Predicting the Macroscopic Fracture Energy of Epoxy Resins from

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Predicting the Macroscopic Fracture Energy of Epoxy Resins from Atomistic Molecular Simulations Zhaoxu Meng,† Miguel A. Bessa,‡ Wenjie Xia,† Wing Kam Liu,‡ and Sinan Keten*,†,‡ †

Department of Civil and Environmental Engineering and ‡Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3111, United States S Supporting Information *

ABSTRACT: Predicting the macroscopic fracture energy of highly cross-linked glassy polymers from atomistic simulations is challenging due to the size of the process zone being large in these systems. Here, we present a scale-bridging approach that links atomistic molecular dynamics simulations to macroscopic fracture properties on the basis of a continuum fracture mechanics model for two different epoxy materials. Our approach reveals that the fracture energy of epoxy resins strongly depends on the functionality of epoxy resin and the component ratio between the curing agent (amine) and epoxide. The most intriguing part of our study is that we demonstrate that the fracture energy exhibits a maximum value within the range of conversion degrees considered (from 65% to 95%), which can be attributed to the combined effects of structural rigidity and postyield deformability. Our study provides physical insight into the molecular mechanisms that govern the fracture characteristics of epoxy resins and demonstrates the success of utilizing atomistic molecular simulations toward predicting macroscopic material properties. failure.19−21 This ductile response observed in atomistic simulation resembles that of polymer glasses, which also involve multiple deformation stages such as elastic, softening, strain hardening, and final failure.22,23 The localized plastic behavior at the nano- and microscale influences the fracture toughness but does not impede the material to be quasi-brittle macroscopically. Therefore, a better understanding of the plastic deformation near the crack tip for neat epoxy resins is of critical importance to predict and improve the fracture toughness of these materials. For these reasons, molecular scale investigations of neat epoxy resins are crucially needed. Atomistic molecular dynamics (MD) simulations on epoxy resins have been successfully applied to predict various material properties. Several computational algorithms have been developed to generate reasonable cross-linked structures for investigations of their physical properties.24−30 MD simulations have been carried out to predict the glass-transition temperature (Tg)20,31 and provided valuable insights into the effects of strain rate, temperature, and conversion degree on Young’s modulus and yielding behavior,20,32 which are consistent with theoretical and experimental studies.33−35 In a recent study, the complete thermoplastic response of an epoxy for general uniaxial loading conditions was obtained from classical MD simulations, and a scaling law was proposed to predict the quasi-static macro-

1. INTRODUCTION The superior thermomechanical properties of epoxies have led to a wide range of applications, most notably as structural adhesives and matrix materials in fiber-reinforced composites.1−3 However, unmodified epoxies suffer from a major shortcomingbrittle fracture. This performance issue has been intensively investigated during the past decades, and several experimental studies have demonstrated the possibility of improving the fracture toughness of epoxies without significantly compromising other thermomechanical properties by dispersing micro- and nanoelastomeric particles4−6 and other reinforcements.7−10 Nevertheless, a fundamental understanding of the brittle fracture behavior in neat epoxy resins is still missing. Fracture toughness of glassy polymers substantially depends on plastic flow processes that dissipate energy near the crack tip region.11,12 In epoxies, these dissipative processes are suppressed by the presence of cross-links, leading to a quasi-brittle behavior in tension. Despite this fact, fracture energies of neat epoxy resins are still much greater than the predicted theoretical value for purely brittle fracture.13−15 Previous studies based on transmission electron microscopy (TEM) have shown that the deformation of epoxy in the region near the crack tip is significantly plastic but also highly localized.16−18 At the molecular level, recent atomistic simulations have shown ductile deformational response occurring in epoxy, as characterized by yielding and softening behaviors upon loading followed by a nonlinear stiffening response due to chain alignment and scission that lead to © XXXX American Chemical Society

Received: July 13, 2016 Revised: November 6, 2016

A

DOI: 10.1021/acs.macromol.6b01508 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Molecular structure of monomers for the two types of epoxy analyzed: (a) Epon 825: DGEBA/33DDS; (b) 3501-6: TGMDA/44DDS. The reaction sites of each monomer forming the cross-linked bond are highlighted in red.

