Molecular Modeling Approach to Prediction of Thermo

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Molecular Modeling Approach to Prediction of Thermo-Mechanical Behavior of Thermoset Polymer Networks Natalia B. Shenogina,*,† Mesfin Tsige,*,‡ Soumya S. Patnaik,§ and Sharmila M. Mukhopadhyay† †

Department of Mechanical and Materials Engineering, Wright State University, Dayton, Ohio, United States Department of Polymer Science, University of Akron, Akron, Ohio, United States § Propulsion Directorate, Air Force Research Laboratory, Dayton, Ohio, United States ‡

ABSTRACT: Molecular dynamics and molecular mechanics simulations have been used to study thermo-mechanical response of highly cross-linked polymers composed of epoxy resin DGEBA and hardener DETDA. The effective cross-linking approach used in this work allowed construction of a set of stress-free molecular models with high conversion degree containing up to 35000 atoms. The generated structures were used to investigate the influence of model size, length of epoxy strands, and degree of cure on thermo-mechanical properties. The calculated densities, coefficients of thermal expansion, and glass transition temperatures of the systems are found to be in good agreement with experimental data. The computationally efficient static deformation approach we used to calculate elastic constants of the systems successfully compensated for the large scattering of the mechanical properties data due to nanoscopically small volume of simulation cells and allowed comparison of properties of similar polymeric networks having minor differences in structure or chemistry. However, some of the elastic constants obtained using this approach were found to be higher than in real macroscopic samples. This can be attributed to both finite-size effect and to the limitations of the static deformation approach to account for dynamic effects. The observed dependence of properties on system size, in this work, can be used to estimate the contribution of large-scale defects and relaxation events into macroscopic properties of the thermosetting materials.

1. INTRODUCTION Chemically cross-linked thermosetting polymer networks are useful in many applications such as coatings, adhesives and matrix components in high performance composites.1 Epoxy resins cross-linked with amine curing agents are popular materials due to their outstanding thermo-mechanical properties such as high-temperature performance, high stiffness and fracture strength. Optimizing of the processing conditions, designing of the new structures with the required properties as well as better understanding of the physical phenomena in polymers imply extensive trial-and-error experimental studies which can be both expensive and time-consuming. In order to reduce the experimental efforts in synthesis and optimization of material properties, computer simulations of the systems of interest can be very useful. Such simulations allow systematic variation of structural or physical parameters of the materials and can significantly lower experimental costs in predicting the properties of new materials. These approaches © 2012 American Chemical Society

may eventually allow for screening of a greater breadth of potential resin chemistries than those that can be tried by experimental testing alone. Among various computational approaches, finite element simulations2,3 are successfully employed to study large-scale events in polymeric materials. However, such methods are often limited by the lack of realistic parameters for inputs to constitutive relations. Two other popular computational approaches used to simulate polymeric systems are Monte Carlo simulations4−10 and molecular dynamics (MD) simulations using a bead−spring model, where each bead represents one or several polymer groups.11−17 While these methods can predict some general trends in the behavior of polymer networks and generate useful insight into the dependence of Received: April 13, 2012 Revised: May 25, 2012 Published: June 7, 2012 5307

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DETDA). While some authors23,26−28 have recently reported simulations showing the influence of the conversion degree on the properties of thermosetting polymers, there is no previous report in the literature that systematically investigated the effect of resin chain length on the thermo-mechanical properties of epoxy networks to the best of our knowledge. The paper is organized as follows. In the next section, we briefly review our simulation methodology, including the systems of interest and the approach we used in building the polymer networks. We present and discuss our results for physical properties of the system in section 3 and for mechanical properties including the effect of resin chain length in section 4. In the last section, we summarize our results as well as the advantages and limitations of the methods used in this study.

