Although all of the above studies have made significant progressin predicting the mechanical response of thermosetting polymernetworks, many questions still remain unanswered. For example,the influence of degree of conversion and length of strands be-tween cross-links on mechanical properties is still not fully un-derstood, mainly due to difficulties in creating realistic systemswith conversion degree consistent with that typical in experiments.Moreover, small system sizes used in simulations lead to substan-tial amounts of scattering in mechanical properties. Besides, thechoice of force field is found to be critical in obtaining reasonablevalues of elastic constants. It must also be noted that a completerange of mechanical studies are generally not performed usingatomistic simulations. Typically, one or two modes of deformationare simulated with the assumption that elastic constants obtainedfrom those can fully characterize the elastic response of theparticular amorphous polymeric material. However, the longsimulation times needed to reproduce macroscopic mechanicalbehavior 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 alsoprovide a systematic study of available modeling approaches, wehave recently investigated the dependence of thermo-mechanicalproperties of thermosetting polymer DGEBA/DETDA on the extentof curing reaction, length of resin strands and size of simulation cellat two different temperatures [10]. We used the effective cross-linking procedure developed by Accelrys [11] that allows con-struction of highly cross-linked polymer networks having struc-tural characteristics close to those in real systems. The resultingsystems are characterized by high conversion degrees and are freeof internal stresses and geometrical distortions. In our recent paper[10] we found that while properties such as density, coefficient ofthermal expansion and glass transition temperature were found to

Fig. 1. (a) Epoxy resin: DGEBA (diglycidyl ether of bisphenol A) with activated reactive sitessites (amine groups) are highlighted in yellow. (For interpretation of the references to colo

be in good agreement with experimental data available in theliterature [12e17], the values of elastic constants, calculated usingstatic deformation approach, showed notable deviation fromvaluesreported in experiments [16,18e23].

In the present work we are employing a dynamic deformationapproach, which takes into account both the potential energycontribution and the influence of thermal motions in the structureon its mechanical behavior. We explore the influence of tempera-ture, extent of curing and length of epoxy strands on elastic prop-erties of thermosetting materials. To verify that the acquired elasticproperties meet assumptions of linear elasticity, we performeduniaxial, volumetric and shear deformation of the simulation cellsand determined all four elastic constants independently usingsmall deformations. We also proposed a novel methodology ofPoisson’s ratio determination that reduces statistical errors andavoid 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 composedof DGEBA (diglycidyl ether of bisphenol A) epoxy oligomers andaromatic amine hardener DETDA (diethylene toluene diamine). Themolecular structures of the initial components used for the cross-linking reaction are shown in Fig. 1 and, we examined stochio-metrically balanced compositions of reactants permitting a theo-retical conversion of 100%.

Initially, the reactants were randomly distributed in a simula-tion box using the Amorphous Cell module of the Materials Studiocommercial package [11], and all subsequent molecular dynamicssimulations 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 ofthermo-mechanical properties of thermosetting polymers [1,4,10].

The cross-linking method developed by Accelrys [11] was usedto build highly cross-linked polymer networks. With this cross-linking method, it is possible to achieve high extents of reactiontypical of real systems, with no internal stresses and no geometricaldistortions in the structures. More details about this method aregiven in Ref. [10].

To study the influence of the extent of curing reaction on themechanical properties of thermoset networks (DGEBA/DETDAepoxy resin), six conversion degrees ranging from 50% to 95% wereselected for each system. These structures were then equilibrated atroom temperature and at elevated temperature, as discussed indetail in our earlier paper [10], to determine the mechanicalproperties of the obtained networks both in glassy and rubberystates. Elevated temperature was chosen to 480 K which is abovethe range of glass transition temperatures (396e430 K) found forthese structures in our previous study [10]. To examine the effect ofepoxy chain length on the mechanical properties, we constructed

(yellow); (b) aromatic amine hardener: DETDA (diethylene toluene diamine). Reactiveur in this figure legend, the reader is referred to the web version of this article.)