scopic response.36 However, fracture mechanisms of epoxies have been relatively less studied, largely because bond scission events are challenging to be captured in the classical MD framework. Failure behaviors of highly cross-linked polymer networks bonded to a solid surface have been studied recently through coarse-grained MD simulations using bead−spring polymer models, which indicates that both stress−strain curve and failure mode are largely associated with the interfacial bond density.37 Other approaches have involved implementing bondbreaking events in a classical framework based on strain energy arguments, which provide physical insight into the molecular origins of fracture.21 Despite significant progress toward understanding epoxy mechanical response, direct prediction of the fracture toughness of neat epoxy resins from MD simulations remains a particular challenge mainly for two reasons. First, as mentioned before, the complexity of accurately representing bond scission at the molecular level calls for reactive models or density functional theory (DFT) based approaches, which are computationally demanding. Second, while void nucleation processes initiate at the atomic scale, failure mechanisms ahead of the crack tip involve a large process zone that approaches several micrometers in size, which is beyond the length scale normally accessible by atomistic simulations.22 MD simulations with reactive potentials such as reactive force field (ReaxFF) can account for stress induced bond breaking to potentially address the first issue, 38 as recently demonstrated for epoxy nanostructures.39 Here, we propose a solution to the second issue by drawing an analogy to crazing failure in polymer glasses,40−43 which is built upon molecularly informed continuum models that relate fracture toughness to breaking force of fibrils and microstructural variables inside the crazing zone.44 Such models have been used in the past to study the molecular weight dependence of the fracture toughness of polymer glasses.45−48 More recently, Rottler et al. have employed this continuum modeling approach to determine the fracture toughness of glassy polymers under plane strain deformation by associating different deformation stages from MD simulation to different locations of the process zone around the crack tip.22 A similar approach can be formulated to characterize the fracture behavior of highly cross-linked thermoset polymers such as epoxies, which is justified by the fact that the nanostructures inside the process zone exhibit fibril formation and anisotropy that are akin to those seen in polymer crazing, even though no obvious large-scale crazes form during fracture in thermosets.4,49

In the present work, we employ MD simulations to systematically investigate and characterize the thermomechanical responses of two representative epoxy materials. We first generate cross-linked atomistic structures of two representative epoxy resins and then investigate how the conversion degrees and curing agent ratios influence Tg to validate our model against experiments. The general-purpose DREIDING force field is employed for running dynamics during the cross-linking simulations and also for the Tg calculations.50 Subsequently, we present tensile deformation simulations to investigate the fracture responses of different epoxy chemical structures using reactive MD simulations with ReaxFF. By employing a continuum fracture model that shares the same principles as those proposed by Brown and Sha et al. for glassy polymers,44,45 we are able to predict and compare, with no empirical parameters, the fracture energies of neat epoxy systems based on molecular information from reactive MD simulations. Finally, we apply this methodology to investigate how the fracture energies depend on molecular chemistry (i.e., epoxy type, component ratio, and cross-link density), and provide insight into the relative importance of competing molecular level factors such as structural rigidity and postyield deformability.

2. MATERIALS AND METHODS We first generate the monomers of epoxy resins and curing agents in Accelrys Materials Studio. Here, two different representative epoxy systems are considered: (1) an epoxy resin commercially known as Epon 825, consisting of diglycidyl ether of Bisphenol A (DGEBA) with curing agent 3,3-diaminodiphenyl sulfone (33DDS); (2) an epoxy commercially denominated as 3501-6, mainly composed by tetraglycidyl methylenedianiline (TGMDA) with curing agent 4,4diaminodiphenyl sulfone (44DDS). Figure 1 shows the chemical structures of the two systems. The DGEBA is a bifunctional epoxy resin, while the TGMDA is a tetrafunctional epoxy resin. Epoxy monomers in activated form (i.e., CH2−O bonds at the ends of the epoxy molecules are broken and the oxygen atoms form alcohol groups) and respective curing agent monomers are then packed in a periodic box according to their stoichiometric ratio using the Amorphous Cell module in Materials Studio. A typical Epon 825 system consists of 1024 monomers of DGEBA and 512 monomers of 33DDS (i.e., stoichiometric mixing ratio of DGEBA:33DDS = 2:1) resulting in a cubic primitive cell with length ∼9 nm, while a typical 3501-6 epoxy has 512 monomers of TGMDA and 512 monomers of cross-link agent 44DDS (i.e., a stoichiometric ratio of TGMDA:44DDS = 1:1) resulting in a cubic primitive cell with length ∼8 nm. To simulate the cross-linking process, the Polymatic Algorithm developed by Abbott et al. is applied.51 The MD code LAMMPS is used to carry out all the simulations.27 At each step, every root meanB