the physical properties on the cross-link networks, these studies are not capable of providing specific correlation with the chemical structure of the resin system. However thermomechanical properties of thermosetting networks are known to be significantly dependent on the molecular-level details of the structure. Detailed microscopic information on the physical properties of thermosetting polymers can be obtained from atomistic simulations, which may lead to predictions in quantitative agreement with experiments. Several groups have developed atomistic level methods of constructing models of highly crosslinked polymer networks, and used them to predict their properties.18−28 More notably, Yarovsky and Evans19 proposed a cross-linking technique in which all cross-linking reactions were carried out in one step (so-called static approach). The structure obtained using this method had conversion degree much lower than in real curing reactions. Alternatively, Wu and Xu21 used cross-linking procedure that allows the formation of one cross-link per step and calculated elastic moduli of the resulting structures. However, this improved approach could become computationally inefficient with increasing the system size. Heine et al.20 used a dynamic cross-linking approach based on a cutoff distance criterion and relaxation procedure of the cross-linked structure with a modified potential. Komarov et al.23 employed four-step reverse mapping procedure to perform cross-linking at coarse-grained level and study properties of the resulting structures on the fully atomistic level as a function of the conversion degree. Varshney et al.24 combined the dynamic cross-linking approach proposed by Heine et al.20 and the relaxation procedure consisting of cycles of energy minimization and molecular dynamics equilibration stages that was proposed by Wu and Xu,21 used new method of multistep bond formation and calculated some thermodynamic and structural properties. Li and Strachan26,27 proposed the cross-linking procedure with charge evolution in the course of chemical reaction and obtained thermo-mechanical properties using their cross-linked models. Whereas these earlier studies provided a significant progress in atomistic computer simulations of highly cross-linked polymer networks, the systems studied were relatively small in size (less than 15000 atoms). This is a limitation since it is known that the size of the system can substantially influence the properties obtained by molecular dynamics simulations. In the present work we have addressed this issue by constructing a significantly larger set of all-atom models, containing up to 35000 atoms. We have used a method developed by Accelrys, Inc.30 which uses a cross-linking procedure that combines several approaches that allow construction of thermosetting networks having structural characteristics close to those in real systems. These include capture sphere growth approach29 and effective relaxation technique21 in combination with monitoring of local stresses in the resulting networks on-the-fly. This combination of approaches yields systems with high conversion degrees and also free of stresses and geometrical distortions. The resulting polymer networks have been used in the present study to investigate the influence of model size, extent of curing and length of epoxy strands on the thermo-mechanical properties of epoxy-based polymer networks. The focus has been on a resin-hardener system that is widely used in engineering applications: the resin selected is DGEBA (diglycidylether of bisphenol A) and the aromatic amine hardener selected is EPI-Cure-W (diethylenetoluenediamine,

2. METHODOLOGY 2.1. Systems of Interest. The molecular structures of the epoxy monomer and curing agent molecule are shown in Figure 1, parts a and b. To simulate the cross-linking reaction, the

Figure 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. (c) The reactive sites in the epoxy resin are activated by opening the epoxy rings at the ends of the molecule. (d) Amine groups in curing agent react with opened epoxy rings at the ends of resin molecules.

reactive sites in the epoxy resin are activated by opening the epoxy rings at the ends of the molecule as shown in Figure 1c. Each of the amine groups in curing agent can react with two resin molecules as terminal carbon atoms of a resin molecule react with the nitrogen atoms of the amine hardener (Figure 1d). In the present work, we simulate stoichiometrically balanced compositions of the epoxy resin and hardener molecules; i.e., in all our systems, the number of the resin molecules is two times that of the hardener ones, which allows a theoretical conversion of 100%. 5308

