Molecular Scale Simulations on Thermoset Polymers: A Review

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1 JOURNAL OF POLYMER SCIENCE REVIEW Molecular Scale Simulations on Thermoset Polymers: A Review Chunyu Li, Alejandro Strachan Department of Materials Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana Correspondence to: A. Strachan (E- mail: strachan@purdue.edu) Received 27 January 2014; revised 27 March 2014; accepted 28 March 2014; published online 29 April 2014 DOI: /polb ABSTRACT: This article reviews the field of molecular simulations of thermoset polymers. This class of polymers is of interest in applications ranging from structural components for aerospace to electronics packaging and predictive simulations of their response is playing an increasing role in understanding the molecular origin of their properties and complementing experiments in the search for tailored materials for specific applications. It focuses on modeling and simulation of the process of curing to predict the molecular structure of these polymers and their thermomechanical response by all-atom molecular dynamics simulations. Results from Monte Carlo and coarse-grained simulations are briefly summarized. VC 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015, 53, KEYWORDS: coarse-grain simulations; glass transition; mechanical properties; molecular dynamics; Monte Carlo; polymer physics; thermoset INTRODUCTION Thepropertiesofpolymersdependonthe chemistry of their constituent monomers as well as on how these monomers combine with each other into the polymer architecture or molecular structure. Due to its importance in determining properties, polymers are often classified according to their molecular structure. Important classes include linear chain, branched, and networked polymers. This report focuses on thermosetting polymers, a class of synthetic polymers that form a threedimensional networked structure when its constituent molecules undergo chemical reaction upon heating. This process is known as curing and results in an infusible and insoluble material; these features are a direct result of the 3D network involving covalent bonds. Thus, unlike thermoplastics, thermosets cannot be reshaped or remolded with temperature or solvents. Several classes of technologically important polymers are thermosets. Rubbers are obtained by crosslinking highmolecular weight polymer chains in a process called vulcanization. The large molecular weight between crosslinks results in their elastomeric behavior. This article focuses on thermosets obtained from low-molecular weight liquid resins that leads to a high density of crosslinks and, often, high stiffness and strength. These properties together with their lightweight, low cost, and ease of processing make thermosets attractive for a wide range of applications, from electronics packaging to the matrices of fiber composites for structural applications in aircraft and aerospace. This article reviews recent advances in molecular-level simulations of thermosetting polymers. These simulations provide a description of these materials with unparalleled resolution and have contributed and continue to contribute to our understanding of the molecular-level origin of their macroscopic response. Accurate molecular-level simulations, based on first principles and with no adjustable parameters, are also important to understand the response of materials under conditions where experiments are costly or difficult. Examples include the response of nanoscale specimens or under extreme conditions of temperature and pressure. Increasingly, these simulations are also contributing to the design or optimization of new materials; when synergistically combined with continuum models and experiments they can reduce the cost and time in design and deployment cycles. As with many classes of materials, there is a great demand for tailoring the properties of thermosetting polymers for a wide range of applications. Such materials optimization efforts require addressing a host of properties simultaneously. For example, while one may be interested in improving strength for a given application, viscosity of the uncured resin should remain within acceptable margins for processability and the solubility of a variety of molecules should remain low to minimize performance degradation. The traditional approach based mostly on experimental testing leads to long and costly design and certification cycles and unacceptably long timescales between the discovery of a new material and its deployment. At the same time, potentially useful materials and material combinations go unexplored. The recently launched United States Materials Genome VC 2014 Wiley Periodicals, Inc. JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

2 REVIEW JOURNAL OF POLYMER SCIENCE Chunyu Li obtained his Ph.D. in solid mechanics at the Harbin University of Technology, China in He was an Associate Professor at Shijiazhuang Railway University in China before moving to the US. He joined the research group at Purdue University in 2009 and is currently a research associate. He has published more than 60 articles in prestigious journals and more than 20 proceeding articles. His research interests are mainly on polymers and multifunctional composites. Alejandro Strachan is a Professor of Materials Engineering and the Deputy Director of the Center for Predictive Materials and Devices (c-primed) at Purdue University. Prof. Strachan s research focuses on predictive simulations of materials at the atomic and molecular level with quantified uncertainties and their application to materials and devices of interest for nanoelectronics, energy and MEMS, active materials including shape memory and high-energy density materials and structural polymers and composites. Initiative (MGI) for Global Competitiveness (June 2011) highlights the urgent need of accelerating the material design process. The MGI goal is to bring together powerful computational toolsets for an integrated material design with powerful data analysis and experiments to achieve a significant reduction in discovery and development time. The rational design of polymers with properties tailored for specific applications requires quantitative chemistry-structureproperty relationships. We believe that the simulation techniques reviewed in this article will play an increasingly important role in such endeavors. This article is organized as follows. We describe the ideas behind molecular dynamics (MDs) and Monte Carlo simulations and the interatomic potentials used with these techniques to describe polymers. We then focus on techniques specifically designed to model the curing of thermoset polymers and the following two sections summarize what was learned from these simulations regarding network formation and quantitative prediction of thermomechanical properties of thermosets (including glass transition temperature, thermal expansion, and mechanical response) including how molecular structure and specimen size affects them. We also briefly discuss the use of these techniques in materials engineering efforts and conclude with our outlook on the field including opportunities and challenges. ATOMISTIC SIMULATIONS OF MOLECULAR MATERIALS In this section, we discuss simulation techniques that enable the prediction of thermomechanical properties of a molecular system given their atomic/molecular structure. These simulations are key to develop an understanding of the molecular origins of macroscopic behavior and to provide quantitative predictions for materials design. The simulation techniques can be classified in two main categories: MD, based on deterministic classical equations of motion, and Monte Carlo, based on a stochastic sampling of configuration space. Molecular Dynamics MD was first used by Alder and Wainwright in to study hard-sphere systems and further developed by Rahman for soft-sphere systems using Lennard-Jones potentials. 2 From this origin, MD has growth to become an indispensable technique in fields of physics, chemistry, biology, and engineering. Current supercomputers enable multi-billion atom MD simulations 3 and accelerated MD techniques enable microsecond timescales. 4,5 The use of MD to study polymers dates back to 1970s with the majority of the work focusing on thermoplastics. 6,7 The relatively fewer efforts in molecular simulations of thermosets can be partially attributed to the difficulty in creating accurate molecular representations of three-dimensional crosslinked structures and simulating the chemical reactions that form the molecular structure during the curing procedure. Next section focuses on techniques to build polymer network structures; here, we just review MD simulations. MD involves numerically solving the classical equations of motion: : : F i r i5vi ; v i5 : (1) m i where r i, v i,andf i and the position, velocity and force vectors corresponding to particle i. The force is obtained from the gradient of an interatomic potential: F i 52r ri Vðfr j gþ: (2) The interatomic potential, also called force field, provides an expression for the total potential energy of the system as a function of the atomic positions. Potentials for molecular systems will be described in next subsection. The particles in MD simulations can be individual atoms (this is referred to as all-atom MD or simply MD) or small collections of covalently bonded beads (this is referred to as coarse grained or mesodynamics). 104 JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