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Macromolecules square (RMS) distance between eligible nitrogen and carbon atoms is computed, and a covalent cross-linked bond is created for the shortest distance pair, if it is within the cutoff distance of bonding criteria (6 Å). After each step, the structure is optimized by an energy minimization process, and a new coordinate file is generated. For every 16 crosslinked bonds formed, there is a dynamic equilibration run under the isothermal−isobaric (NPT) ensemble for 10 ps. This equilibration is used to remove the residual stresses generated during the newly formation of bonds. Each amine group can react with two epoxy molecules. The cross-linking process proceeds until the desired conversion degree is achieved or no more eligible nitrogen and carbon atoms fall within the cutoff distance during 10 consecutive dynamic runs under NPT ensemble, each for 10 ps. The conversion degree is defined as the ratio between the number of cross-linked bonds created and the maximum possible number of bonds between eligible nitrogen and carbon atoms. In a similar way, epoxy systems with varying curing agent/epoxy resin ratios (also referred herein as amine/epoxide ratio) are generated for both Epon 825 and 3501-6. Also, by choosing the specific coordinate file, we can obtain different conversion degree structures. Taking 3501-6 as an example, we have generated systems with different conversion degrees ranging from 65% to 95% and four different amine/epoxide ratios. Since the stoichiometric ratio of 3501-6 is 1:1, the ratio of 0.8:1 indicates an excess in epoxide group, while the ratios of 1.5:1 and 2:1 indicate an excess of the amine group. The different ratios are obtained by keeping the number of monomers of the epoxy (TGMDA) constant, while varying the number of amine monomers (44DDS). The Tg of specific epoxy system can be determined from the density change as the system is cooled down.20,52 A typical cooling rate of 0.5 K/ps is used in this study. Specifically, the system is cooled down from 600 to 250 K at a temperature step of 25 K. At each temperature, an equilibration of ∼10 ps is performed using NPT ensemble at atmospheric pressure (∼1 atm), and the density of the system is calculated. The densities are plotted against temperatures and the change of slope in the density against temperature curve marks the Tg. Then, an existing force-field parameter set of ReaxFF (ReaxFF_Mattsson) is used to simulate the plane strain uniaxial deformation behavior and characterize the failure response of the epoxy materials.53 Our results for the elastic and yield response of the material using ReaxFF are consistent with those using DREIDING. Specifically, the Young’s modulus, yield stress, and Tg results do not differ by more than 10% between DREIDING force field and ReaxFF. For the fracture simulations, the epoxy network is first relaxed using NPT at ∼300 K with 1 atm in three dimensions using the Nosé− Hoover thermostat and barostat. Then the structure is uniaxially stretched up to complete failure while the other two dimensions normal to the stretch direction are held fixed. During the tensile deformation, the temperature is controlled at ∼300 K using the Nosé− Hoover thermostat. A constant engineering strain rate of 5 × 108 s−1 is applied, and the tensile stress of the entire system of atoms in the deformation direction is computed using the virial theorem. A typical “elastic-yielding-hardening-failure” behavior is observed for all epoxy systems, which is similar to the observation in other studies of similar epoxy chemistries,19,21,54 and is also similar to the behavior of polymer glasses in MD simulations.22 We define the yield stress S as the plateau stress after the curve softens or at the obvious “knee” in the stress− strain curve if the stress keeps increasing with strain, which is consistent with those in previous studies.35,36 The maximum stress Smax is calculated as the ultimate largest stress before final failure of the structure (i.e., tensile stress decays to 0). The atomic configurations during the whole process are visualized using the Visual Molecular Dynamics (VMD) package.55 During the uniaxial deformation, there is a cavitation process occurring in the epoxy structure. An open source code Zeo++ is adopted to analyze the void size evolution,56,57 which is based on Voronoi cell decomposition to calculate the void space and its distribution. More details about the calculation of those characteristic parameters can be found in the Supporting Information.