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between reactive atoms whenever a valid atom falls within the sphere of another one. In the case of densely cross-linked polymers, the network potential energy has a significant influence on the properties of the material. Hence, to create thermosetting polymer with realistic material properties, additional measures were taken to ensure undistorted geometry of the molecules with low internal stresses. In order to minimize the presence of internal stresses and geometric distortions during cross-linking, energetic and structural information are analyzed on-the-f ly after each cross-linking cycle. Unrealistically high stretching energy of any of the individual bonds in the system can be considered as a sensitive measure of the distortion. It usually happens when a defect is introduced in the course of the cross-linking reaction. In the present work, when a dramatic increase in the maximum bond stretching energy is detected at any stage of the cross-linking procedure, the structure is rejected in favor of others with low internal stresses. In actual chemical reactions, the reactive groups diffuse through the mixture and bond as they approach to each other at the appropriate distance. Equilibrating the mixture at high temperature enhances the movement of the reactive groups toward each other. However, in the case of a dramatic increase in molecular weight that usually occurs in the course of the cross-linking reaction, the diffusion takes much longer time and is currently beyond the reach of all-atom molecular dynamics approach. To circumvent this limitation and to relieve network stresses within a reasonable computational time, the equilibration can be supplemented with energy minimization of the structure that allows quick changes in geometry which otherwise would take long time in dynamic simulations. Thus, to mimic the diffusion in actual cross-linking reactions our model systems are subjected cascades of both energy minimization and constant volume and temperature equilibration after each cross-linking cycle.21 Cross-Linking Procedure. In the present study, the initial amorphous structures of the uncross-linked DGEBA/DETDA mixtures were built using Accelrys amorphous builder which uses a Monte Carlo packing algorithm based on the rotational isomeric states model.33 The DGEBA and DETDA molecules in stoichiometrically perfect proportions were added to a cubic periodic simulation box by growing the molecules segment by segment, taking into account both energy of interaction with all atoms in the box and chain conformations. As both epoxy and amine hardener molecules contain aromatic rings, a special care was taken to check ring spearing during the construction of the amorphous mixture. The formation of thermosetting polymer network starts with the equilibration of reactant mixture. In order to enhance molecular mobility in the course of chemical reaction and hence accelerate the network formation the curing is carried out at elevated temperature of 480 K. The cross-linking cycle begins with determination of the available reactive sites, i.e., terminal carbons of the activated epoxy molecules and nitrogen atoms of the amine hardener. During the simulation, distances between these reactive sites are calculated and those reactive sites falling within the current reaction radius are identified. The initial cutoff of chemical reaction is set to 5 Å and increase the cutoff radius by a small value (0.5 Å) after each crosslinking cycle. New bonds between the identified reactive atoms are created and surplus hydrogen atoms are removed from both epoxy and amine group reactive sites. The partial charges and atom types of the atoms that participated in the chemical

To study the effect of system size on the thermo-mechanical properties of the thermosetting polymers, we constructed seven different system sizes ranging from (32, 16) to (512, 256), where the numbers in the parentheses represent the number of the epoxy resin and hardener molecules respectively, which corresponds to approximately 2200 to 35100 atoms in the simulation cells. Furthermore, to investigate the effect of resin chain length on the physical and mechanical properties of this class of materials, we also built the structures using short resin oligomers with various degree of polymerization, consisting of one, two and four monomer epoxy molecules (mono-, di-, and tetramers, respectively). In addition, in order to examine the role of the degree of curing on the properties of the system, structures with degrees of conversion ranging from 50% to 100% are selected during the curing process described below. We would like to also add that for better prediction of the mechanical properties of the thermosetting material under investigations using molecular dynamics (MD) simulation, we generated up to 70 topologically independent structures for each configuration with a given system size, degree of conversion and resin chain length. The details of this approach will be discussed in the mechanical properties section. 2.2. Simulation Details. The initial mixtures of reactants were built by packing activated epoxy resin and curing agent molecules into a cubic simulation cell followed by a geometry optimization using the Amorphous Cell module of the Materials Studio commercial package.30 All subsequent MD simulations were performed using the Discover module of the Materials Studio software. In all our simulations, 3D periodic boundary conditions were imposed to a cubic simulation cell in order to avoid introducing artificial surface effects. Interatomic interactions are described using the second-generation COMPASS force field31 due to its high accuracy in predicting the properties of polymeric materials. Wu and Xu21 validated COMPASS force field for predicting the elastic moduli of highly cross-linked polymer networks. These properties were found to be in better agreement with experiment than previous calculations of these properties using DREIDING force field. In addition, Tack and Ford 32 showed that density prediction of thermosetting polymer using COMPASS is better than that using the cff91 force field. 2.3. Building of Polymer Networks. Earlier attempts to construct realistic models of highly cross-linked polymer networks encountered difficulties resulting from high internal stresses and unrealistic geometry distortions (see, for example, Yarovsky and Evans19). As a consequence, these models produced markedly different predictions with properties of thermosetting polymers that are often far from the experimental values. In this study we used the methodology that is currently most reliable for creating models of free of stresses and geometrical distortions and yet reaching the levels of degree of conversion consistent with those typical in real systems. To build polymer networks with high degree of conversion, we used Accelrys software.30 This software employs the capture sphere growth approach that was originally elaborated by Eichinger29 and was successfully used for generating the topology of lightly cross-linked elastomers. In this approach, a cross-linked system is developed by growing the radius of a sphere around each cross-link site and bonds are formed 5309