3 JOURNAL OF POLYMER SCIENCE REVIEW There are three main stages in an MD simulation of a thermosetting polymer: (i) building a molecular model with atomistic details to be used as an initial condition, (ii) integrating the equations of motion to obtain the atomic positions and velocities at various time, and (iii) extracting the desired properties from the trajectories using statistical mechanics. The core of a MD simulation is the second stage in which trajectories are generated by numerical integration of Newton s equations of motion with very small time intervals (order of femtoseconds for all-atom MD) using methods such as Verlet s algorithm 8 or Gear s predictor-corrector algorithm. 9 Direct numerical integration of eqs (1) and (2) leads to trajectories consistent with the microcanonical ensemble (constant number of particles, volume and energy, NVE). Comparison with experiments often calls for simulations under isothermal or isobaric conditions. During the 1980 s pioneering work by Andersen, Nose, Hoover, Rahman, Parinnello, and others led to dynamical simulations consistent with other thermodynamic ensembles. Thermostats and barostats based on stochastic approaches 10 or by extended Lagrangian techniques that modify the classical equations of motion are common in MD simulations. Several powerful software packages for MD simulations are available. Among the most widely used, freely available codes are: LAMMPS, 14,15 GROMACS, 16 NAMD, 17 and AMBER. 18 These simulation codes make use of parallel processing and accelerators to achieve large-scale simulations. Typical polymer simulations range in size from a few thousands to a few millions atoms; resulting in simulation cells ranging from a few nanometers to a few tens of nanometers on the side. Much larger simulations are possible in today s supercomputer, but there is always a tradeoff between system size and simulation time for a given computational budget. Typical simulation timescales are in the nanosecond regime. Periodic boundary conditions are routine practice to avoid free surfaces and mimic a macroscopic sample. Interatomic Potentials The interaction between atoms can be described from first principles by quantum mechanics. Unfortunately, these electronic structure calculations are computationally intensive and so-called ab initio MD (MD simulations where forces are computed from an ab initio electronic structure calculation) remain restricted to relatively small systems and short times. Thus, large-scale MD simulations are performed using molecular force fields that describe the interaction between atoms via a series of energy terms designed to capture covalent, van der Waals, and electrostatic interactions. Force fields for molecular systems can be divided into nonreactive ones (where covalent bonds cannot be broken or formed during the simulations) and reactive force fields that enable chemical reactions. The vast majority of polymer simulations have been performed using nonreactive force fields, where the molecular connectivity is specified prior to the simulation and cannot be modified. Reactive potentials, such as ReaxFF 19 and AIREBO, 20 use the concept of bond order to enable chemical reactions, and, we believe, will find increasing use in polymer simulations. Here we describe some of the most widely used non-reactive force fields for polymer simulations. Several generic molecular force fields have been applied to polymer simulations, examples include the consistent valence force field (CVFF), 21 Chemistry at HARvard Macromolecular Mechanics (CHARMM), 22 DREIDING, 23 polymer consistent force field (PCFF), 24 and condensed-phase optimized molecular potentials for atomistic simulation studies (COMPASS). 25 All these force fields decompose the total potential energy of a molecular system into covalent interactions, designed to describe the effects of chemical bonds, van der Waals interactions, to describe mid-range attraction due to London dispersion and short range repulsion originating from the Pauli exclusion principle, and long range electrostatic interactions originating from partial atomic charges. Nonreactive potentials use atom types to distinguish between the same element in different bonding conditions (e.g., a sp 2 bonded carbon atom is described with parameters different from a sp 3 bonded one). Covalent interactions in these valence-based force fields describe the effects of directionality in bonding caused by the electronic structure via a series of terms describing bond stretch, angle bending, torsional potentials, and out of plane deformations (also called improper torsions). Harmonic terms are typically used for bond stretch; angle bending is described also by harmonic terms either on the angle or (more appropriately) the sine of the angle, and sinusoidal terms for the torsion angles. The general-purpose DREIDING force field uses common force constants based on hybridization rules bond distances based on covalent radii. Force fields such as CHARMM and AMBER have additional parameters are designed to provide highly accurate results for the set of molecules they are designed for. The CFF91, CFF93, 26 PCFF, and COMPASS force field family contain, in addition to the terms described above, cross-terms between covalent interactions which are useful, among other things, to tune vibrational energies and have been parameterized using a combination of ab initio data and experiments. Coarse-Grained (CG) MD Many applications do not require full atomistic detail or they may not need it at all times. In such cases, CG models, where groups of atoms are represented by beads or mesoparticles, are an attractive alternative to all-atom MD. While atomistic detail is lost in these approaches, a molecular description is maintained. Two main factors make coarse grain representations computationally more efficient than all-atom MD. In the first place, the number of particles in the simulation is reduced; second, high frequency degrees of freedom (e.g., CAH vibrations) are not explicitly described in the dynamics and consequently larger timesteps can be used. CG molecular models by systematic coarse-graining atomistic details date back to the 1970s and have increased in popularity and accuracy since. The various approaches differ in the degree of coarsening over an all-atom description. At the JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

4 REVIEW JOURNAL OF POLYMER SCIENCE lowest level of coarsening, united atom models that group nonpolar carbons with their bonded hydrogen atoms into single particles are common practice. Several force field parameterizations for united atom descriptions were developed over the years. Examples include the AMBER unitedatom force field, 27 GROMACS. 28 Mid-level CG models replace larger groups of atoms, a few united atoms or monomers, by a single particle, CG bead. These include residue-based CG models, 29 shape-based CG models 30 for biopolymers using the popular MARTINI force field. 31 At the highest level of coarsening, a part of polymer chain or even entire chains are grouped as a single CG bead. 32 An interesting connection between coarse-grain particle approaches and continuum mechanics is provided by Hoover and Hoover in Ref. 33. The choice of the coarse-graining level should be dictated by the application, taking into account minimal features of the model required to reproduce the desired material properties and the predictive power required. Key features of the models to consider include the shape and volume of a molecule, intramolecular interaction energies, flexibility, and connectivity of a polymer chain, ability to form a crystal or amorphous condensed phase and so on. For thermosetting polymers, low or mid-level coarse-graining is usually employed. Basic steps of CG simulations for thermosetting polymers are: (i) selecting a coarse-gaining level based on application; (ii) mapping atomistic details into CG particles; (iii) Parameterizing bonded and nonbonded interactions between CG particles; (iv) constructing crosslinking networks based on certain criteria for bond creating; (v) evaluating material properties of the crosslinked thermoset. Monte Carlo simulations MC simulations are also widely used for polymer simulations. The first MC study of a polymer dates back to 1950s 34 only a few years after the celebrated Metropolis et al. 35 algorithm was developed at Los Alamos Scientific Laboratory (now Los Alamos National Laboratory), in New Mexico. MC simulations generate an ensemble of configurations of a model polymer that satisfy a desired statistical distribution function using stochastic methods. Although most MC simulations are applied to study systems in equilibrium, they also have the capability to study nonequilibrium thermodynamic phenomena under certain assumptions. MC methods include on-lattice or off-lattice CG polymer models, 36,37 but fully atomistic descriptions are becoming popular in recent years. 38 One main advantage of MC simulations over MD simulations is the huge speedup, usually many orders of magnitude, in achieving equilibration. Readers interested in the topic are referred to the excellent review by Theodorou 39 for recent progress in the application of MC techniques to polymers and Refs. 40 and 41 and for the general theory and implementation. BUILDING POLYMER NETWORKS Creating well-equilibrated molecular models of thermosetting polymers is challenging and significant efforts have been devoted to developing such methods. Research on the formation of polymer networks can be traced back to the end of 1970s. 42,43 These pioneering attempts used MC simulations of lattice or CG models. Fully atomistic studies began in the mid-1990s when molecular force fields and computer power enabled such endeavors. Here we review state-of-the-art methods that use MD or MC simulations to predict the molecular structure of thermoset polymers; these include both atomistic and CG models. Network Formation By MD Simulations In the early 1990s, Grest and Kremer 44 studied polymer networks by randomly and instantaneously crosslinking wellequilibrated linear polymer chains represented by beadspring model with Lennard-Jones interactions. In the mid- 1990s, the first fully atomistic MD simulation was carried out by Hamerton et al. 45 In the study, three-dimensional polymer networks with less than 200 atoms were generated for trifunctional polycyanurate. The predicted elastic modulus and glass transition temperatures were in reasonable agreement with experimental data despite the small size of simulation system. In the late 1990s, Doherty et al. 46 allowed progressive polymerization reactions during a MD simulation. In the crosslinking process, monomers were allowed to react with each other. The criterion for forming chemical bonds was based on the distance between predefined reactive sites on the monomers or the growing polymer chains. Their crosslinking procedure involves packing and equilibrating monomers, repetitive crosslinking until a certain conversion is achieved, and relaxation of final structures via energy minimization. With few modifications, this general approach to create polymer networks is widely used today; current approaches differ in its details in terms of the description of interactions during bond creation, capture distance to create structures (typically between 0.4 and 1 nm), etc. Yarovsky and Evans 47 developed a computational procedure for constructing molecular models of crosslinked polymer networks and applied it to low molecular weight, watersoluble, epoxy resins cured with various crosslinking agents. The PCFF was used and the simulations predicted volume shrinkage. Heine et al. 48 simulated the structure and elastic moduli of end-crosslinked poly(dimethyl-siloxane) networks using a united atom force field. The networks were formed dynamically with reactions to occur when two united atoms are within cutoff distance 0.65 nm. The resulting topology was relaxed using a modified potential; harmonic at short (bonding) distances and linear at larger distances to avoid large forces and instabilities when a new bond is formed. Wu and Xu 49 developed a method to construct polymer networks for epoxy resin system based on DGEBA (diglycidyl ether bisphenol A) and IPD (isophorone diamine). They used the DREIDING force field with charge equilibration to build the structure but the COMPASS for property predictions. Komarov et al. 50 reported a computational method where the polymer network is polymerized at a CG level and then mapped into a fully atomistic model. MDs were then carried out with the OPLS force field. Varshney et al. 51 studied 106 JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