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several systems using short epoxy oligomers of one, two or fourmonomer resin molecules, referred hereafter as mono-, di-, andtetramers, respectively. The system based on epoxy monomersconsisted of 512 epoxymonomers and 256 cross-linkers andwill bedenoted hereafter by (512,256). Similarly, dimer and tetramerbased structures are denoted by (256,128) and (128,64), respec-tively. Note that similar system sizes were used for all cases.

For better prediction of the elastic response of polymer net-works and to reduce statistical scattering due to nanoscopicallysmall simulation cells, data obtained from five topologically inde-pendent structures were averaged for each extent of reaction andepoxy chain length.

2.2. Choice of deformation approach

In an earlier paper [10], we used the so-called static deformationapproach [25], where uniaxial and shear deformations of a smallmagnitude are instantly applied to a simulation cell in differentdirections and energy minimization subsequently performed. Thisapproach was introduced by Theodoru and Suter to study smalldeformations of polymers at relatively low temperatures. Due to itscomputational efficiency, this methodology allows the analysis of alarge number of nanoscopically small volume elements tocompensate for large scattering in mechanical properties data andto partially circumvent size limitations of the molecular dynamicssimulations method. However, this approach takes into accountonly potential energy contribution to the mechanical response ofthe material, neglecting the contribution of thermal motions. It isknown that local motions in polymers occur even at temperatures afew degrees above absolute zero and become significant in therubbery state. For this reason, deformation in polymers should beconsidered an intrinsically dynamic event that involves thermalactivation of molecular rearrangements and, therefore, the effect oftemperature on the mechanical properties cannot be ignored. Thisassertionwas confirmed in the course of our previous study, wherewe found that static approach did not accurately predict the me-chanical response, especially at elevated temperatures.

In the present study, wemodeled stressestrain behavior using adynamic approach, taking into account potential energy as well asentropic and vibrational contributions to the elastic response ofthermosetting polymers. The deformation of a given structure wasapplied in stepwise fashion at a rate of 108 [1/s], which is typical forMD simulations. At each step the structure was deformed by 0.1%followed by energy minimization and equilibration at constanttemperature and volume for 10 ps.

Elastic moduli were then calculated as initial slopes of stressestrain curves obtained using appropriate components of stress andstrain tensors. More specifically, Young’s modulus was obtained byapplying tension and compression uniaxial strains individually ateach coordinate direction of the simulation cell and calculated assii=εii, where sii and are diagonal elements of the stress and straintensors, respectively.

Bulk modulus describes the material response to uniformpressure. In the present study it was obtained by simultaneouslyapplying equal compression or dilatation strains in all three di-rections and determined as the initial slope of the curve repre-senting the average of stress tensor diagonal components vs.volumetric deformation as:

B ¼�½1=3ðsÞ�ii þ sjj þ skk

�εii þ εjj þ εkk

(1)

Similarly, shear modulus was obtained by applying shear defor-mation in each direction and calculated as sij=εij, where sij and εijare off-diagonal elements of the stress and strain tensors. Poisson’s

ratio was obtained in uniaxial deformation mode. The details of itsdetermination are discussed in the next subsection.

Keeping in mind that nanoscopically small structures investi-gated using MD simulations are not perfectly isotropic and homo-geneous, we performed uniaxial and shear deformations ofsimulation cells in the three different directions and subsequentlyaveraged the acquired data.

It is also important to keep in mind that one of the characteristicfeatures 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 thesematerials can be observed only for very small deformations, typi-cally below 1%. At such low deformation, the statistical scattering ofthe stresses computed from MD simulations can be significant.Nevertheless, the slope of the stressestrain curve at infinitesimaldeformation could be estimated by taking into account that itsmoothly changes at small deformations from compression totension deformation mode. Hence, to a first approximation, theslope in the vicinity of zero deformation can be calculated as anaverage over the slopes of tension and compression stress-straincurves measured within a few percent of deformation. In thiswork, we deformed simulation cells up to 1.5% in uniaxial, volu-metric and shear deformation modes, and εii elastic moduli weredetermined as an average over the slopes in tension andcompression modes.