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αG and αR correspond to the volumetric coefficients of thermal expansion (CTE) in glassy and rubbery regions, respectively. Glass transition temperatures were determined at the point of the change in the slope (represented by the intersection of the solid lines in the Figure 2). To validate the temperature ranges in which we perform linear fits, we varied the boundaries of these ranges and found that linear fits obtained are within the error bars of the volume-temperature plot and bring negligible changes in glass transition temperature. Furthermore, for mechanical characterization of the material both in glassy and rubbery states the systems should be equilibrated to the temperatures of interest. In this work we have chosen two temperatures, 298 and 480 K, which are above and below the glass transition region of a given structure, respectively. First, we determined densities at these two temperatures from volume−temperature plots that were generated by averaging data from five randomly picked structures for each extent of the reaction. Then all the structures were equilibrated to these two predefined temperatures and corresponding densities for mechanical properties characterization of the material under investigation.

reaction process are also modified after the reaction. The cycle recurs until the specified maximum cutoff radius is achieved or all available sites are reacted. Unreated epoxy rings remain open. After each cross-linking cycle a relaxation procedure is applied as described in the above section and structural and energetic characteristics of the obtained configuration are evaluated. If the maximum bond stretching energy detected after relaxation remains unusually high, the reaction stops. Otherwise, the cross-linking cycle continues. 2.4. Determining Thermal Properties of the Network. As described above, to enhance molecular motions and reduce stresses in the models, a curing reaction was conducted at an elevated temperature of 480 K that is above the experimental glass transition region for this material. To study thermal response of the constructed networks, we applied stepwise constant pressure cooling procedure performing a sequence of constant pressure and temperature molecular dynamics simulation runs at a set of temperatures. Here we implemented Andersen thermostat and Berendsen barostat to control temperature and pressure in our simulations. Cooling of the model systems was carried out in steps of 10 K with 100 ps equilibration run at each temperature within the temperature range of 623 to 223 K. The volume at each temperature was then computed by averaging results from at least five randomly picked epoxy structures and is used to construct volume− temperature plots from which a set of physical properties are calculated as described below. Figure 2 shows a representative specific volume−temperature plot of the cooling process. As can be seen in Figure 2, the

3. PHYSICAL PROPERTIES 3.1. Results. A. Glass Transition Temperature. The results of our glass transition temperature investigation are presented in Figure 3a where the glass transition temperature is plotted as a function of the extent of the reaction for the seven different system sizes ranging from (32, 16) to (512, 256) DGEBA/ DETDA molecules represented by different colors. As expected34 we observe an increase in glass transition temperature with increase in degree of curing. In general, the dependence of Tg on the degree of curing shows better defined trends with increasing the system size giving about 50 K divergence in values for the smallest and the largest system at high conversion degrees. However, even for the largest system size, small sampling size is prone to larger error bars resulting in higher standard deviations. These range from 3 K at low curing degrees to 21 K at 95% of curing. As the confidence interval for high extents of the reaction is wide, it makes it very hard to draw any definite conclusion about the shape of the curve in Figure 3a at high degrees of curing. The experimental values of the glass transition temperature for this material reported by different groups under different conditions35−38 also have wide variation of about 35K (441 to 476 K) and are shown in Figure 3a as open symbols. We see that our simulation results for extent of reactions higher than 90% are within the range of the reported experimental values. B. Volumetric Coefficient of Thermal Expansion. Figure 3b shows the results of the coefficients of thermal expansion (CTE) calculations for the glassy and rubbery regions at different degrees of reaction for the seven different system sizes mentioned above. In both the glassy and rubbery regions, the CTE monotonically decreases with increase in extent of curing. In addition, CTE shows well observable dependence on system size and, on average, it increases with increase in number of atoms in the model. Experimental values for fully cured structure at glassy and rubbery states38 are also shown in Figure 3b as open symbols. The CTE values obtained from our simulations for high degrees of curing are slightly below but in a reasonable agreement with experiment and will be discussed later. C. Density. In Figure 3c, the dependence of density on the extent of curing reaction at temperatures below and above Tg