3. RESULTS AND DISCUSSION 3.1. Glass Transition Temperature. We first present physical properties predicted from our simulations and compare them with existing experimental data to validate our modeling approach. From Figure 2a, the characteristic slope

Figure 2. (a) Density as a function of temperature for the two epoxy resins at maximum conversion degrees (>95%) and stoichiometric amine/epoxide ratios. (b) Tg as a function of conversion degree and DiBenedetto equation (eq 1) fitting for both 3501-6 and Epon 825 systems. (c) Tg and cross-link density as a function of amine/epoxide ratio for the 3501-6 case. The error bars are introduced due to both uncertainties in identifying the slope variations of the linear fit and different runs.

change can be clearly observed in the density vs temperature curve. For Epon 825 in our simulations, Tg is predicted to be approximately 480 K, while for epoxy 3501-6 it is about 515 K. Both epoxies are evaluated at a high conversion degree (approximately 95%) and using the stoichiometric amine/ epoxide ratio. Tg predictions obtained through density measurements from MD simulations are normally higher than C


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Figure 3. (a) Schematic diagram showing the process zone and crack tip. (b) An epoxy system in a typical MD simulation at the maximum stress state, which represents the deformation state of epoxy near the crack tip. (c) Schematic illustrating the void diameter in the epoxy system compared with fibril spacing in linear chain polymer.

studies.63 As for the amine/epoxide component ratio, our results show that the Tg is the highest at the stoichiometric ratio 1:1 (Figure 3c), as expected. Our results also show that there is a close relationship between the cross-link density and Tg. 3.2. Fracture Energy Prediction. Various fracture mechanics models have been proposed for crazing and failure in glassy polymers.43−47 These models share several common points. First, they assume that the energy dissipated inside the crazing zone (i.e., process zone) is a major contributor to the macroscopic fracture energy. Second, within the process zone, polymer chains group into fibrils as they align parallel to the stretching direction. These fibrils are separated by voids but are linked transversely by secondary fibrils called cross-ties. In early studies, Brown proposed a continuum mechanics model based on this generic fibril/cross-tie microstructure to relate fracture energy with microstructure quantities such as the draw stress, the fibril spacing, and the chain breaking force.44 Specifically, the fracture energy can be predicted as follows:

experimental values, which can be attributed to the higher cooling rates.20 The cooling rate effects have been estimated using the Williams−Landel−Ferry (WLF) relations,58 which suggest that there is an increase of about 3 K in Tg per order of magnitude increase in the cooling rate. As a result, Tg calculated from MD simulations should be about 30 K higher than the experimental value. However, this is likely an upper bound to the difference as it is unclear whether the relation holds over such orders of magnitude for the cooling rate. Taking into account the cooling rate effect, our MD results are in reasonable agreement with experimental results reported in the literature, reporting Tg to be around 450 K for Epon 82552 and 466−483 K for epoxy 3501-6.59 In both experiments and simulations, tetrafunctional epoxy resin 3501-6 achieves greater cross-link density and consequently a higher Tg compared to bifunctional epoxy resin Epon 825, corroborating the importance of monomer functionality.60 Varying the conversion degree and the amine/epoxide ratio of the epoxy system enables a comprehensive understanding of the effect of chemical structure on Tg. Note that increasing the conversion degree for the same amine/epoxide ratio will lead to an increase in the cross-link density, whereas changing from the stoichiometric ratio to a different amine/epoxide ratio for the same conversion degree will lead to a lower cross-link density. Here, we define cross-link density in terms of the inverse of average molecular weight between cross-linked bonds: Dc = n/ Mt, where Dc and Mt denote the cross-link density and the total molecular weight, respectively, and n is the number of crosslinked bonds. Figure 2b shows that increasing conversion degree can increase Tg, which is more pronounced as the curing process proceeds. The effect of conversion degree on Tg can be well captured by the DiBenedetto equation:61,62 Tg(α) =

φα (Tg ∞ − Tg0) + Tg0 1 − (1 − φ)α

Gc = 2πD

Smax 2 ⎛ 1 ⎞⎟ C22 ⎜1 − λ ⎠ C66 S ⎝

(2)

