Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading

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1 Journal of Algorithms & Computational Technology Vol. 2 No Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading Markus J. Buehler 1, *, Alan Cohen 2, and Dipanjan Sen 1,3 1 Laboratory for Atomistic and Molecular Mechanics, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Room 1 235A&B, Cambridge, MA 02139, USA 2 Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA 3 Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA Received: 01/10/2007; Accepted: 02/11/2007 ABSTRACT We report a novel multi-paradigm multi-scale approach based on a combination of the first principles ReaxFF force field with an empirical Tersoff potential. Our hybrid multi-scale simulation model is computationally efficient and capable of treating thousands of atoms with QM accuracy, extending our ability to simulate the dynamical behavior of a wider range of chemically complex materials such as silicon, silica and metal-organic compounds. It is implemented in the Python based Computational Materials Design Facility (CMDF). We exemplify our method in a study focused on a systematic comparison of the fracture dynamics in silicon under mode II shear versus mode I tensile loading. We find that the mode II crack tends to branch at an angle of approximately 45 degrees once the crack speed approaches 38% of the Rayleigh-wave speed. In contrast, the mode I crack continuously propagates in the direction of the initial crack, and only makes a slight change of direction towards 10 degrees once fracture instabilities occur. Our results reveal fundamental differences of fracture dynamics under mode I versus mode II loading. *Corresponding author, mbuehler@mit.edu, Phone: , Fax:

2 204 Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading Key words: Mechanics; chemistry; elasticity; fracture; silicon; mode I; mode II. 1. INTRODUCTION Brittle fracture occurs in materials such as glass, concrete or silicon is characterized by continuous breaking of atomic bonds leading to formation of two new materials surfaces. As macroscopic fractures develop, thousands of bonds are stretched under the applied load and eventually break, giving rise to spreading of cracks and eventually failure of entire structure [1]. Most existing atomistic models of fracture assume an empirical relationship between bond stretch and force. Whereas such empirical relations provide reasonable approximations for the behavior of materials under small deformation, these models fail when materials are deformed strongly and individual bonds are stretched significantly [2, 3]. This is because breaking of bonds in real materials can be an extremely complicated process whose description often requires a quantum mechanical (QM) treatment of bond behavior. Thus far, these processes can only be captured with sufficient accuracy by using QM methods such as Density Functional Theory (DFT), which are limited to approximately 100 atoms [4 6]. However, typically thousands and more atoms participate in the complex fracture patterns as cracks spread throughout structures by continuous rupture of atomic bonds. Due to these numerical limitations, treatment of fracture problems with QM level accuracy in many materials such as silicon, silica or other chemically complex materials remains elusive. As a consequence, few rigorous theories have been put forward that describe the fracture mechanics in these materials. Here we present a new theoretical concept based on building a multi-scale simulation model completely derived from QM principles, while being computationally efficient and capable of treating thousands of atoms with QM accuracy, implemented in a Python based simulation environment. Our method is based on the idea to combine the first principles based reactive force field ReaxFF [7] with an inexpensive potential, as shown schematically in Figure 1. The ReaxFF reactive force field has been developed to reproduce the barriers and structures for reactive processes from QM, but at a computational cost many orders of magnitude smaller. The ReaxFF reactive potential for silicon [7, 8] is used for a modest region of a few thousand atoms close to the

3 Journal of Algorithms & Computational Technology Vol. 2 No Figure 1. Geometry used for simulating fracture in a silicon single crystal. The whole systems contain 85,284 atoms. At the tip of the crack, several thousand atoms are modeled with ReaxFF (indicated in yellow); the rest of the system is treated using a nonreactive potential (indicated in blue). In the right part of the figure, we indicate the crystal orientation used for all studies reported in this paper. Determination of highenergy surface atoms for definition of the reactive domain is done inside the search region (indicated by the grey dashed line). crack tip while a computationally inexpensive but nonreactive Tersoff potential is used elsewhere. The Tersoff-type potential and ReaxFF lead to similar materials behavior for small strains, but deviate strongly at large strains [9]. Since both descriptions overlap for small strains, this enables a smooth handshake between the two methods [9]. We exemplify our new method in modeling fracture of silicon. The fracture mechanics of silicon has received tremendous attention in the past, in particular due to its complexity of bond breaking and due to interesting failure dynamics observed experimentally [10 14]. Atomistic modeling of the fracture of silicon