Figure 2. Representative specific volume−temperature plot for epoxy system containing (512, 256) DGEBA/DETDA molecules cured to 90%. The small error bars are evidence of small variation of the specific volume between the 5 structures used to generate the data.

volume continuously decreases with decrease in temperature and a linear analysis was then used to predict the behavior observed at high and low temperature regions. To determine the properties of interest we perform linear fits of these two regions using: VG = VG° + αGT (223−323 K)

(1)

VR = VR ° + αR T (473−623 K)

(2)

where T indicates temperature, VG and VR are specific volumes in glassy and rubbery regions, while the slopes of these regions, 5310

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increase shows noticeable dependence on temperature. In addition, at a given extent of reaction the density, on average, decreases with increase in system size. The density at the highest extent of the reaction is in excellent agreement with experimental data (less than 2% difference).39 D. Properties of the Structures with Different Chain Length of the Epoxy Strands. The dependencies of glass transition temperature, coefficients of thermal expansion, and density on the extent of the reaction for different chain length of the epoxy strands are shown in Figure 4, parts a−c. In this work we compared three kinds of systems which were constructed using 512 monomers, 256 dimers and 128 tetramers of the epoxy resin DGEBA of respectively (512, 256), (256, 128) and (128, 64) systems. All properties under consideration show observable dependence on the resin chain length. The slopes of the conversion dependencies are maximum ones for the structures cured with epoxy monomers. Besides, better correspondence of densities and coefficients of thermal expansion with experimental values is observed for longer epoxy strands. 3.2. Discussion. A. Cooling Rate Effect. The transition between liquid and glass is not a transition between two states that are in thermodynamic equilibrium. It is a dynamic transition from an ergodic to a nonergodic state. Glass is a kinetically locked state of the material and its properties (density, CTE, Tg) are strongly dependent on its thermal history. The properties presented above strongly depend on the rate of cooling which reflects the fact that highly cross-linked polymer structures do not immediately respond to changes in temperature. In particular, exceptionally high cooling rates used in MD simulations usually result in densities that are lower than experimentally observed values39 as the structure freezes before its molecules get into more compact equilibrium configuration. For the same reason, the CTE values from simulation are usually lower than those estimated by experiment.39 It was shown in many DSC experimental studies (see, for example, review by Wunderlich40) that glass transition temperature depends not only on the rate of temperature change but also on the sign of its change. In other words, glass transition occurs at different temperatures at cooling and heating of the same sample giving lower Tg values at cooling and higher ones at heating. Moreover, it was shown by experiments that hysteresis increases with increasing the rate of temperature change. Thus, we could expect that cooling cycles done using molecular dynamics simulations, as in the present work, may also yield lower glass transition temperature than that of heating cycles. B. System Size Effect. Glass transition temperatures obtained in our simulations (Figure 3a) show strong dependence on system size at high extents of the reaction. Indeed, the smaller the volume, the lesser structural rearrangements it can accommodate. So glass transition temperatures of smaller model structures (with periodic boundaries, i.e. without free surfaces) are higher than that of larger ones. The same reason is valid for the size dependence of the coefficients of thermal expansion. An increase in the coefficients of thermal expansion is observed when the simulation cell size is increased. Similarly, as larger volumes in atomic scale take more time to equilibrate, the densities of larger systems obtained from our simulations are slightly lower. C. Effect of Chain Length of the Resin Strands. The dependence of Tg on resin chain length is not as clear as the trends seen for CTE and density. The observed dependence of