5 JOURNAL OF POLYMER SCIENCE REVIEW FIGURE 1 Flowchart of the MD crosslinking process described by Li and Strachan. (Reproduced from Ref. 59, with permission from Elsevier.) molecular modeling of thermosetting polymers with special emphasis on crosslinking procedure. They described different approaches to build highly crosslinked polymer networks and proposed a multistep relaxation procedure for relaxing newly formed topology in the crosslinking process. Using the CVFF, they predicted density, glass transition temperature, and thermal expansion coefficient of an epoxy-based thermoset (EPON-862/DETDA). Lin and Khare 52 presented a singlestep polymerization method for the creation of atomistic model structures of crosslinked polymers using the AMBER force field. A simulated annealing algorithm was used to identify pairs of reacting atoms that minimize the sum of the distances of the proposed bonds; after this optimal connectivity is chosen all crosslinking bonds are created in a single step. Bermejo and Ugarte 53 introduced a method for building fully atomistic models of chemically crosslinked poly(vinyl alcohol). Their crosslinking procedure also combines periodic crosslinking based on a distance criteria followed by structural relaxation. Similar to the approach of Yarovsky and Evans, 47 Bandyopadhyay et al. 54 proposed an efficient method of creating united-atom molecular models of a crosslinked epoxy system where the simultaneous breaking of CH 2 AO bonds in the epoxide ends of the EPON-862 molecules and NAH bonds of the DETDA molecules results in activated CH 2 ends capable of forming crosslinks with activated N atoms when the root mean square distance between them is within a cutoff. An important aspect during polymerization is the evolution of partial atomic charges during chemical reactions. Electrostatic interactions play an important role in the overall binding of these systems and an accurate calculation of charge distribution is essential for the prediction of their structure and properties. Thus, an accurate simulation of the crosslinking process should include a procedure for updating charges to take into account changes in chemistry and topology. Geometry dependent, self-consistent charges can be obtained by minimizing an expression for the total electrostatic energy using methods like Charge equilibration (QEq) 55 or electronegativity equalization method (EEM) These methods are accurate and transferable and have been applied to polymerization 49 but they remain computationally intensive especially for large-scale simulations. Varshney et al. 51 used a charge update scheme based on partial charges obtained from ab initio calculations on a small model molecule topology is similar to the newly formed crosslink of EPON862/DETDA. Bandyopadhyay et al. 54 maintained the neutrality of the EPON862/DETDA system by assigning negative or positive charges to specific atoms during bond breaking and formation. A rather thorough study of charge evolution during polymerization was conducted by Li and Strachan. 59 They used EEM to predict partial atomic charges evolution during primary and secondary reactions involved in the polymerization of an EPON862/DETDA system. They found that atomic charges evolve significantly only during chemical reactions and that this evolution is predictable from the chemistry of the reaction alone and insensitive to the actual atomic configuration of the process. Based on these observations, they proposed a computationally efficient charge updating approach, denoted electronegative equalization-based charge assignment, which enables largescale polymerization and crosslinking simulations. This crosslinking methodology has been adopted by the MAPS (Materials Processes and Simulations) platform, commercial software for molecular modeling of polymers. 60 A flowchart of the crosslinking procedure is reproduced in Figure 1. Network Formation By MC Simulations MC studies on polymer network formation can be classified along two lines. One line is based on theories of network JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