4 206 Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading Figure 2. The figure indicates the two loading conditions, mode II shear loading (subplot (a)) and mode I tensile loading (subplot (b)). has been the subject of several studies using empirical force fields [14 18]. In contrast to many metals, a proper description of fracture in silicon has proved to be far more difficult, requiring a more accurate treatment of the atomic interactions. There have been several previous attempts to describe fracture of silicon using atomistic methods. Early attempts to model fracture in silicon, using Tersoff s classical potential [19] and similar formulations such as the Stillinger-Weber [20], MEAM method [18] or EDIP potential [17] failed due to an incorrect description of the bond breaking process. It has become evident that in order to obtain an accurate description of the dynamics of fracture of materials, the accuracy of QM for atoms near the propagating crack tip is necessary.

5 Journal of Algorithms & Computational Technology Vol. 2 No Concurrent methods that enable the multiscale analysis of deformation and stress of brittle and ductile materials have been reported earlier. Typically such methods communicate through handshaking approaches often featuring regions where two distinct methods are used to implement the transition from one method to another. These methods include schemes to couple Tight- Binding MD to empirical potential and continuum domains [15, 21, 22], methods to bridge atomistic to continuum domains [23 25] as well as the quasicontinuum method [26 28]. Despite these major advances, the integration of chemistry and mechanical properties remains an issue that has not been satisfactorily addressed, in particular in systems at finite temperature and when thousands and more reactive atoms are present. Chemistry at the atomic scale is well handled by quantum mechanical methods such as QC and DFT methods. However, the number of reactive atoms in these approaches remains limited to a few hundred at most, which constitutes a severe limitations in modeling defect structures and deformation phenomena that emerge at larger scales. The first principles based reactive force field ReaxFF has been shown to provide the versatility required to predict catalytic processes in complex systems nearly as accurate as QM at computational costs closer to that of simple force fields, including the capability to describe charge transfer during chemical reactions at a computational cost that is a fraction of that of DFT or TB approaches. ReaxFF has a wider range of applicability to other types of atomic structures, such as hydrocarbons, proteins and semiconductors. The uniqueness of our hybrid ReaxFF-Tersoff approach is the ability to simulate large reactive regions at finite temperature, consisting of thousands of atoms. Earlier studies of modeling fracture using a ReaxFF based approach have focused on mode I tensile loading of silicon [9, 29]. Here we report studies of fracture of a single crystal of silicon under mode II shear loading. The mode II and mode I loading conditions are shown schematically in Figure 2. The plan of the paper is as follows. In section 2, we introduce the ReaxFF- Tersoff coupling scheme and CMDF, and provide theoretical details about the energy and force based coupling. Section 3 is dedicated to a detailed description of the atomistic model. In section 4, we report results of our method in studies of crack propagation in single crystal of silicon. We compare two cases, a cracked crystal under mode I tensile loading and a mode II shear loaded crystal. We conclude in section 5 with a systematic comparison of the two cases.

6 208 Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading Figure 3. Schematic representation of the CMDF computational framework. Various python modules are linked together via a central Extended OpenBabel (XOB) data structure, and made accessible via a Python scripting environment. For the results presented in this article, only the Tersoff and ReaxFF force field modules are used, together with the integrator module, temperature control functions and handshaking methods. In principle, CMDF can handle additional or alternative force fields that can be used to probe the effect of other force fields on the fracture mechanics. 2. HYBRID REAXFF-TERSOFF MODEL The hybrid ReaxFF-Tersoff model is implemented in the Computational Materials Design Framework (CMDF) [30], a set of computational tools that allows for straightforward integration of simulation methods across various length scales through a simple Python scripting environment. We provide a brief summary of the CMDF concept in the next section CMDF Framework and Implementation CMDF utilizes a Python scripting layer to integrate different computational tools to develop multi-scale simulation applications [30]. CMDF represents a set of software tools that combine DFT quantum mechanics methods, the first principles ReaxFF reactive force field, empirical all atom force fields such as DREIDING or UFF, mesoscale and continuum methods. The central data structure Extended OpenBabel (XOB) plays a critical role in serving as glue between applications. Figure 3 depicts a schematic overview over the different computational methods linked within the Python scripting environment.