Figure 3. (a) Glass transition temperature, (b) volumetric coefficients of thermal expansion in glassy state (squares) and rubbery state (circles), and (c) density in glassy state (squares) and rubbery state (circles) as a function of the extent of the reaction for the atomic structures ranging from (32, 16) to (512, 256) DGEBA/DETDA molecules. In all cases the open symbols represent experimental values and are taken from the following: (a) square, Jansen et al.;35 triangle, Ratna et al.;36 circle, Shen et al.;37 rhomb, Liu et al.;38 (b) in glassy (square) and rubbery (circle) states; 38 (c) in glassy state (square).39 The color codes are as follows: orange (32, 16); cyan (64, 32); black (96, 48); red (128, 64); green, (256, 128); blue (432, 216); magenta (512, 256).

for seven different system sizes is shown. As expected the density increases in the course of the reaction and the rate of 5311

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Note that the structures built using epoxy monomers contain about 35000 atoms, while dimer structures have 27500 atoms and tetramer structures have 24000 atoms. As these systems are not of the same size, quantitative comparison of absolute values of the properties is not advisible since they are influenced by both chain length and size of the simulation cell. However, comparison of the individual slopes is still informative, and worth noting.

4. MECHANICAL PROPERTIES OF THE HIGHLY CROSS-LINKED POLYMERIC NETWORKS 4.1. Approach Used in the Mechanical Properties Calculations. Any small volume element of an amorphous material in atomic scale can be characterized by a unique distribution of matter within it and consequently displays unique properties that sometimes can be pretty far from the macroscopic properties of the sample. It means that a macroscopically homogeneous amorphous material can be viewed as heterogeneous at the nanoscale. We can thus partition the macroscopic sample into many small elements and take an average of the properties of the individual elements to predict the macroscopic properties of the material. In this work we estimate elastic constants of the thermosetting polymer by calculating elastic constants for a number of nanoscopically small simulation cells with subsequent averaging of the obtained properties. Simple averaging of the stiffness and compliances matrices gives so cold Voigt and Reuss bounds representing upper and lower bounds of the elastic constants of the material. However, these bounds are often pretty broad for polymer networks while the aim of this study is to be able to distinguish between properties of very similar materials, e.g., materials with extent of reaction differing by 5%. For this purpose, in the present work we used two different bound estimation techniques to study statistical variation of elastic constants. The simplest, but not necessarily the best, is rough estimation technique giving Voigt and Reuss bounds mentioned above. The second technique proposed by Hill and Walpole41−44 is more sophisticated but gives significantly narrower bounds. To calculate elastic constants, we used Accelrys software30 which implements the static approach.45 First, three tensile and three shear deformations of a small magnitude were applied to the systems in three directions with subsequent energy minimization. The obtained stress tensor is then used to calculate stiffness and compliance matrices Cij and Sij of the simulation cells followed by estimation of Voigt−Reuss and Hill−Walpole bounds. Finally, assuming isotropic symmetry of the model, these stiffness and compliances matrices are used to calculate two Lamé elastic constants from which Young’s, shear, and bulk moduli and Poisson’s ratio are calculated.44 Because of high computational efficiency, this methodology allows one to analyze the elastic constants of a large number of nanoscopically small volume elements giving narrow bounds of the material properties. Such statistical treatment allows mimicking the effect of nanoscopic heterogeneities that are always present in real macroscopic samples and partially overreach the size limitation of the molecular dynamic simulation method. For this purpose we generated batches of up to 70 topologically distinct thermosets for each extent of the reaction. To the best of our knowledge, we are not aware of any all-atom molecular modeling that treated such a great number of statistical data (while using large simulation cells of up to 35000 atoms) to calculate elastic constants.