2.3. Poisson’s ratio calculation

Poisson’s ratio can be measured as the ratio of lateral to longi-tudinal strain in uniaxial tests. However, due to the viscoelasticbehavior of polymers in the course of uniaxial deformation, lateralstresses approach equilibrium zero values over a finite period oftime and lateral contractions are strain rate dependent. Keeping inmind that experimental and MD simulations strain rates differ by10e12 orders of magnitude one can expect simulated Poisson’sratio values to be lower than experimental values.

To address this issue and to obtain realistic values of Poisson’sratio, we employed the following procedure to calculate this elasticconstant. We applied stepwise uniaxial tension and simultaneouscompression deformation in the transverse directions, corre-sponding to certain values of Poisson’s ratio. Several Poisson’s ratiosranging from 0.0 to 0.5 were probed for each structure both at roomand at elevated temperature and lateral stresses were monitored inthe course of deformation. At constant lateral conditions, whentransverse shrinkage does not occur during uniaxial tension(n ¼ 0.0), positive lateral stresses are developed during tensiledeformation yielding positive slope of the lateral stress-axial straincurve sii=εkk. Such a behavior in simulated systems is predictable astypical experimental values of thermoset Poisson’s ratio fall in therange of 0.33e0.40 at room temperature and rise to about 0.5 in therubbery state [19e22]. In another limiting case of incompressiblematerial (n ¼ 0.5), lateral stresses approach equilibrium zero valuesat elevated temperature (rubbery state), giving negligible sii=εkkslopes. At room temperature, these slopes are negative in the caseof tension, denoting too-large lateral contraction for glassy state.Probing up to six values of Poisson’s ratio reveals linear dependenceof the slopes of the lateral stress-axial strain curves sii=εkk on theprobed Poisson’s ratios.

Fig. 2 represents such a dependence, which characterizes lateralstresses developed in the structure during uniaxial deformation atvarious probed Poisson’s ratios. Such dependences were plotted foreach extent of reaction, both at room and at elevated temperature.Each point on the plot represents an average over five structuresand all three directions in tension and compression simulations toreduce statistical errors and take into account anisotropic effects. To

Fig. 2. The ratio of lateral stress sii to axial strain εkk , developed in the course ofuniaxial deformation of monomer-based structure at room temperature and cured to95%, at various probed Poisson’s ratios. sii and εkk are diagonal elements of the stressand strain tensors, respectively. Indexes denote transverse (i) and longitudinal (k) di-rections. The red line represents a linear fit of the data. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version ofthis article.)

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estimate the Poisson’s ratio at which lateral stresses are notdeveloped during deformation of a given structure, a linear fit tothe data was used (Fig. 2) to interpolate the Poisson’s ratio value atthe intersection of the fit with the horizontal axis. This Poisson’sratio value is identified as true Poisson’s ratio of the given structure.This method was applied for each extent of reaction to determinethe 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’sratio found using this approach was then used in the direct mea-surement of Young’s modulus, i.e., by keeping the Poisson ratio ofthe structure to this value during uniaxial deformation.

2.4. Comparing simulation results with the predictions of linearelasticity theory

The theory of linear elasticity [27] has been successfully used todescribe the mechanical response of materials to infinitesimal de-formations. The most common experimental means of polymercharacterization are uniaxial tension and shear deformations, whilebulk (volumetric) response and Poisson’s ratio experimental dataare limited due to difficulties associated with making precisemeasurements of very small deformations. Nevertheless, in ex-periments, having any two elastic constants available from directmeasurements of an amorphous polymer at small deformation isconsidered sufficient to fully characterize the mechanical proper-ties of the system, since the remaining constants are often obtainedby using the theory of linear elasticity with the assumption thatstructures are homogeneous and isotropic.

However, nanoscopically small simulation cells of thermoset-ting polymers used in molecular dynamics simulations are lesshomogeneous and less isotropic than macroscopic samples used inexperiments, as each simulation cell is characterized by a uniquedistribution of matter and therefore generates unique mechanicalproperties, causing significant scattering in the data. Furthermore,different deformationmodes such as uniaxial, volumetric and shearstrains may result in significantly different molecular rearrange-ments in the structure. Therefore, starting with some deformation

level within the range typically employed in MD simulations ofamorphous polymers, elastic constant values obtained using twodifferent deformation modes could be different beyond statisticalerrors [28]. It is thus very important to verify the compatibility ofthe results obtained using the different deformation modes.