7 Journal of Algorithms & Computational Technology Vol. 2 No Figure 4. Example CMDF Python script. In line 1, the first step of the Velocity Verlet integration is performed. In lines 3-5, a new system object is created that contains the collection of atoms OB1 and OB2 (XOB objects). This system object is used to update the new positions of atoms in the collection of atoms OB1 and OB2. In line 8-12, new forces are calculated with two distinct computational engines (ReaxFF and Tersoff). The forces calculated by these engines are then combined into the global collection of atoms OBtot in line 17 after creating a system object in lines Finally, in line 19 the second step of the Velocity Verlet integration is performed. Since all simulation tools and engines can be called from a Python scripting level, scale agnostic combinations of various modeling approaches can be realized straightforwardly. This strategy enables complex simulations to be simplified to a series of calls to various modules and packages, whereas communication between the packages is realized through the CMDF central data structures that are of no concern to the applications scientist. Figure 4 shows a simple example CMDF script that illustrates how complex simulation tasks can be developed Hybrid Formulation: Mixed Hamiltonian We couple different domains within CMDF using handshake regions. The role of the handshake regions is to ensure that the correct boundary conditions are applied to each side, and that each side senses proper continuation in the distribution of density and forces. Different regimes are linked by an overlap region in which the Hamiltonians of the two methods are mixed. Figure 5 illustrates the mixed Hamiltonian approach implemented in a smooth transition region.

8 210 Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading Figure 5. Schematic showing coupling of different interatomic potentials using the concept of mixed Hamiltonians by using a smooth, sinusoidal interpolation scheme. The transition layer serves as a handshake region between the two distinct methods (this can be extended to describe multiple regions treated with different numerical accuracy using a similar concept). In the example discussed in this paper we couple the Tersoff and ReaxFF force fields. The transition region is described by two parameters, R trans for the width of the transition region, and R buf for the width of the ghost atom region. In order to bridge distinct computational engines we utilize spatially varying weights w i to determine the weighting of the energy contribution from different simulation engines. Each computational engine i has an associated specific weight w i (x) (the variable x is a spatial coordinate). Therefore, for two computational engines, the total Hamiltonian of the system is written as: where Htot = H + H + H, ReaxFF Tersoff ReaxFF Tersoff = ( ) + ( ( )) Tersoff H w x H 1 w x H ReaxFF Tersoff ReaxFF ReaxFF ReaxFF (1) (2)