Figure 4. (a) Glass transition temperature, (b) volumetric coefficient of thermal expansion, and (c) density as a function of the extent of the reaction for the atomic structures built using monomers (black), dimers (red), and tetramers (green) of the epoxy resin. The open symbols in part b represent experimental values in glassy (square) and rubbery (circle) states,38 and in part c, the open square represents experimental value.39

CTE and density on the chain length of the resin strands can be understood using following considerations. For infinitely long strands the cross-linking density tends to zero and any of the properties discussed above should not show any observable dependence on the degree of conversion. Moreover, it is known that small amount of resin oligomers is always present in commercial products. Our results for density and coefficients of thermal expansion for oligomer-based structures match better with experimental values and clearly show the correct trend. 5312

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Figure 5. Elastic moduli at 298 K(squares) and 480 K (circles) as a function of the extent of the reaction for the atomic structures ranging from (96, 48) to (512, 256) DGEBA/DETDA molecules: (a) Young’s modulus; (b) Poisson’s ratio; (c) bulk modulus; (d) shear modulus. Error bars represent Hill−Walpole bounds. Color codes are black (96, 48), red (128, 64), green (256, 128), blue (432, 216), and magenta (512, 256).

4.2. Results. The dependence of the elastic moduli on the extent of reaction at two temperatures for five different system sizes ranging from (96, 48) to (512, 256) DGEBA/DETDA molecules are shown in Figures 5a-d. As expected Hill-Walpole bounds are significantly narrower than Viogt and Reuss ones so in all our elastic moduli plots we show only Hill-Walpole bounds, where the size of the error bars reflect the width of the bounds. The elastic moduli for small systems containing (32, 16) and (64, 32) molecules are not shown here due to considerable scattering and extremely wide bounds. As expected, the results from our simulation demonstrate the pronounced increase in Young’s, bulk and shear moduli with extent of curing at both temperatures. However, the values of the Young’s modulus determined from our simulation are above the experimental value (2.71 GPa) at high extents of the cross-linking reaction.46 The Young’s, bulk and shear moduli values are lower at the elevated temperature, which is evidence of the reduced stiffness of the polymer with increase in temperature. Besides, these elastic constants demonstrate systematic dependence on the system size giving more accurate shapes of the curves and narrowing the bounds significantly with increase in system size. We do not detect any dependence of the Poisson’s ratio on the extent of the reaction or system size. From our simulations we determined an average Poisson’s ratio value of 0.31 at ambient temperature for all extensions and sizes of the simulation cell, which is in good agreement with experimental values for epoxy materials.47 The dependence of mechanical properties on the length of the epoxy strands in the material are shown in Figure 6, parts

a−d. All elastic constants show a pronounced dependence on the length of the epoxy strands. The dependence of the Young’s, bulk and shear moduli on the degree of conversion shows different slopes with the maximum slope corresponding to the structures with the shortest epoxy strands. For the structures with the shortest epoxy strands these moduli have higher value at high extent of the reactions. As expected, Poisson’s ratio increases for the structures with longer epoxy strands showing the correct trend. 4.3. Discussion. A. Size Effect. In this study, we managed to successfully build models that are free of stresses and distorted geometry (bonds, angles etc.). However, these defects are not the only ones present in real polymers. For example, big voids and large-scale cooperative motions in macroscopic samples can lower the measured stiffness of the material. This phenomenon was observed experimentally e.g. by Shen et al.37 and Possart et al.48 The authors measured micrometerscale Young’s modulus and compared it with macroscopic values obtained from tensile tests. In both studies Young’s modulus measured at micrometer scales using nanoindentation test is higher than integral values measured at macroscale by tensile test. At nanoscale, the deviation from integral values is expected to be even more pronounced. Therefore, it is not surprising that simulation predictions at nanoscale are higher than those measured experimentally (at micro and macro scales). All the models we constructed for our MD simulations are nanoscopically small and nearly perfect. As a consequence, such structures cannot accommodate either big defects or large5313

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Figure 6. Elastic moduli at 298 (squares) and 480 K (circles) as a function of the extent of the reaction for the atomic structures built using monomers (black), dimers (red) and tetramers(green) of the epoxy resin: (a) Young’s modulus; (b) Poisson’s ratio; (c) bulk modulus; (d) shear modulus. Error bars represent Hill−Walpole bounds.

properties dependence on this parameter. The shorter the strands the more sensitive the material property to changes in the conversion degree and the higher the slope of the dependence of the mechanical property on extent of reaction. Furthermore, the shortening of the distance between crosslinking sites increases the stiffness of the polymer network reducing the Poisson’s ratio and raising the values of the remaining three moduli of the material.