There is a set of experimental papers [28e32] in which the au-thors examined the experimental limitations of deriving elasticconstants frompropertiesmeasureddirectly. Similar to these studies,we conducted independent simulations of uniaxial, volumetric andshear deformations to acquire all four elastic constants.We thenusedthe theory of linear elasticity to calculate elasticmoduli fromany twoconstants obtained by means of direct simulations. Finally, thecalculated moduli were compared with the moduli obtained fromdirect simulations. The similarities and differences between moduliwill be discussed at the end of the next section.

To the best of our knowledge, this is a first reported compre-hensive study of the mechanical response of thermosetting poly-mers which compares elastic constants acquired by directsimulations with the corresponding material functions computedusing the theory of linear elasticity.

3. Results and discussions

3.1. The role of deformation approach

Fig. 3 shows the elastic constants of monomer-based structuresobtained from direct deformation simulations using both static(reported in Ref. [10]) and dynamic approaches.

Young’s, bulk and shear moduli show monotonic increase withthe degree of cure both in static and dynamic simulations,reflecting the expected increase in material stiffness. The slopes ofdynamic curves, however, were found to be notably lower than thatof static curves, as can be seen in the figure. The values of all threeelastic moduli at both temperatures, obtained using dynamic sim-ulations, show excellent improvement compared to those foundusing static simulations, a clear justification that one must fullyaccount for dynamic effects. The Young’s modulus of the dynamicsimulations is in very good agreement with experimental data atroom temperature [18]. Though no experimental data exist for theother moduli of the DGEBA/DETDA thermoset, the values obtainedusing dynamic simulations are within the range for commonthermosetting polymers [19e23].

The simulation results for elastic moduli can be understood ifwe recall that epoxy polymers demonstrate nonlinear behavioreven at very low deformations. In Fig. 4, a representative stressestrain curve obtained from tensile simulations is shown. In thisseemingly stepwise curve, regions of increasing of internal stressesalternate with regions of stress relaxation. The relaxation in thestress is due to thermally activated molecular relaxations and re-sults in decreasing Young’s modulus values. In contrast to this, thedeformation level in static simulations was less than 0.1%, wheremolecular relaxations are less intensive, causing high stress deriv-ative and unrealistically large elastic moduli values. Similarbehavior can be observed at the initial 0.3% deformation portion ofthe stress-strain curve. Moreover, extremely high deformationrates used in molecular dynamics simulations result in less inten-sive molecular relaxations in polymeric systems and may increasethe elastic moduli values.

Poisson’s ratios (Fig. 3d) obtained using dynamic simulationsshow different behavior from previously obtained static simulationresults (also shown in the figure) both qualitatively and quantita-tively. While the results of the static simulations show no depen-dence of Poisson’s ratio on degree of cure, the dynamic results showa monotonic decrease in Poisson’s ratio with degree of cure. Thestatic approach gives unrealistically low values of Poisson’s ratio

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 forGDEBA/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]; upperborder 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.)

N.B. Shenogina et al. / Polymer 54 (2013) 3370e33763374

and will not be discussed further. The values of Poisson’s ratio athigh temperature are higher than that at room temperature, dis-playing the behavior typical in rubbery and glassy states. ThePoisson’s ratio value at high degree of cure at room temperature isfound to be within the range of experimental values for similarmaterials [20e22].

3.2. Relationships between elastic constants

Fig. 5 shows a comparison of the elastic constants obtained fromdirect simulations and by applying linear elasticity theory both at

Fig. 4. Stressestrain curve obtained at uniaxial tension dynamic deformation ofmonomeric structure at 108 [1/s] deformation rate.

room and elevated temperatures. Almost all properties are found inexcellent mutual agreement. However, both Poisson’s ratio andbulk modulus calculated using direct Young’s and shear simulationresults, represented as yðE;GÞ and BðE;GÞ, respectively, are charac-terized by significant scattering of the data (Fig. 5b, d). A closeexamination of the following equations:

B�E;G

�¼ EG

3ð3G� EÞ (2)

yðE;GÞ ¼ E2G

� 1 (3)

used to calculate bulk modulus and Poisson’s ratio, respectively,reveals that these calculated values are sensitive to the ratio ofYoung’s modulus to shear modulus. In Eq. (2), when the Young’smodulus is about three times higher than shear modulus, the un-certainty in the calculated bulk modulus value grows dramaticallyandmay grow to several orders of magnitude. In addition, since G issmall, especially at elevated temperature, the ratio of E and G valuesin the calculation of Poisson’s ratio using Eq. (3) causes significantscattering of the data.