9 Journal of Algorithms & Computational Technology Vol. 2 No In eq. (2), the w ReaxFF (x) is the weight of the reactive force field in the handshaking region. Forces on individual atoms are negative of the partial derivatives of H tot with respect to atom coordinates. This is straightforward in the single potential regions, where the derivatives give the same force as pure potentials, either ReaxFF or Tersoff. In the handshaking region, however, due to a gradual change of weights with position, we obtain ( ( ) Tersoff ) = ( ) + ( ) w ReaxFF ( x) ( H H ) F w x F 1 w x F ReaxFF Tersoff ReaxFF ReaxFF ReaxFF x ReaxFF Tersoff If w ReaxFF Tersoff varies slowly in the spatial domain from zero to one at the edges of the handshaking regions (that is, if the gradient w ReaxFF (x)/ x is small), the last term in eq. (3) can be neglected. Alternatively, if the Hamiltonian energies for the reactive force field and the Tersoff method are almost the same (that is, if the difference H ReaxFF H Tersoff is negligible in the transition region), the last term in eq. (3) can be neglected. This leads to a simpler expression for interatomic forces inside the transition region: ( ) = ( ) + ( ( )) F w x F 1 w x F ReaxFF Tersoff ReaxFF ReaxFF ReaxFF Tersoff. Thus, this force handshaking algorithm must satisfy either one of two conditions: (i) a large transition region leading to a slowly varying interpolation function, or (ii) that both force fields give a similar approximation of bond energies in the transition region. In our numerical scheme we satisfy both conditions to ensure that our method is robust and still works even if one condition fails: Condition (i) is satisfied if the transition parameter is large, that is, R trans is several times larger than the bond distance R bond (R bond is typically on the order of 1 2 Å). Condition (ii) is satisfied if both force field expressions are valid for the material state inside the transition region. Typically this implies that the local strains have to remain small. A simple strategy to ensure this is to choose the transition region sufficiently far away from any defects. We note that by explicitly considering these terms, one could obtain a computationally more efficient model since the transition region could be chosen much smaller. We leave the implementation of this approach to future work. The hybrid model is in principle not limited to two methods, but can be generalized to N different computational methods: (3) (4)

10 212 Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading F w x H hybrid, = ( ) N i i i= 1 N (5) where the N weights always add up to one: wi ( x) = 1. Similarly as shown in eq. (4), for the forces we obtain i F w x F = ( ) hybrid, N i i. i= 1K N (6) (7) The different simulation regimes are coupled to one another by smoothly interpolating between different engines by using smoothly varying weighting functions. The width of the transition region R trans depends on the nature of the system, but it is generally a few times the typical atomic distance in a lattice or in an organic molecule. The width of the buffer layer R buf describing the ghost atoms is approximately 10% larger than any long range cutoffs to rule out possible boundary effects. The important point here is that the atoms at the interface to the ghost atoms (which still contribute a small amount to the total force) should not sense the existence of the boundary of the buffer layer at any point, thus R buf > R cut, (8) where R cut is the cutoff distance for the force interactions for the given force field. The frequency at which the calculation regimes or force field weights w i are updated depends on the nature of the problem Algorithm for Determination of Reactive Regions The reactive region is determined based on a simple algorithm that filters all high-energy atoms whose energy corresponds to that of surface atoms (critical atomic energy is larger than 4.5 ev). Subsequently, each surface atom is embedded into a cylindrical reactive region with radius 10 Å with transition and ghost radii of 5 Å. The union of all reactive atoms represents the entire reactive domain. All other atoms in the system are treated using the nonreactive force field. We update the topology of the calculation domains every 20 integration steps. We choose a sinusoidal handshaking weight function of the form w i (x) = {cos ((x R)/R trans π) + 1}/2 (see Figure 5). As discussed in the previous section, this force handshaking algorithm requires a large transition region and a slowly varying interpolation function.

11 Journal of Algorithms & Computational Technology Vol. 2 No We found that a size of the reactive region of larger than 10 Å is sufficient to avoid any large strains within the transition region. For sizes of the reactive region larger than 10 Å, the simulation results are not influenced by this parameter. This indicates that the transition region is far enough away from the large stresses and strains at the tip of the crack. Since only reactive atoms ahead of the crack tip and in a relatively small region behind the crack tip are important for the fracture process, we have further implemented a constraint that limits the reactive atoms to a region of approximately 30 Å behind the crack tip (determination of position of crack tip see Section 3.2). This constraint ensures that atoms that would be tagged as reactive at the surface behind the crack tip are not included in the reactive domain. 3. MODEL GEOM7ETRY, LOADING CONDITIONS AND CRYSTAL PROPERTIES We consider a perfect crystal with an initial crack of length a serving as the failure initiation point (Figure 1). The crystal that is referred to as (111) 1 1 system is oriented so that the x-y-z directions are (,, ),, 2, 2, 0, creating a (111) fracture plane with initial [112] fracture direction. The crack is inserted by removing atoms in a wedge-like shape, with an initial crack length a =180 Å. The slab dimensions L x L y L z are approximately 570 Å 800 Å Å. The system contains 85,284 atoms. The reactive region size varies from approximately 600 (early stages) to 3,000 atoms (later stages). In the simulations we have carried out we have not observed a dependence of the fracture phenomena on the initial crack length. We use periodic boundary conditions in the out-of-plane z-direction, rendering the systems to be plane strain. The thickness of the systems is one unit cell. We note that this constraint may influence the possibility of nucleating dislocations from the tip of the crack or other defects; however, this geometry enables us to focus on the purely brittle crack dynamics in silicon. To apply load, we continuously strain the system according to mode I or mode II by displacing the boundaries [2, 3] according to a prescribed homogeneous strain field (either according to pure shear as in mode II, or pure tensile strain as in mode I). The boundaries represent a domain of 7 Å at the outermost left and right layers of the system, which are held fixed during the simulation. They only move when external strain is applied. The temperature is controlled at 300 K using a Berendsen thermostat [31]. The Velocity Verlet integration time step is t = 0.5 fs. We have also carried out simulations with an NVE ensemble; in this case, the system tends to heat up more rapidly. It is noted that in general the use of a thermostat for such