scale motions and thus resulted in higher than experimentally measured Young’s modulus values. B. Mechanical Properties at Temperatures above Tg. At temperatures above glass transition we should expect to observe high value of Poisson’s ratio that is close to 0.5 as well as a rubbery plateau modulus that is two or 3 orders of magnitude lower than the modulus at glassy state. However, the Young’s moduli determined at 480 K from our simulations are slightly lower than the corresponding Young’s moduli at room temperature, while Poisson’s ratio takes distinctly lower values than that for perfectly incompressible material. Nevertheless, these results could be interpreted in terms of frequency dependence of elastic constants in amorphous polymers. Large-scale cooperative motions of big segments of the molecules characterize the rubbery state. However, static method of deformation does not accurately take into account the dynamic effects that are especially noticeable at high temperatures. Moreover, typical sizes of the simulation cells used in atomistic simulations could not accommodate these kinds of motions and thus resulting in higher Young’s modulus values. Besides, these kinds of structure rearrangements could take macroscopic-scale time. So, one could say that elastic constants simulated in this study correspond to elastic constants measured at extremely high frequencies where relaxation motions are frozen. C. The Role of Epoxy Chain Length. The explanation given in section 3.1 on the observed dependence of material properties on the length of the resin strands composing the network is generally valid in explaining the mechanical

5. SUMMARY AND CONCLUSIONS In the present study, we were able to generate a set of stressfree thermoset models with high degree of cure containing up to 35000 atoms. These models were used to predict the dependence of the thermo-mechanical properties of highly cross-linked polymers on several parameters: the extent of curing reaction, temperature, size of the model, and chain length of the resin strands. It was seen that densities, coefficients of thermal expansion, and glass transition temperatures of the systems are in good agreement with experimental data. However, some of the elastic constants from our model systems were found to be higher than in real macroscopic samples. The advantages and limitations of the methods used in this study are summarized as follows. A. Cross-Linking Method. The capture sphere growth approach employed in this study to build polymer networks allows one to achieve high extents of the reaction close to the real systems. The effective on-the-fly monitoring of the model quality and analysis of energetic and structural information 5314

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enables to minimize the presence of internal stresses and geometric distortions in the produced structure. The relaxation procedure used to build thermoset structures has successfully helped to overreach time-scale limitations of the MD method. B. Static Deformation Approach for Mechanical Properties Calculation. Large number of samples averaged in the analysis of our data allows us to compensate for large scattering in mechanical properties data that is usually caused by unique distribution of matter in each nanoscopically small simulation cell. While this deformation method makes it possible to distinguish between properties of very similar materials that have minor differences, it may have difficulties in correctly predicting mechanical properties at high temperatures. Though simulated structures were equilibrated at predefined temperatures, this approach does not accurately take into account dynamic effects that become more important at elevated temperatures. C. Size and Time Scale Effects on Thermo-Mechanical Properties. Since systematic size effect on the material properties is observed one may extrapolate to predict properties at macroscopic sizes. However, due to size limitations of atomistic simulations, simulation cell could not accommodate large-scale events such as large voids and cooperative motions that are important in macroscopic-sized polymers. Therefore, the elastic constants obtained can be considered as properties for “perfect” highly cross-linked polymer networks with no large-scale events. Moreover, time scales used in all-atomistic molecular dynamics simulations of thermosetting polymers do not allow monitoring macroscopically long structural rearrangements, which are characteristic features of amorphous polymers. In this sense, the properties obtained in the present study allow to estimate contributions of dynamic effects as well as large-scale defects and relaxation events into the macroscopic properties of thermosetting materials.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: (N.B.S.) [email protected]; (M.T.) [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Low Density Materials Program of the Air Force Office of Scientific Research Grant Number: FA9550-09-1-0358. The authors gratefully acknowledge Dr. Charles Lee (AFOSR) for valuable discussions, and the Air Force Research Laboratory DoD Supercomputing Resource Center High Performance Computing for computer time.

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REFERENCES

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dx.doi.org/10.1021/ma3007587 | Macromolecules 2012, 45, 5307−5315