where C22 is the elastic tensile modulus in the 2 direction (i.e., stretching direction, Figure 3a) and C66 is the in-plane shear modulus. We note that eq 2 is a simplified expression obtained by assuming that the elastic constant in the stretching direction (C22) is much larger than those in other directions. The original form and derivation can be found in those early studies.22,44 We elaborate later in our study and Supporting Information that the results change little with the original expression either simplified or original expression. S and Smax are the yield stress and maximum stress of the defect-free material, respectively. λ is the average stretch ratio of the process zone, and D denotes the fibril spacing, i.e., the distance between adjacent fibrils. Rottler et al. later showed that MD simulations could be used to characterize the stresses inside the process zone to determine the key parameters of the continuum model that predicts the fracture energy.22 Specifically, they utilized a bead−spring model, which has an analytic bond potential that enables bond breaking to model a volume element that would be representative of the large-deformation regions near a crack tip and formulating continuum parameters from MD simulations. We propose that predicting the fracture energy of neat epoxy resins from molecular simulations can be achieved using the same generic continuum model that idealizes the microstructure into fibrils connected by cross-ties. We start by illustrating how this continuum mechanics model can be

(1)

where φ is an adjustable parameter between 0 and 1 that is related to the chain rigidity and functionality,63 α is the conversion degree, and Tg0 and Tg∞ denote the glass transition temperatures for conversion degrees of 0% and 100%, respectively. We find that φ = 0.16 fits our simulation data best for epoxy 3501-6. An experimental value for this particular epoxy has not been reported before in the literature. However, for Epon 825, we find that φ = 0.31, which lies in the range of values (0.22−0.38) reported in the literature.61,63 We also observe this decreasing trend of φ with increasing functionality and chain rigidity, which is consistent with experimental D


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perpendicularly to the stretching direction, i.e. mean length of the short semi-axis of the elliptic void) as a more representative measure. The MVD grows with deformation similarly to fibril spacing, given that the total free volume increases with deformation when the other two directions are nearly undeformed, and reaches a plateau in the maximum stress regime, as discussed in detail in the Supporting Information. In this sense, the MVD in epoxy is similar to the fibril spacing in thermoplastic glasses, where both metrics characterize the nanostructural conditions at the maximum stress state near the crack tip.44 The relation between void diameter and fibril spacing is illustrated in Figure 3c. We calculate the MVD at strain levels in the maximum stress regime and determine the average value as the parameter D. We choose the stretch ratio λ at the strain level where the stress is equal to (S + Smax)/2. The elastic constants (C22 and C66) are calculated at different strain levels inside the strain hardening stage. This is because these parameters are related to the averaged physical properties of the process zone rather than at the extreme stress state before failure. Our results on the variation of the elastic constants justify the anisotropic assumption, and the scale difference also fulfills the requirement to use the simplified expression (eq 2) to predict the fracture energy, similar to the results reported by Rottler et al.22 More justifications and sensitivity analysis of these characteristic parameters can be found in the Supporting Information, illustrating clearly that the main results are robust with the parameters defined and calculated herein. Although the choice of stretch ratio λ is empirical in this study, the consistent definition for all cases enables this parameter λ to represent the deformability consistently for different epoxy systems. The elastic constants C22 and C66 gradually increase with strain in the hardening stage. However, the parameter (C22/C66)1/2 is independent of the strain in this stage, and the standard deviation is only within 10% for the epoxy systems studied herein as shown in the Supporting Information. This suggests that this parameter is representative of the characteristic anisotropy of the process zone. In addition, our results suggest that the parameter (C22/C66)1/2 exhibits a variation that is less than 20% for different epoxy chemistries, indicating it only plays a minor role in eq 2. Consequently, we can conclude that the key parameters governing the fracture toughness for epoxy resins based on eq 2 are the void diameter D, stretch ratio λ, and the ratio between the maximum stress and yield stress, Smax2/S. D and λ mainly represent the postyield deformability that dissipates energy through the cavitation process, which are inversely related to cross-link density. Smax2/ S quantifies the structural rigidity given that a more rigid structure (higher cross-link density) usually results in a higher value of Smax2/S. For un-cross-linked glassy polymers that fracture by crazing, the fibril spacing D (∼20−30 nm) is expected to be about 10 times larger than the void diameter of epoxy,22 while Smax2/S does not decrease too much due to the low yield stress for un-cross-linked polymers. Thus, the fracture toughness of un-cross-linked glassy polymers is generally expected to be larger than that of typical epoxy materials.11 Dispersing nano- and microelastomeric particles will also give rise to stronger cavitation processes that lead to larger values for D and λ along with greater energy dissipation near the crack tip and thus significant enhancement in fracture toughness.5 From reactive MD simulations, the stress−strain responses for the two types of epoxy using stoichiometric ratio at maximum conversion degree (>95%) are presented in Figure