6 REVIEW JOURNAL OF POLYMER SCIENCE formation, either the statistical/percolation theory 61 or the kinetic theory. 62 The second is based on direct MC simulations of structure growth in configuration spaces. 63 Early studies of lattice and off-lattice percolation-based MC simulations of polymer networks provide important insight. The off-lattice simulations especially gave some successful applications, as summarized by Dusek. 64 However, MC simulations based on the kinetic theory became more popular latterly because the kinetic method directly accounts for the history of network formation. For examples, Somvarsky and Dusek developed a MC simulation procedure for kinetically controlled polymer structure growth including network formation determined by Smoluchowski-type differential equations. 65,66 Cheng and Chiu developed a very general algorithm of MC simulations for network formation with complex chemical reaction mechanism 67 and this algorithm was applied to systems of epoxides cured with amines 67,68 and systems of epoxy/phenol resins cured with imidazole. 69 A concise flowchart of typical kinetic MC simulations can be found in Ref. 69. Direct MC simulations are more versatile and can in principle be applied to any polymer network formation. The pioneering work by Leung and Eichinger 63 was subsequently extended in a series of works at the University of Washington. 70 This type of MC simulations has been used for conducting crosslinking reactions between linear and/or branched polymers and crosslinking agents, including endlinking polymers (reactive groups only at the chain ends) and random-crosslinking polymers (reactive sites on their repeat units and crosslinking reactions can take place at random locations along the chains). In the implementation of direct MC simulations, usually, crosslinking molecules are treated as rigid structures but polymer molecules are assumed to be flexible. The coordinates of the reactive sites on the crosslinking molecules and polymer molecules are chosen stochastically from a pre-determined probability density function. The crosslinking reactions proceed based on the capture sphere concept that around each reactive site there exists a capture sphere volume in which the reactive site can diffuse through. If another reactive site is within this sphere and is allowed to react with the site of interest, a reaction between these two sites is enforced. After two sites react, the components containing these two reactive sites move toward each other along the line joining the reacting sites. During the crosslinking reactions, the capture radius is increased in a stepwise manner so that a higher extent of reaction can be achieved in the following rounds. The crosslinking process ends when the capture radius is roughly half the simulation cell. EVOLUTION OF POLYMER NETWORK The response of polymers is determined both by chemistry and molecular structure. As discussed above, a fundamental understanding of the evolution of polymer networks during curing is not only essential to establish structural-property relationship but also to understand how properties evolve FIGURE 2 RDF between reactive atoms at various conversion degrees. (Reproduced from Ref. 76, with permission from Elsevier.) during curing and understand processing. Over past decades, several experimental techniques have been developed for the characterization of polymer networks. 71 For examples, the crosslinking density can be quantified by swelling experiments, 72 nuclear magnetic resonance 73 or infrared spectroscopy (FTIR) 74 and the gel point can be measured by dynamic mechanical analysis. 75 Relating the experimental observables in these techniques to molecular structure remains challenging and requires the use of complex models; this process results in uncertainties and ambiguities in the conclusions. Thus, powerful computer simulations are becoming an indispensible companion to such experiments. In the case of molecular simulations, the structure is known precisely and the challenge is assessing how representative they are of the real systems. Structure Evolution and Gel Point As a thermosetting polymer is cured the chemical reactions cause changes in thermo-mechanical response of the resin; the increase in molecular weight leads to an increase in viscosity, glass transition temperature and density. Simulations that mimic such processes provide valuable information to understand and ultimately tune processing conditions of thermosets and their composites. A variety of signatures have been used to develop a molecular picture of the crosslinking process. The evolution of the radial distribution function (RDF) between reactive atoms during cure (see Fig. 2) for an epoxy phenol novolac (EPN) and bisphenol-a (BPA) system, 76 shows a shift towards larger distances with increasing conversion degree. As nearby reactive atoms undergo chemical reactions the first peak of the RDF is reduced. The crosslinking proceeds initially at a fast rate and then slowing down significantly at 60 70% conversion degree; 59 this is due to an increase in the mean separation between reactive sites (see Fig. 3) and reduced molecular mobility with increasing conversion degree. The range of conversion degrees where 108 JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

7 JOURNAL OF POLYMER SCIENCE REVIEW FIGURE 3 Initial crosslinking distance as a function of conversion degree (85% conversion is reached by 12 crosslinking cycles in this simulation for a DGEBA/33DDS system with 65K atoms, higher conversion degree can be further reached by increasing the cutoff distance). crosslinking slows down significantly corresponds to a steep increase in the molecular weight of the systems (see Fig. 4), 51,76,77 which marks the gel point. The gel point marks the state where the largest molecule in the system percolates throughout the entire sample. The cure degree that corresponds to the gel point depends on stoichiometry, functionality, reactivity of functional groups as well as possible side reactions. In molecular simulations, the gel point can be estimated from the inflection point of the molecular weight of the largest molecule with conversion degree or the incipient formation of secondary cycles, which are defined as intramolecular reaction within a same group. 78 The weight-averaged reduced molecular weight (RMW), which is defined as the weight-average molecular weight of all reacting groups except the largest one, was also proposed for estimating the gel point. 79 It has been shown that both the maximum in RMW and the inception of the secondary cycles are good indicators for characterizing the gel point, 51,78 as shown in Figures 4 and 5, respectively. FIGURE 4 Molecular weight changes with conversion degree: largest group (circles), second-largest group (squares), and weight-averaged RMW (diamonds). (Reproduced from Ref. 51, with permission from American Chemical Society.) Of course, approximations in the model can lead to inaccuracies. Yarovsky and Evans 83 conducted MD simulations on the curing of a phosphated epoxy resin with two different crosslinkers (CYMEL 1158 and CYMEL 1172). Their simulations predicted a volume reduction of 512% for several systems with different conversion degrees cooled down from 600 to 300 K. Varshney et al. 51 tracked the volume changed during crosslinking of their EPON862/DETDA systems and observed a 7% volume reduction. The volume shrinkage shows a linear relationship with the crosslinking degree, which is consistent with the PVT experimental observation for the chemical shrinkage of an epoxy resin cured at higher temperatures ( K). 82 Yang and Qu 76 conducted MD simulations on an epoxy EPN-1180/BPA system and equilibrated simulation cells generated along the trajectory of the polymerization simulation with different conversions at 300 K. Volume Shrinkage During Crosslinking Chemical reactions that take place during the crosslinking process pull end groups together and gradually transform a liquid into a crosslinked solid. The formation of polymer networks leads to a volumetric shrinkage, which often exceeds 6% 80 and can be as high as 10% for some resins. 81 This cure-induced volume shrinkage in thermosets causes warpage and residual stresses in thermoset composites. Experimentally, the cure induced volume shrinkage can be measured by a pressure-volume-temperature (PVT) analysis. However, the chemical shrinkage is not easily obtainable because the overall volume change is affected by many factors 82 like thermal expansion and because it is difficult to quantify the degree of cure accurately. Molecular simulations, however, enable a direct assessment of chemical shrinkage. FIGURE 5 Estimation the gel point from the inception of secondary cycles. (Reproduced from Ref. 78, with permission from Wiley.) JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

8 REVIEW JOURNAL OF POLYMER SCIENCE They also observed an almost linear volume reduction upon increasing crosslinking degree. A maximum volume shrinkage of 4.2% was predicted for the highest crosslinked system (90%). Interestingly, an increase in volume with increasing conversion has been reported in some systems for high-conversion degrees. MD simulations with a united atom model for an epoxy system (EPON862/DETDA) of Bandyopadhyay et al. 54 showed that a 63% cured system is denser than its 76% counterpart at all temperatures and a 54% cured system has a larger volume shrinkage than the 63 and 76% cured. This somewhat surprising result was attributed to the ability of lightly cross-linked epoxy systems to pack into dense configurations. A similar trend was also observed experimentally, where the isothermal density for epoxy resins pass through a maximum in the glassy state with increasing chemical conversion and was explained by the packing efficiency falling off for the highly crosslinked networks. 84 These results show that the physics of polymer networks especially over the gel point and in the glassy state is not fully understood. Polymer viscosity and stiffness as a function of cure degree are key information in modeling processing-induced residual stresses in composite materials. Knowledge of the internal stress in the polymer matrix is critical to understand the ultimate properties of composites. In most cases, chemical shrinkage is assumed to be proportional to conversion degree; 85,86 while not applicable to all systems, simulation and experimental results support this assumption for a wide class of polymers. An important, unanswered, question is whether the proportionality constant relating chemical shrinkage and cure degree is dependent on temperature. Molecular simulations can provide an answer to such a question since, contrary to experiments, cure degree and temperature can be independently controlled. We performed such analysis based on our prior simulation results for an epoxy resin system (EPON862/DETDA). 59,77 Figure 6 shows the specific volume of the epoxy resin containing 17,000 atoms as a function of conversion degree at different temperatures. As expected, the specific volume decreases as the conversion degree increases. Volume depends approximately linearly with conversion degree, especially for lower temperatures. Interestingly, the slope (which represents the chemical volume shrinkage coefficient) increases with increasing temperature. At higher temperatures, a slight nonlinearity is observable for the volume shrinkage with conversion. The simulations indicate that volume shrinkage is not a constant but a temperature dependent quantity that varies in a wide range; from 5 to 20% for an admittedly large temperature range from 300 to 600 K. These results show the importance of processing temperature on volume shrinkage and, consequently, on the development of internal stresses in composites. PREDICTIONS OF MATERIAL PROPERTIES FIGURE 6 Volume shrinkage as a function of conversion for an EPON862/DETDA system (MD simulations: EPON862/ DETDA 5 256/128, total 17,000 atoms, data from Refs. 59 and 77 ). Of interest in most polymer applications is the determination of basic thermomechanical properties including glass transition temperature, elastic constants, and ultimate mechanical properties such as yield stress and post yield behavior. These properties can be obtained directly from molecular simulations. As discussed in prior sections, the accuracy of the predictions depends on the interatomic potentials used but other simulation details also affect the prediction of some of these properties. The time and length scales achievable in MD simulations are orders of magnitude smaller than in most experiments 87 affecting properties that depend on time or specimen size. Despite these limitations, molecular simulations are playing an important role in the development of a fundamental understanding of physical and mechanical behavior of polymers and they have been shown to capture nontrivial trends such as the dependence of yield and postyield on temperature, thermal history, deformation rate and loading path, see, for example, Refs. 88 and 89. In addition, physics-based constitutive models 90 can be used to map MD results to the scales of interest in applications; in many cases, this requires extrapolating MD results to the time and spatial scales of interest in the application and is a topic of ongoing research. Input parameters for constitutive models can often be obtained directly or indirectly from experiments but with the increasing interest in the use of predictive materials simulations to decrease the time and cost involved in the development and deployment of new materials, 91,92 it would be highly desirable to obtain these fundamental material parameters directly from first principles. Connecting MD results with continuum polymer models remains a scientific challenge and key step developing predictive models for polymers. Thermal Properties Glass Transition Temperature One of the most important thermal properties of amorphous polymers is their glass transition temperature T g. As a polymer is cooled down through T g, it transforms from a rubber (for crosslinked systems over the gel point) or liquid (when chains are not covalently bonded to each other as in the 110 JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