Note that a good agreement between all four elastic constantsdetermined by different deformationmodes is expected only undercertain conditions, since molecular mechanisms contributing todifferent deformation modes are not the same. For instance, uni-form compression involves local motions of molecular segments,while shear deformation involves both local and extensive molec-ular rearrangements. Such diversity in mechanical responses cancause significant variation in the values of elastic constants ob-tained by different deformation modes if large strains are imposed.Mutual agreement between all four elastic constants in our simu-lations confirms that slopes of the stressestrain curves at

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 usinglinear 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) eblue 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 fitsto the data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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N.B. Shenogina et al. / Polymer 54 (2013) 3370e33763376

infinitesimal deformations are calculated correctly and the ac-quired results were within elastic limits.

3.3. The role of epoxy chain length

Fig. 6 represents elastic constants of monomer-, dimer- andtetramer-based structures obtained from direct deformation sim-ulations using the dynamic approach. The dynamically obtainedYoung’s, bulk and shear moduli of all oligomer-based structuresshow monotonic increase with the degree of cure, while Poisson’sratio values demonstrate notable decrease with extent of reaction,particularly at high temperature. It can be seen that elastic prop-erties of oligomer-based structures reveal the same trend: struc-tures with shorter distance between cross-links tend to showmorepronounced dependence on degree of cure as can be observed fromthe slope of the fitted lines.

4. Conclusions

This study uses a proven cross-linking procedure on largepolymer systems containing up to 35,000 atoms, to generate stress-free thermoset networks with high degree of cure. A dynamicdeformation approach has been used to simulate the elasticresponse of the generated structures and predict their mechanicalproperties. The dependence of elastic constants on process pa-rameters such as temperature, degree of conversion and length ofresin strands has been investigated in these simulations. Young’s,shear and bulk moduli, as well as Poisson’s ratio, were obtaineddirectly at small deformations from atomistic simulations usinguniaxial, volumetric and shear deformation modes. A novel algo-rithm to calculate the Poisson’s ratio was proposed and found tosuccessfully reduce statistical errors and circumvent the time-scalelimitations of molecular dynamics simulations. Finally, all simula-tion results of individual parameters were shown to comparefavorably with values calculated using linear elasticity theory.

The dynamic deformation approach has provided realisticvalues for mechanical properties, both in glassy and rubbery states.Values of Young’s, shear and bulk moduli and Poisson’s ratio at highextents of curing reactionwere found to be in very good agreementwith experimental data of actual cured polymers and show excel-lent improvement compared to elastic constants calculated usingthe static deformation approach. This finding supports that thermalmotions have significant influence on the mechanical response ofhighly cross-linked polymers, both in glassy and rubbery states.Further insight into the role of extent of reaction and length of resinstrands on elastic properties of thermosets was attained showingrealistic mechanical response. To the best of our knowledge, this isthe first reported paper that predicts all four elastic coefficients of athermosetting polymer by direct simulation and then successfully

compares themwith the corresponding values computed using thetheory of linear elasticity.

The approaches used in this study exhibit significant promise intheir ability to predict the mechanical behavior of highly cross-linked polymeric materials.

Acknowledgment

This work was primarily supported by the Low Density Mate-rials Program of the Air Force Office of Scientific Research (gs1)Grant Number: FA9550-09-1-0358. The authors gratefullyacknowledge Dr. Charles Lee (AFOSR) for valuable discussions,Wright State University for partial salary support, and the AirForce Research Laboratory DoD Supercomputing Resource CenterHigh Performance Computing for computer time.

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