applied to epoxies by drawing analogies between epoxies and polymer glasses. First, previous experimental investigations have shown that, microscopically, the process zone exists even in the most tightly cross-linked epoxies.18 The process zones at the crack tip of polymer glasses and epoxy resins both extend transversely to the loading direction and share the same shape.49 Then, at the molecular level, the tensile deformation mechanisms of epoxy and polymer glass are similar, both including elastic, yield, plastic flow, strain hardening, and void formation followed by failure, as shown in recent studies.19,22 As a result, during crack propagation, the energy dissipation process within the process zone in the epoxy is analogous to that in polymer glasses. Moreover, epoxy resins exhibit substantial ductility microscopically as well, suggesting similar localization of deformation in the process zone near the crack tip.16,18 Therefore, it is justified that the dominant contribution to the fracture energy in epoxy resins is the work done to propagate the process zone ahead of the crack tip, as illustrated in Figure 3a. This is also the basic assumption adopted in eq 2. In addition, eq 2 is formulated based on two other key observations that must hold for epoxy resins in order to apply this model. The first one is that the microstructure in the process zone is anisotropic, consisting of cross-tie fibrils that transfer stress between main fibrils such that the fibrils directly ahead of the crack tip reach the breaking stress of chains.44 The anisotropy arises from the fact that alignment of chains to form fibrils increases the modulus and strength in the loading direction, and the presence of the cross-ties leads to less loadbearing capacity and stiffness due to different but non-negligible transverse properties that influence the fracture process.44 The process zone ahead of the crack tip in epoxy resins exhibits anisotropy microscopically, given that the chains are more aligned along the loading direction (i.e., direction 2) in a similar fashion to deformation-induced microstructure evolution in polymer glasses. The second key observation is that the length of the process zone is much larger than its width. This can be justified by previous experimental observations on neat epoxy resin fracture tests, although the fracture process zone observed is smaller than that of thermoplastics.16−18,64 Previous experiments have also shown that by introducing cross-links into polymer glasses, the process zone size decreases.42,65 Based on these observations, the deformations at the crack tip of both polymer glasses and epoxy resins are analogous, and the essential requirements of eq 2 are satisfied for epoxy resins. Therefore, we can apply eq 2 to estimate the fracture energy of highly cross-linked epoxy resins. Indeed, our reactive MD simulations reveal that the nanostructure of epoxy evolves to resemble crazes in linear chain polymer glasses as stress-induced chain scission and reorientation events lead to formations of coarsened fibrils, voids, and cross-ties in the molecular network. Specifically, fibril-like structures having aligned chains form in the loading direction, separated by large elliptical voids (Figure 3b). The void formation process is consistent with what has been seen in a recent study, where significant voids form at large strains beyond the yield strain.21 The nanostructures of epoxy resins and polymer glasses can thus be idealized in the same way. However, to apply eq 2 for epoxy nanostructures, we need to redefine the parameter D, which quantifies the fibril spacing. This is because highly cross-linked thermosets such as epoxies prevent the elliptical voids from becoming interpenetrated, which limits fibril formation as observed experimentally.4,49,66 Here, we take the mean void diameter (MVD) (measured E


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Figure 4. (a) Stress−strain curves for the TGMDA-based epoxy 3501-6 and DGEBA-based epoxy Epon 825. The parameters for predicting the fracture energy are also highlighted in the figure. (b) Void diameter distribution comparison between the two epoxy systems before total failure. The MD simulation snapshots show the tensioned structure before failure for (c) the 3501-6 system at 100% strain and (d) the Epon 825 system at 200% strain.

and thus more plastic energy will get dissipated during the crack growth, leading to a higher fracture toughness (or fracture energy).33 With all the necessary parameters determined from the MD simulations (Table 1), the fracture energy can be calculated