9 JOURNAL OF POLYMER SCIENCE REVIEW value significantly. Figure 7 shows a typical densitytemperature plot from MD simulations. Clearly, the density increases with decreasing temperature and polymer exhibits two linear regions corresponding to the glass and rubber; the change in slope marks T g. FIGURE 7 Density as a function of temperature (the percentage represents crosslinking degree, (1024, 512) stands for monomer numbers of epoxy and curing agent). (Modified from Ref. 99, with permission from Elsevier.) thermoplastics) to a glass (an amorphous solid trapped in a non-equilibrium state). T g is governed by local chain dynamics and represents an intrinsic signature of the molecular structure. Below T g the motion of polymer chain segments dramatically slows down and the mechanical properties of polymers become very different from those above it. Thus, T g is a key property to determine processing and application temperature ranges for a specific polymer. Although determining T g experimentally by dilatometric technique is currently a standard practice for material engineers, the origin of glass transition is not fully understood and predictive simulations are a valuable tool. 93 Given the importance of T g from basic and applied science points of view it is not surprising that significant efforts have been devoted to using MD and MC simulations to uncover the nature of the transition and to quantitatively predict it for polymers of interest. Several thermodynamic properties, including density (or specific volume), internal energy, specific entropy, can be used in molecular simulations to determine T g. 97 An abrupt change in the temperature dependence of these properties indicates the polymer cannot maintain its equilibrium state and the phase transformation. Common practice is to use the temperature dependence of density (or specific volume) to determine T g. But the temperature dependence of energy terms, while showing the glass transition, can also provide insight into the different roles of these energy components in the glass transition process. 98 To determine T g via MD simulations it is customary to cool down and/or heat up a model system at a constant rate with the use of a thermostat to control temperature and under isobaric conditions. The cool-down or heat-up is normally conducted in a stepwise manner using fixed temperature decrements/increments for a pre-determined amount of time; the thermal history of the sample is critical to understand the glass transition and heating/cooling rates affect its Quantitative Predictions of T g and Role of Rate In recent years, there have been an increasing number of molecular simulations on the prediction of T g. Table 1 lists selected predictions together with available experimental data and Figure 8 compares the predictions with experimental data. Most of these simulations are by all-atom MD but some use coarse grained approaches. We note that slightly different conversion degrees were reported in these simulations and the conversion degree of experiments was not reported; typical values are believed to be around 90% or higher. The role of conversion degree on T g will be discussed below; at this point we will ignore the small differences in cure degree between simulations and experiments. This data clearly shows the predictive power of these simulations since the parameters are not tuned to reproduce T g. Yet, some predictions show significant errors. We note that most of the all-atom predictions overestimate T g by 20 K. Interestingly, one would expect this trend due to the high heating/cooling rates in the simulations. Rates in standard PVT experiments are of the order of 10 K/min but those in MD simulations are typically 10 K/nanosecond or faster; that is, 10 orders of magnitude faster. This much higher cooling rate shifts T g towards higher values, at a rate of about 3 K per decade according to the well-established Williams-Landel-Ferry (WLF) equation, 119,120 which describes the relation between relaxation time and temperature measured from a reference value and was obtained empirically from extensive experimental measurements on various glass formers from polymers to organic liquids. Soldera and Metatla 121 confirmed the validation of WLF equation to the atomistic simulations of polymers. Therefore, a good estimation of T g from molecular simulations should take into account the adjustment from the difference of cooling rates. Role of Molecular Structure on T g Besides the significant role of rates on T g, it is of importance to understand how molecular structure and chemistry of the constituent molecules and network architecture affects T g. Experiments have shown that the T g of an epoxy network can be shifted more than 140 K by just using different crosslinkers. 122 Atomistic simulations have also found that the T g values decrease with an increase in the chain length of the crosslinkers and this trend qualitatively agrees with the experimental observations. 123 The molecular weight dependence of T g was studied for thermoplastics by MD simulations 124 but there have been few studies of chain length effect on T g for thermosets. 106 Recent MD simulations explored the role of crosslinker functionality and character basic thermomechanics properties but significant more efforts on this topic will be needed to a complete picture to emerge. JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

10 REVIEW JOURNAL OF POLYMER SCIENCE TABLE 1 Predicted and Experimental Glass Transition Temperatures of Thermosets Model System Simulation Data Experimental Data Resin Crosslinker T g (K) Conversion Method Refs. T g (K) Refs. PVA 1,2-Ethanediol % MD VE Styrene % MD DGEBA TMAB % MD DGEBA POP MD DGEBA 33DDS % MD DGEBA DETDA % CG MD a EPON862 DETDA % MD EPON862 DETDA % MD 51 EPON862 DETDA % UA MD EPON862 DETDA % MD EPON862 DETDA 380 MD 112 EPON862 TETA % MD Polyester HMMM % CG MD EPN1180 BPA % MD UP Trigonox42PR % MC Phenol Methylene % MD 118 TYG 4-MHHPA % MC/MD a CG MD: coarse-grained MD; UA MD: united atom MD. Effect of Cure Degree on T g From Figure 7 we see that the MD simulations predicted an increase in T g with degree of conversion. This trend is consistent with experiments and MD simulations 77 found that the dependency of T g with conversion becomes more pronounced as the curing proceeds and a significant increase in the slope of T g vs. conversion degree occurs at around 60% conversion, which corresponds to the gel point. Normally, the conversion degree >90% is difficult to achieve for molecular simulations. But the T g close to the fully cured systems can be estimated by extrapolating the T g prediction of simulations at lower conversion degree. Two equations have been extensively used to describe the relationship between T g and conversion degree. One is the DiBenedetto s equation, The latter is believed to be more generally applicable. 128 In both cases, k is an adjustable parameter that describes the nonlinearities between T g and conversion and has been related to the ratio of the step changes in heat capacity at T g between the fully cured and uncured system (k5dc p1 =Dc p0 ), n stands for the degree of conversion, T g is the glass transition temperature at the degree of conversion n; finally, Tg 0, Tg 1 are the glass transition temperatures for uncured and fully cured systems, respectively. Only a few examples of fitting simulation data of T g by using these equations were reported. Li and Strachan 77 used a least squares method to obtain k and Tg 1 for the EPON862/DETDA systems from their MD predictions T g for conversion values up to 86%. Their predicted result of k falls in the range of reported in the literature. Li et al. 99 fitted their MD T g data below 85% conversion for DGEBA/33DDS systems T g 2Tg 0 kn Tg 1 5 2T0 g 12ð12kÞn : (3a) Another is the Venditti-Gillham equation, 128 ln ðt g Þ2ln ðtg 0Þ ln ðtg 1Þ2ln ðt0 g Þ 5 kn 12ð12kÞn : (3b) FIGURE 8 Comparison of simulation predictions and experimental results of T g. 112 JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