12 214 Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading highly nonequilibrium system should be checked carefully and may influence some of the simulation results. The strain rates used are ε ij = 1E-5 per integration step, for both shear strain (mode II) and tensile strain (mode I). We apply a pre-tensile strain of 0.5% in mode I in both cases prior to beginning of the simulation to account for thermal expansion of the lattice Reactive Force Field ReaxFF To simulate the atoms around the crack tip we employ the first principles ReaxFF force field [7]. As described in [8], this potential has been tested against QM for a wide range of processes [6], including Si-Si bond breaking in H 3 Si SiH 3 and Si=Si bond breaking in H 2 Si = SiH 2, equations of state for 4-coordinate silicon (diamond-configurations) and 6-coordinate silicon phases ß-tin), and for simple cubic crystals. This force field is also capable of treating interactions of Si with O and H, including dissociation of H 2 O molecules at Si surfaces (Fig. 12 in [8]). Prior to the fracture studies, this earlier ReaxFF was supplemented by including to the training set the stability and equation of state of a 5-coordinate Si-condensed phase and re-evaluated the Si-Si bond and Si- Si-Si angle parameters to improve the fit to the energy and equation of state of this phase. All parameters are determined based on a training set of QM calculation results. Because of the complexity of the mathematical expressions describing the partial bond orders, energies, and charges, ReaxFF is 1 2 orders of magnitudes more expensive than the Tersoff potential but several orders of magnitude faster than QM. The Tersoff-type potential [19] and ReaxFF [8] lead to similar materials behavior for small strains (see Figure 3(c) in reference [9]), but deviate more strongly at large strains. This property ensures that the potential coupling as described in Section 2 can be implemented, since H ReaxFF H Tersoff 0 inside the transition region Numerical Determination of Crack Speed The algorithm used to determine the reactive region is used to extract information about the current position of the crack tip by calculating the surface atom with largest y-value inside a search region (the search region encompasses only the interior region of the slab that excludes the boundary region; see Figure 1 dashed line). We take averaged time derivatives of the position vector; the magnitude of this quantity is the crack speed.

13 Journal of Algorithms & Computational Technology Vol. 2 No Figure 6. Snapshots of dynamical crack propagation in a perfect, single crystal of silicon, for the (111) crack orientation, slowly increasing mode II loading (blue=tersoff region, yellow=reaxff region). The plot shows crack initiation followed by an initial regime during which the crack generates an atomically flat, clean fracture surface in the direction of the initial crack orientation. This initial regime is followed by a sudden change in crack direction at an angle of approximately 45 degrees to the left. The snapshots correspond to times of 0 ps, 2 ps, 3.5 ps and 3.5 ps Theoretical Limiting Speed of Cracks The Rayleigh-wave speed is 4.68 km/sec [21] for the (111) crystal orientation considered here. This wave speed is important since the maximum speed of cracks is controlled by these material properties. Both mode II and mode I cracks can propagate at speeds between zero and the Rayleigh-wave speeds [1]. Mode II cracks can also propagate at speeds between the shear wave speed and the longitudinal wave speed. However, there exists a forbidden velocity gap between the Rayleigh wave speed and the shear wave speed that can only be overcome by mother-daughter crack mechanisms [32].