4a. Statistical replications of the simulations can be found in the Supporting Information, and our results indicate that the variations from different simulations are small. The observed tensile stress−strain curves involve multiple stages, including elastic response, yield/plastic flow, hardening, maximum stress, and ultimate failure, which are consistent with other MD studies on similar epoxies as well as linear chain polymer glasses.19,20 The plots in Figure 4a also show some quantitative differences between the two types of epoxy. Epon 825 (red dotted line) has a lower cross-link density and exhibits a more pronounced plastic region prior to strain hardening and a larger strain at the maximum stress compared to epoxy 3501-6. To illustrate the differences in the structural evolution of both systems during the tensile deformation, Figure 4b shows the void size distribution at the strain level of maximum stress. Comparing the deformed structure of the two epoxy systems at those strain levels, we observe that Epon 825 can be stretched more substantially before total failure, and the voids in deformed Epon 825 are more obvious than those in deformed 3501-6 as shown in Figure 4c,d. More information about the deformation characterization, including potential energy evolution, segment elongation and reorientation, void development, and bond scission, can be found in the Supporting Information. Higher maximum stress Smax and yield stress S are observed for the epoxy 3501-6. The higher cross-link density for tetrafunctional epoxy 3501-6 explains the higher yield stress and maximum strength of the material and also the lower postyield ductility when compared to Epon 825. Generally speaking, lower yield stress will induce a larger process zone,

Table 1. Parameters Obtained from MD Simulations to Calculate Fracture Energy for Both Epon 825 and 3501-6 S (MPa) Smax (MPa) λ D (nm) (C22/C66)1/2

Epon 825

3501-6

205 ± 10 890 ± 22 2.1 ± 0.02 3.37 ± 0.12 3.22 ± 0.1

360 ± 10 1880 ± 30 1.55 ± 0.02 1.84 ± 0.08 2.68 ± 0.06

using eq 2. The predicted fracture energy for the Epon 825 is approximately 139 ± 10 J/m2, which is within the range of experimental values ranging from 80 to 200 J/m2 for similar DGEBA-based neat epoxies.4,9,67−69 The prediction for the epoxy 3501-6 is 109 ± 8 J/m2, which is also in reasonable agreement with the manufacturer reported value of 128 J/m2 (∼12% error). The errors of the fracture energy prediction come from not only the variance obtained for each parameter (Table 1) but also the choices of the parameters, such as the stretch ratio λ. In addition to the approximate nature of the continuum model, the complete details of the real chemical structure of all the curing agents (particularly the 3501-6) are not available yet, which may result in greater uncertainty. Nonetheless, the overall trend that the fracture energies F


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Figure 5. (a) Stress−strain curves of epoxies with different conversion degrees. (b) Predicted fracture energy of epoxies with different conversion degrees. (c) Stress−strain curves of epoxies with different amine/epoxide ratios at maximum conversion degree (>95%). (d) Predicted fracture energy of epoxies with different amine/epoxide ratios. The epoxy type is 3501-6.

a characteristic of epoxy systems, when conversion degree is considered as a primary design parameter.63 Next, we investigate how the initial amine/epoxide component ratio influences the fracture properties. Different ratios have a direct effect on the final network structure of the epoxy,70 and the network structure is closely related to the mechanical properties such as modulus and fracture toughness.71,72 Figure 5c,d shows the fracture energy predictions for different amine/epoxide component ratios (0.8:1, 1:1, 1.5:1, and 2:1) at maximum conversion degree (>95%) of epoxy 3501-6. Again, the fracture energies predicted using other definitions of λ are shown in Figure S8, and the conclusions still hold. These results further reveal that the fracture energy of the epoxy increases with increasing ratio of curing agent. This is consistent with the trend observed in experiments, in which the fracture toughness of a tetrafunctional epoxy resin similar to 3501-6 increases with increasing amount of amine curing agent.72 Another key observation from our simulations is that increasing the curing agent from a ratio of 0.8:1 to 1:1 does not necessarily decrease the fracture energy. As discussed above, the fracture energy depends on two competing factors: structural rigidity and postyield deformability. The maximum stress obtained for the 1:1 ratio increases significantly compared with the 0.8:1 case due to the fact that nearly all the functional groups in the system with a stoichiometric ratio of 1:1 have reacted to form cross-linked bonds, resulting in a more rigid structure. However, only meager increases of the values for D and λ are observed for the case with the 0.8:1 ratio. As a result, the fracture energy of 1:1 ratio epoxy resin is higher than the 0.8:1 ratio because the decrease in postyield deformability is more than counteracted by the increase in structural rigidity.