11 JOURNAL OF POLYMER SCIENCE REVIEW CTE. 99 The estimations of CTE at the same conversion as experiments were extrapolated from MD results at lower conversions. In this example, the predicted CTE for the glassy state is 20% higher than the experimental value while the CTE for the rubbery state is 10% lower. A similar trend of CTE dependence on conversion was also reported by Shenogina et al. 106 for DGEBA/DETDA systems. The difference between simulation CTE results and experimental CTE data could be partially due to differences in cooling rates 129 but no systematic study of such has been carried out. FIGURE 9 Normalized volume as a function of temperature for DGEBA/33DDS thermosets with different conversion degrees (the slope of a linear fitting is the volumetric CTE). and gave Tg K and k50:58 using the DiBenedetto s equation, and Tg K and k50:65 using the Venditti- Gillham equation. Based on these fittings, the T g at the experimentally measured 92% conversion is estimated to be 525 K, which is only 7.4% higher than the experimental data after adjustment by cooling rate effect. The fitting parameter k is also very close to the ratio Dc p1 =Dc p given by their independent heat capacity calculations. Coefficient of Thermal Expansion (CTE) Thermal expansion of polymers is a critical physical property that affects the polymer residual stress buildup during processing. Despite its importance, there have been few reports of CTE predictions as compared with T g. The volumetric CTE a can be defined as: a5 (4) V P There are very few studies on the effect of molecular structures on the polymer CTE based on computational simulations. Soni et al. 123 recently investigated the effect of crosslinker length on the volumetric CTE and reported that the CTE for both the rubbery and glassy states increases with increasing crosslinker length. It was attributed to the formation of tighter polymer networks in the systems containing shorter crosslinkers. Shenogina et al. 106 compared the CTE of epoxy resins with different chain lengths and the results were non-conclusive. More efforts are needed to develop a clear understanding of how molecular structures affect CTE. Heat Capacity Specific heat is another fundamental thermal property of polymers that depends on their chemistry, molecular structure, and state. The value depends on the number of degrees of freedom capable of absorbing energy at a given temperature. Thus, the specific heat exhibits a discontinuous change at the glass transition. Molecular simulations can provide information about these degrees of freedom and complement experimental measurements. 130 Relatively few studies have been performed to characterize specific heat of polymers at where V is the volume of simulation cell at temperature T, V 0 is the volume of simulation cell at a reference temperature (e.g., room temperature). The linear CTE equals to a=3. Almost all simulation results of the volumetric CTE are from MD simulations and the deviation from experimental measurements range for 2 to 40%. 51,59,77,99,108,123 Typical data for determining the volumetric CTE is shown in Figure 9. The volumetric CTE can be obtained by fitting the MD volume-temperature data to straight lines above and below T g as the temperature dependence within the glassy and rubbery regions is small. Similar to T g, the CTE is also strongly dependent on the conversion degree. Figure 10 displays the effect of conversion degree on the volumetric CTE for DGEBA/33DDS thermosets, together with the experimental data obtained from TMA measurement of the coefficient of linear thermal expansion multiplied by three and PVT measurement of the volumetric FIGURE 10 Volumetric CTE below and above T g for DGEBA/ 33DDS (1024, 512) systems (MD results are obtained from NPT cooling simulations and experimental results are from TMA and PVT, shadowed points are estimations based on the extrapolation of MD data), (Reproduced from Ref. 99, with permission from Elsevier.) JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

12 REVIEW JOURNAL OF POLYMER SCIENCE the molecular level and compare the results with experiments. Soldera at al. 131 extracted the constant volume and pressure specific heats (C v and C p ) of PMMA from MD simulation trajectories using the fluctuations of internal energy and enthalpy according to statistical mechanics. These predictions (that ignore quantum effects) show significant differences with experiments. The classical MD predictions underestimate the experimental results; this is surprising result since classical statistical mechanics leads to an overestimation of the specific heat. Quantum mechanically highfrequency modes, those with quantum of energy (hw, where w is the corresponding frequency) larger than kt cannot absorb energy and do not contribute to the specific heat. However, classically all modes absorb energy at any temperature. Li et al. 99 computed the constant-volume specific heat C v of a thermoset using quantum statistical mechanics and the quasi-harmonic vibrational density of states (DoS) obtained from the power spectrum of atomic velocities in MD simulations. The authors obtained the constant-pressure specific heat C p based on the thermodynamic relationship: C p 5C v 1a 2 BVT (5) where a is the volumetric thermal expansion coefficient, V the volume, T the temperature and B the isothermal bulk modulus. A comparison between the simulation results and their own experimental data was also made, as shown in Figure 11. From Figure 11, the difference in terms of the specific heat discontinuity at the glass transition is seen relatively small. The magnitude of the specific heat C p jump at T g for a DGEBA/33DDS predicted by Li et al. 99 is roughly 0.22 JK 21 g 21. It is comparable to their own experimental data 0.36 JK 21 g 21 and a previous experimental data 0.35 JK 21 g 21 for a similar thermoset DGEBA/DDM. 132 The Gr uneisen parameter of the glass c5ðabv=c v Þ T5Tg for the same thermoset DGEBA/33DDS was calculated to be 0.62, which is also close to the experimental value for polyvinylacetate and polystyrene (PS). 133 However, a clear quantitative difference in the specific heat over most of the large temperature range is still noticeable and it suggests more efforts are necessary. Elastic Properties Elastic constants describe a materials response to small deformations and are critical to assess the performance of thermoset polymers and their use in composites. They are also key input parameters for constitutive models used in macroscopic simulations. 134 Due to the symmetry of isotropic materials, such as amorphous polymers, knowledge of two independent elastic constants enables the construction of the entire elastic constant tensor and the description the elastic response of the material under any loading condition. However, it is important to take into account that finite sized molecular samples are only approximately isotropic and it is advisable to compute elastic response along various directions and/or for multiple samples. Elastic constants can be calculated for atomistic simulations using several approaches; they can the divided in three main FIGURE 11 Specific heat capacity C p of DGEBA/33DDS as a function of (T 2 T g ). (Reproduced from Ref. 99, with permission from Elsevier.) categories: (i) explicitly deformation of the sample and monitoring its response; (ii) from equilibrium simulations in the undeformed state if analytical second derivatives are know; and (iii) from fluctuations of cell dimensions under constant stress or fluctuations of stress under constant cell dimension conditions using relationships from statistical mechanics. The fluctuation method 135 requires long simulation time ( ns) and is very time consuming thus seldom used. The static method for obtaining elastic constants was proposed by Theodorou and Suter in their pioneering work for atomistic modeling of polymeric glasses. 136 Basically, small strains are applied to a simulation cell experienced an initial energy minimization. Following an energy minimization after straining, the stiffness constants in a matrix form are obtained by calculating the second derivatives of potential energy with respect to strain. Lame s constants can in turn be calculated from the stiffness constants. Then other elastic constants can be derived from Lame s constants. The dynamic method was proposed by Berendsen et al. 137 for nonequilibrium MDs and was adapted and extended for polymers by Brown and Clarke. 138 In this method, elastic constants are obtained from stress-strain curves resulting from a deformation process similar to that in a laboratory experiment. The deformation can be performed via strain control or stress control. The desired elastic constants are then obtained by fitting the linear part of stress-strain curves depending on loading pathways, such as tension, shearing or volumetric expansion. 89,147 There have been numerous studies of the elastic constants of thermosetting polymers using molecular simulations. Several influential factors, including conversion degree, strain rate, temperature, accuracy of molecular force fields, have been investigated. A detailed comparison of elastic constants at room temperature is listed in Table 2. A plot is also given in Figure 12 for a visual comparison. In spite of the strain 114 JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