14 216 Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading Figure 7. Snapshots of dynamical crack propagation in a perfect, single crystal of silicon, for the (111) crack orientation, slowly increasing mode I loading (blue=tersoff region, yellow = ReaxFF region). The plot shows crack initiation followed by an initial regime during which the crack generates an atomically flat, clean fracture surface in the direction of the initial crack orientation. This initial regime is followed by a regime during which crack motion is increasingly erratic and several microcracks are formed. The snapshots correspond to times of 0 ps, 2 ps, 2.75 ps and 3.5 ps. 4. SIMULATION RESULTS Figure 6 depicts snapshots of dynamical crack propagation in a perfect, single crystal of silicon, for the (111) crack orientation, for slowly increasing mode II loading. The plot shows crack initiation followed by an initial regime during which the crack generates an atomically flat, clean fracture surface in the direction of the initial crack orientation. This initial regime is followed by a sudden change in crack direction at an angle of approximately 45 degrees to the left. Under mode I loading, crack dynamics is rather different, as shown in Figure 7. The plot shows crack initiation followed by an initial regime during which the

15 Journal of Algorithms & Computational Technology Vol. 2 No Figure 8. Crack speed as a function of the y-position of the crack, for both mode II and mode I loading. The plot illustrates that soon after nucleation of the crack, the speed quickly approaches 2 km/sec indicating that intermediate velocities are not observed. The mode II case indicates a slightly more smooth slower acceleration in the early fracture stages than the mode I case. The speed of the mode II crack approaches approximately 3.2 km/sec, whereas the mode I crack eventually reaches 3.5 to 4 km/sec. crack generates an atomically flat, clean fracture surface in the direction of the initial crack orientation. This initial regime is followed by a regime during which crack motion is increasingly erratic. Figure 8 depicts the crack speed as a function of the y-position of the crack, for both mode II and mode I loading. The plot illustrates that soon after nucleation of the crack the speed quickly approaches 2 km/sec. The mode II case indicates a slightly smoother and reveals a slower acceleration than the mode I case. After the initial jump in crack speed, the speed continuous to increase; however, it increases more slowly. The final speed of the mode II crack eventually approaches approximately 3.2 km/sec, whereas the mode I crack reaches 3.5 to 4 km/sec. These results show that the mode II crack approaches approximately 68% of the Rayleigh wave speed, whereas the mode I crack reaches 74% to 86% of its theoretical limiting speed. Since the mode II crack is much below the limiting speed, development of a secondary intersonic crack is unlikely. This is because for the nucleation of such a secondary crack, the primary crack must move at a

16 218 Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading Figure 9. Crack tip position trajectory over the simulation time span, for both mode II and mode I case. The mode II crack propagates straight for approximately 15 Å, and then suddenly changes direction towards the left, forming a sharp 45 degree angle relative to the initial direction. The mode I crack propagates in a direction at an angle of approximately 10 degrees to the right. However, unlike in mode II, no sudden change in direction is observed. speed close to the limiting speed in order to develop a large shear stress concentration that can nucleate the secondary shear crack [32]. A significant observation is that the trajectories of the cracks are quite different under mode II versus mode I loading. Figure 9 depicts the crack tip position trajectory over the simulation time span, for both mode II and mode I case. The mode II crack propagates straight for approximately 15 Å, and then suddenly changes direction towards the left, forming a 45 degree angle relative to the initial direction. The mode I crack propagates in a direction at an angle of approximately 10 degrees to the right. However, unlike in mode II, no sudden change in direction is observed. The sudden change in direction towards the 45 degree angle in mode II occurs at a crack speed of approximately 1.8 km/sec, corresponding to 38% of the Rayleigh wave speed.