decrease with increasing the number of functional groups is consistent with the experimental observations.68 Varying the conversion degree has a more complex influence on the fracture toughness of epoxies, which is different from other thermomechanical properties. Previous results in section 3.1 have shown that Tg always increases with increasing conversion degree, since the cross-links contract the structure and decrease the segment mobility. 20,31 However, the investigations by Marks et al. suggest that the fracture toughness of amine-cured epoxy thermosets exhibits a maximum value for conversion degrees between 65% and 95%, depending on the type of curing agent.63 Following the same procedure outlined in the previous section, we predict the fracture energy for the epoxy 3501-6 at varying conversion degrees as shown in Figure 5a,b. Increasing the conversion degree (i.e., increasing the cross-link density) is associated with an increase in both yield and maximum stress but a decrease in the postyield deformability, indicated by the decrease of D and λ. These competing factors give rise to a maximum fracture energy at moderate conversion degree (around 75% conversion degree) (Figure 5b). Figure S8 in the Supporting Information shows that the characteristic trend of predicted fracture energy is conserved regardless of how λ is defined in the strain hardening regime although actual values of predicted fracture energy can exhibit up to 30% difference. As the simulations reveal, the decrease in plastic deformability with increasing cross-link density near the crack tip causes the more brittle response of highly cross-linked epoxies. This observation also implies that although higher conversion degree can enhance many thermomechanical properties, such as Tg and modulus, it may not necessarily enhance fracture toughness. The hallmark trade-off between strength and toughness is thus G


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Further increasing the curing agent decreases both the crosslink density and the number of branches in the network structure, leads to large enhancement of postyield deformability, and thus facilitates more dissipation of energy at the crack tip and increases the fracture energy. Therefore, although the cross-link density and Tg maximize for the stoichiometric ratio case, our results show that the fracture toughness of a typical epoxy system can be enhanced by increasing the initial curing agent ratio beyond stoichiometric ratio. With this in mind, one can design the desired thermal-mechanical property of the final epoxy product by controlling the initial epoxy resin and curing agent ratio.

Miguel A. Bessa: 0000-0002-6216-0355 Sinan Keten: 0000-0003-2203-1425 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge support from the Ford Motor Company with funding from the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE), under Award DE-EE0006867. W.X. gratefully acknowledges the support from the NIST-CHiMaD Postdoctoral Fellowship. M.A.B. and W.K.L. acknowledge the support by AFOSR (FA9550-14-1-0032) and by IRSES-MULTIFRAC. M.A.B. also acknowledges the support from the Portuguese National Science Foundation (SFRH/BD/85000/2012) and the Fulbright scholarship. In addition, the authors thank support from the Department of Civil and Environmental Engineering and Mechanical Engineering at Northwestern University. A supercomputing grant from Quest HPC System at Northwestern University is also acknowledged.

4. CONCLUSIONS The results obtained in the present study indicate that the DREIDING force field adequately captures the Tg differences resulting from different epoxy types, conversion degrees, and component ratios, as verified by experimental results for the two epoxy systems considered herein. An existing parameter set of ReaxFF not only agrees with DREIDING in small deformation for neat epoxy resins studied herein but also yields reasonable failure behavior of epoxies at the molecular level. The plastic deformation and cavitation processes observed from reactive MD simulations provide further evidence for the plastic deformations at the crack tip region that contribute to the fracture toughness during macroscopic fracture propagation events. Our work proposes a multi-scale modeling framework to link the deformations seen at the molecular scale with the continuum fracture properties by drawing an analogy to the crazing behavior observed in polymer glasses. Through this analysis, we show that the fracture toughness of epoxy resins is closely related to their molecular architectures and two competing factors, the structural rigidity and postyield deformability, which influence the fracture process simultaneously. Specifically, fracture toughness is predicted to decrease with increasing functionality of the epoxy resins studied herein. In addition, for the tetrafunctional epoxy resins, the fracture toughness reaches peak value at a moderate conversion degree (between 65% and 95%), and it can also be enhanced by adding more curing agent. The predictions of Tg and fracture energy in this study are in good agreement with experiments, demonstrating the viability of utilizing atomistic simulation to predict key material properties and trends. The proposed framework shows promise in accelerating the material-bydesign process for thermosets by incorporating data from molecular models. The framework can be further extended to examine moisture, fillers, and confinement effects on the thermomechanical properties of epoxy and other glassy polymers, such as Tg and fracture toughness.73−77

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b01508.

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REFERENCES

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Figures S1−S8; Tables S1 and S2 (PDF)

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Predicting the Macroscopic Fracture Energy of Epoxy Resins from

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