13 JOURNAL OF POLYMER SCIENCE REVIEW TABLE 2 Comparison of Elastic Constants of Thermosets Model System Simulation Data Exptl. Data Resin Crosslinker Conversion Method a E (GPa) m Ref. E (GPa) m Ref. PVA 0% MD/static VE Styrene 98% MD/static DGEBA IPD MD/static DGEBA DETDA 80% MD/dynamic DGEBA 33DDS 85% MD/dynamic DGEBA DETDA >90% CG MD/static EPON862 DETDA 100% MD/static EPON862 DETDA 76% UA-MD/dynamic EPON862 DETDA 86% MD/dynamic ,144 EPON862 DETDA MD/dynamic EPON862 TETA 61% MD/fluctuation Polyester HMMM 100% CG-MD/dynamic EPN1180 BPA 90% MD/dynamic Phenol Methylenes 92% MD/fluctuation a CG-MD: coarse-grained MD; UA-MD: united atom MD. rate used in molecular simulations (usually in the order of magnitude of s 21 ) most simulation results of elastic moduli for higher conversions, especially results obtained with the dynamic method, are in good agreement with available experimental results. Elastic moduli generally show a significant increase with conversion degree. This can be seen in Figure 13 that shows the Young s modulus of EPON862/DETDA at room temperature. This increase is found to be more pronounced for polymer systems with shorter resin or crosslinker chain length. 106 As expected for a solid, the elastic constants of glassy polymers are not very sensitive to strain rate, 76,77 thus explicit MD simulations can provide accurate results given appropriate molecular structures and force fields. However, above the glass transition the wide range of relaxation timescales lead to significant rate effects and affects the comparison of MD predictions and experiments. The effect of temperature on elastic constants, i.e. higher temperature reduces stiffness, has been rather accurately captured by molecular simulations for thermosets in glassy state. 76,77,99 The simulations also capture the significant softening as a glassy thermoset turns into a rubber at the glass transition temperature and the entropy-dominated stiffening of the rubber with temperature. However, the stiffness of the rubber is overestimated due to the high-rates of deformation in the simulations. 99 FIGURE 12 Comparison of Young s modulus. FIGURE 13 Predicted Young s modulus as a function of conversion degree (strain rate s 21, the black line showing the trend and experimental values are from Refs. 109 and 145). (Reproduced from Ref. 77, with permission from Elsevier.) JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

14 REVIEW JOURNAL OF POLYMER SCIENCE Ultimate Mechanical Properties Two important ultimate mechanical properties of polymers are yield strength and fracture strength. Both atomistic and CG molecular simulations have been used to study these ultimate properties of polymers. Rottler 7 has given a comprehensive overview of the key results of molecular modeling on the failure mechanism of glassy polymers, including shear yielding, creep, physical aging, strain hardening, crazing and fracture. While not without limitations, molecular simulations are capable of capturing the mechanical behaviors in polymer glasses observed in macroscopic experiments and provided important insights into the underlying molecular processes of polymer failure. The general effects of loading conditions, temperature, and strain rate on yield strength have also been systematically investigated and found to be consistent with experimental observations. However, the majority of molecular simulations have focused on linear homopolymers. Much less attention has been paid to simulations of ultimate properties of highly crosslinked thermoset polymers. A number of simulation studies on the ultimate mechanical properties of thermosets used CG MD. For example, Stevens 148 investigated the failure of highly cross-linked random and ordered polymer networks between solid walls using CG MD. The simulations relate failure strain (in tension and shear) to the shortest distance between the solid surfaces through the network. The author further simulated the interfacial fracture between highly crosslinked polymer networks and a solid surface using the same model and found that interfacial bond density determines failure location either adhesively at the interface or cohesively through polymer chain scission. 149 Stevens et al. 150,151 also studied the effect of crosslinker functionality and curing degree on interfacial fracture and found that the failure stress decreases with decreasing functionality or curing degree while failure strain increases with decreasing functionality or curing degree. Using a similar approach, Mukherji and Abrams 152,153 studied the mechanical behavior of a polymer network with random bonding and found that the formation of microvoids, without bond breaking, constitutes the microscopic origins of strain hardening after yielding. They also suggested that flexible cross-linkers might help introducing more ductility into polymer networks. Panico et al. 154 reported CG simulations of tensile failure in glassy polymer networks and relate failure for the scission of primary bonds. They also found that the ultimate tensile strength and the brittle behavior are enhanced by increasing crosslinking density, which limits the nucleation and growth of microscale cavities. FIGURE 14 (a) Predicted yield strength of EPON862/DETDA at 300K as a function of conversion degree (experimental data from Refs. 109 and 155); (b) Strain at yield peak as a function of conversion degree. (Reproduced from Ref. 77, with permission from Elsevier). Role of Cure Degree on Yield Stress Most recently, all-atom MD simulations were used to study the yield strength of a thermoset EPON82/DTDA. 77 The authors found that the yield strength increases linearly with conversion degree approximately but found the corresponding yield strain to be relatively insensitive to conversion degree, 77 as shown in Figure 14. The importance of this finding is that it lends support to strain-based yield criteria. 156 An energy-based criteria also demonstrated the importance of strain to the yield of thermoplastic polymers under a variety of loading paths ranging from pure volumetric to pure deviatoric (or shear) deformations. 89 Since a strong sensitivity of strength with strain rate is firmly established, 157 the available experimental values are significantly smaller than the MD predictions. Two main factors influence the increase in yield strength with conversion degree: (i) an intrinsic one associated with increased network connectivity and (ii) an extrinsic or indirect effect associated with the increase in T g. The network connectivity effect is relatively easy to understand: higher conversion results in higher density of covalent bonds and stiffer networks. The role of T g is indirect: T g increases with conversion degree and consequently the quench depth (difference between testing temperature and T g ). Thus, increasing conversion leads to lower effective temperature that may increase the yield stress. Li and Strachan 77 examined the room temperature yield strength as a function of T 2 T g for 116 JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