17 Journal of Algorithms & Computational Technology Vol. 2 No DISCUSSION AND CONCLUSION We have developed and applied a novel hybrid modeling scheme integrating the ReaxFF force field with Tersoff potentials that enables us to model elastic, plastic and fracture mechanics of chemically complex materials, including silicon, silica and other materials. This method represents an advance in simulation technology that allows one to handle simulation of chemically complex materials during mechanical deformation of materials, considering charge transfer and continuous changes of the energy landscape under chemical reactions between organic and inorganic atoms. Such effects can not easily be captured by methods such as Tersoff, EAM or MEAM, in particular due to the limitations of these methods to describe organic phases and adhesion of such molecules on various surfaces. Our results indicate that the Python-based CMDF framework provides a suitable platform for development of new multi-scale modeling schemes in general, as it constitutes a test bed for probing new simulation methods. We have carried out a set of simulations to probe differences in crack dynamics of fractures under mode II versus mode I loading. We find that the loading condition significantly influences the crack dynamics. Whereas the mode II crack changes direction towards a 45 degree angle, the mode I crack remains in almost perfect straight forward direction. This behavior was illustrated in Figure 9 that shows an analysis of the crack tip trajectory. The reasons for this behavior may be a change in local asymptotic stress field, as suggested in earlier work [2]. We leave further analysis of this to future work. Our findings are important to understand machining processes of single crystals of silicon. In such technologies, a good understanding of the directional preference of fractures is vital to engineer machining tools in order to create smooth surfaces in a controlled fashion via methods such as drilling, milling or grinding. ACKNOWLEDGEMENTS This research was supported by the Army Research Office (ARO), grant number W911NF , program officer Dr. Bruce LaMattina. All computations were carried out at MIT s Laboratory for Atomistic and Molecular Mechanics (LAMM). AC acknowledges support from MIT s UROP office. REFERENCES [1] Freund, L.B., Dynamic Fracture Mechanics. 1990: Cambridge University Press, ISBN [2] Buehler, M.J. and H. Gao, Dynamical fracture instabilities due to local hyperelasticity at crack tips. Nature, : p

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19 Journal of Algorithms & Computational Technology Vol. 2 No [21] Bernstein, N. and D.W. Hess, Lattice trapping barriers to brittle fracture. Physical Review Letters, (2): p [22] Broughton, J.Q., et al., Concurrent coupling of length scales: Methodology and application. Physical Review B, (4): p [23] Xiao, S.P. and T. Belytschko, A bridging domain method for coupling continua with molecular dynamics. Computer Methods in Applied Mechanics and Engineering, (17 20): p [24] Wagner, G.J. and W.K. Liu, Coupling of atomistic and continuum simulations using a bridging scale decomposition. Journal Of Computational Physics, (1): p [25] Park, H.S., et al., The bridging scale for two-dimensional atomistic/continumn coupling. Philosophical Magazine, (1): p [26] Tadmor, E.B., M. Ortiz, and R. Phillips, Quasicontinuum analysis of defects in solids. Phil. Mag. A, : p [27] Knap, J. and M. Ortiz, An analysis of the quasicontinuum method. Journal Of The Mechanics And Physics Of Solids, (9): p [28] Dupuy, L.M., et al., Finite-temperature quasicontinuum: Molecular dynamics without all the atoms. Physical Review Letters, (6). [29] Buehler, M.J., et al., Threshold Crack Speed Controls Dynamical Fracture of Silicon Single Crystals. Phys. Rev. Lett., 99: p [30] Buehler, M.J., et al., The Computational Materials Design Facility (CMDF): A powerful framework for multiparadigm multi-scale simulations. Mat. Res. Soc. Proceedings, : p. LL3.8. [31] Allen, M.P. and D.J. Tildesley, Computer Simulation of Liquids. 1989: Oxford University Press. [32] Gao, H., Y. Huang, and F.F. Abraham, Continuum and atomistic studies of intersonic crack propagation.j.mech. Phys. Solids, : p

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