15 JOURNAL OF POLYMER SCIENCE REVIEW Size and Confinement Effects Advances in computer power and algorithms enable multimillion-atom MD simulations of polymers; trillion atom MD simulations have achieved, but for simple short-range potentials and short timescales.3 These heroic calculations that require parallel computing using multiple processors remain restricted to spatial scales below 100 nm. Thus, size effects on the predictions must be taken into account when comparing atomistic predictions and experiments. The periodic boundary conditions often imposed in such simulations restrict the wavelength of any perturbation in the system; thus, phenomena with characteristic size exceeding the simulation cell size will be incorrectly described. Thus, the relatively small system sizes in MDs have different effects on different properties; a simulation cell large enough to provide converged results for elastic constants would likely exhibit significant size effects on yield stress predictions. Size Effects on Bulk Thermomechanical Response The study of Shenogina et al. 106 indicated that T g shows a strong dependence on system size for highly crosslinked systems; they observed increasing T g with decreasing model size for very small systems, ranging from 35,000 to 2,000 atoms. Similarly, an increase in CTE was observed with increased simulation cell size. However, considering the 95% confidence interval, the MD results of Li et al. 77,99 for relatively larger systems (16,000 to 65,000 atoms) showed only weak size effects on yield strength and T g. FIGURE 15 Room temperature yield strength for EPON862/ DETDA with different conversion degrees (full circles) compared with the yield strength of the system with 86% conversion at various temperatures as a function of T 2 T g. (Reproduced from Ref. 77, with permission from Elsevier.). systems of varying conversion degrees together with the yield stress of a system with 86% conversion at various temperature (T 5 300, 350, 400 K). The simulations revealed that for low conversion degrees there is a rapid increase in yield strength with decreasing T 2 T g but for samples with over 60% conversion the increase in yield strength is reduced. This change in mechanical behavior coincides with the formation of a percolating polymer network (gel point) and indicates that for highly crosslinked thermosets increasing the degree of curing increases their thermal stability more than their ultimate mechanical properties as compared with conversions degrees below the gel point. A comparison of the room temperature yield strength for various conversions with the temperature dependent yield strength as functions of T 2 T g (Fig. 15) showed that as the conversion is increased over the gel point the strengthening mainly originates from an increase in T g while below the gel point the yield strength increasing mainly comes from network connectivity. 77 For the same reason as in the case of T g size effects on elastic response of polymers are weak as the system is deformed relatively uniformly. On the other hand, ultimate mechanical properties are governed by strain localization via the formation of shear bands or crazes. 158 Furthermore, such strain localization may be triggered by fluctuations in the polymer network or defects. Under such circumstances size effects would be significant. We believe that the consistent overestimation of yield stress in glassy polymers in MD simulations is partially due to the lack of strain localization caused by the small cell sizes as well as the large deformation rates. 159 Size Effects on the Response of Nanoscale or Confined Specimens Understanding how size effects on the response of nanoscale polymers (either confined between substrates or free-standing) is very important in many fields of nanotechnology including the development of nanofilms and nanocomposites. In such cases, molecular simulation is an ideal tool to uncover the physics behind size effects and quantify how they affect properties. Most of the work to date has focused on thermoplastic polymers 160 and work by Keddie et al. 161 and Forrest et al. 162 provided key insight into the confinement effect. Fewer efforts have been devoted to nanoscale thermosetting polymers. Recent work 163 modeled the curing and thermo-mechanical response of thin films of DGEBA/33DDS using MD. Interestingly, the authors found that the presence of the free surfaces significantly slow down the process of curing; curing a 5 nm thick film can take twice as a long as the corresponding bulk sample. The authors also found that T g gradually decreases as the film thickness decreases. The maximum T g depression for the thinnest film in their study is about 60K from the bulk value, as shown in Figure 16. The main reason for the T g depression for the overall film is believed from the lower T g of the polymer near the free surface, as evidenced by previous studies. 164 Another contributing factor for the T g depression is the lower conversion degree in ultrathin thermoset films. They also found that the empirical function T g ðhþ5t g ðbulkþ½12 ða=hþ d Š for T g as a function of film thickness (h) based on experimental works on PS films 161,162 is capable of capturing the T g change in thermoset films rather accurately. Both Young s modulus and yield strength were found to decrease with film thickness relative to the bulk values, as JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,

16 REVIEW JOURNAL OF POLYMER SCIENCE strength and toughness. These types of simulations can help the design and optimization of reversible or adaptable networked polymers. 169 Unlike traditional thermosets, these materials can be re-configured and are attractive from recycling and re-use points of view. In general, the correlations between chemistry, molecular architecture and property obtained from molecular simulations discussed throughout this review are playing an increasingly important role in design and engineering applications. 170,171 We foresee this trend to continue to increase as more accurate simulations become possible and more rigorous validation against experiments are attempted. CONCLUSIONS FIGURE 16 Glass transition temperature versus thermoset film thickness. (Reproduced from Ref. 163, with permission from American Chemical Society.) shown in Figure 17. But the yield strain was found to be very insensitive to film thickness. The reduction of film mechanical response with decreasing size was attributed to three factors: (1) Slightly lower conversion degree for thinner films: mechanical properties decreasing with decreasing conversion degree; (2) Lower T g for thinner films: thinner film samples with smaller quench depths would be expected to exhibit depressed moduli and strength. (3) Intrinsic size effects: low density surface layers having much lower stiffness and strength reduce overall film mechanical properties. MD simulations show that the later two effects contribute approximately equally to the reduction in stiffness and strength in thin film thermosets. COMPUTATIONAL MOLECULAR DESIGN The use of molecular simulations to design polymers optimized at the molecular level has been a longstanding goal of the research community and progress is being made. Such efforts involve either searching for new chemistries 165 or optimized molecular architectures Increasing compute power and efficient simulation software that can take advantage of such resources are making molecular simulations and indispensable tool in the fields of polymer science and engineering. At the same time, advances in in physics-based modeling are making quantitative predictions possible. While thermoset simulations lag behind those of thermoplastics, significant efforts are being devoted to the characterization of the molecular processes that govern their behavior and quantitative predictions on systems of interest. In this review we summarized the current state of the art in molecular-level simulations to predict the structure and properties of thermosetting polymers. Despite significant lingering challenges, molecular simulations are capable of providing not just insight but semi-quantitative predictions of several properties. Properties such as density, elastic properties and glass transition temperature fall within this category. However, phenomena that are affected more dramatically by the intrinsic length and time limitations, such as yield strength or fracture, remain more challenging for quantitative predictions; however, molecular simulations are providing important insight and can capture non-trivial effects. A significant remaining challenge in the field is bridging the timescales between MD simulations and experimental ones Christensen and D Oyen, of the Boeing, used a multiscale approach to relate intrinsic properties of thermoset polymers to the ultimate mechanical response of composites used for structural applications and used MD simulations to obtain optimal formulations. They explored a large number of epoxy and amine systems and found a good correlation between the theoretical predictions and experiments regarding the properties that determine the engineering performance of the systems. The authors estimate that time required to arrive at a decision regarding a new formulation for aerospace can be reduced by a factor of 20 given appropriate computational resources. Molecular simulations can also provide insight into optimal molecular architectures. For example, Salib et al. 168 explored optimal combination of strong and weak bonds to improve FIGURE 17 Mechanical properties versus thermoset film thickness. (Reproduced from Ref. 163, with permission from American Chemical Society.) 118 JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2015, 53,