5 Multiscale Modeling and Simulation Methods

Transcription

1 5 Multiscale Modeling and Simulation Methods This chapter is dedicated to a discussion of multiscale modeling and simulation methods. These approaches are aimed to provide a seamless bridge between atomistic and continuum approaches, sometimes by introducing intermediate mesoscopic methods of simulation. This chapter gives a brief overview over available models and approaches, along with a discussion of the historical development. The quasicontinuum method and a hybrid ReaxFF method are discussed in more detail. Several case studies are reviewed. 5.1 Introduction The motivation for multiscale simulation methods is that it is not always necessary to calculate the full atomistic information in the whole simulation domain. Based on this insight, several researchers have articulated the need for multiscale methods [20, ] by combining atomistic simulations with continuum mechanics methods (for instance, finite element methods). A variety of different methods have to be developed to achieve this. An important motivation for this is to save computational time and by doing that, to extend the lengthscale or timescale accessible to the simulations. It is common to distinguish between hierarchical multiscale methods and on-the-fly concurrent multiscale methods. In hierarchical multiscale methods, a set of different computational tools are used sequentially. First, the most accurate method (e.g., quantum mechanics) is used to determine parameters for the next computational approach (e.g., via force field fitting to generate interatomic potentials). Molecular dynamics simulations with interatomic potentials are then used to determine constitutive equations, or criteria for plasticity, which are utilized as parameters in finite element approaches. These approaches can be carried out for a variety of computational methods. In onthe-fly concurrent multiscale methods, the computational domain is divided into different regions where different simulation methods are applied. A critical issue in both methods is the correct mechanical and thermodynamical

2 158 Atomistic Modeling of Materials Failure coupling among different models. This applies to different geometric regions in a concurrent approach and to different simulation methods in a hierarchical approach. There are many computational challenges associated with these schemes, as some of the computational engines may require more computational effort than others, so that load balancing becomes an important issue. There exist several review articles on multiscale modeling of materials in the literature [18, 176, 177], including a mathematical perspective. In addition to methods aimed to extend accessible lengthscales, several methods have been developed to cover larger timescales. In the following sections, we provide a discussion of selected multiscale approaches. First, we focus on methods to span vast lengthscales, and second, we focus on methods to span vast timescales. 5.2 Direct Numerical Simulation vs. Multiscale and Multiparadigm Modeling Fig. 5.1 A summary of a hierarchical multiscale scheme that can be used to develop an understanding of the behavior of materials across scales in length and time Material deformation is a phenomenon that cannot be understood at a single scale alone. It requires the consideration of multiple scales to capture the progression of the elementary physical mechanisms. How can one capture multiple scales? One possibility is to simply simulate all particles in a system. Another possibility is to use a combination of methods with different accuracy or resolution. This idea is motivated by the fact that

3 5 Multiscale Modeling and Simulation Methods 159 in many problems, a high resolution and a high accuracy is only required in regions that are small compared with the overall specimen size. This is for instance the case in the example of a crack-like defect, where only atoms close to the crack tip experience large stresses, whereas atoms further away undergo only small deformation. The connection of multiple simulation techniques, coupled by handshaking or parameter passing, is illustrated in Fig We will discuss this figure later in much more detail and also illustrate in selected case studies how the handshaking between various computational techniques can be achieved. The systematic integration of models that range from the quantum mechanical to macroscopic scales can enable one to make quantitative predictions of complex phenomena with few (or without, in some cases) empirical parameters. Figure 5.2 depicts an overview over the process of predictive multiscale modeling. Quantitative predictions are enabled via the validation of key properties, which then enables to extrapolate and predict the behavior of systems not included in the initial set of parameters used to develop the model. Fig. 5.2 Overview over the process of predictive multiscale modeling. Quantitative predictions are enabled via the validation of key properties, which then enables to extrapolate and predict the behavior of systems not included in the initial training set 5.3 Differential Multiscale Modeling Computational modeling can be used to predict quantitative numbers of material properties. Since parameter or geometry changes can typically be implemented quite easily, it can also be used to perform a procedure referred to as differential multiscale modeling. Differential multiscale modeling, in contrast to predictive multiscale modeling focuses on the differential aspects of how macroscopic properties change due to variations of microscopic properties and microscopic structures. In this approach, it is not required that each individual simulation provides a quantitative predictive capability of the phenomena. Rather, it is the change of properties that is predicted. It has been argued that this approach, in some ways a weak form of the predictive method, may provide a robust set of results and can thereby provide

4 160 Atomistic Modeling of Materials Failure important insight into the physical basis of the simulated phenomena. The use of differential multiscale methods is particularly fruitful in the analysis of very complex processes or phenomena, in which there is no a priori known theoretical guidance about the behavior. It can also be very helpful in developing physics-based theories to describe phenomena, as it is possible to center the attention on the core properties at distinct lengthscales. A combination of differential and predictive tools can be used for the study of problems, providing an advantageous method in the analysis in particular of very complex mechanisms. For example, quantitative methods can be used to help establish an atomistic model in the regime of interest, thereby forming a reference system with proper variables and parameters. Then, differential methods can be used to explore the behavior in the vicinity of this reference system. It is noted that the reference system is sometimes also referred to as the control system in the spirit of the analysis of biological laboratory experiments. Notably, these laboratory methods were developed based on the difficulty of obtaining quantitative results. However, one is able to effectively control the boundary conditions and thereby being able to study the correlations between microscopic parameters and the system behavior. 5.4 Detailed Description of Selected Multiscale Methods to Span Vast Lengthscales In this section, the aim is to provide a brief review of selected multiscale methods used for modeling mechanical deformation in crystalline materials in particular metals along with their advantages, potentials, and drawbacks Examples of Hierarchical Multiscale Coupling An example for hierarchical multiscale modeling is a study of the shear strength of crystals [178]. The authors investigate the shear strength of crystals based on a multiscale analysis, incorporating molecular dynamics, crystal plasticity, and macroscopic internal state theory applied to the same system. The objective of the studies was to compare different levels of description and to determine coupling parameters. Further studies of climb dislocations in diffusional creep in thin films [50] and mesoscopic treatment of grain boundaries during grain growth processes [179,180] have also employed hierarchical simulation approaches. The development of hierarchical multiscale approaches has also contributed to new methods and simulation strategies in a variety of fields. For example, mesoscopic dislocations dynamics simulations treat dislocations as particles embedded in a linear elastic continuum (see, for instance [95, 96, 181, 182]). Dislocation particles interact according to their linearelastic fields and move according to empirical laws for dislocation mobility. All

5 5 Multiscale Modeling and Simulation Methods 161 Fig. 5.3 Example for implementation of a hierarchical multiscale method, where parameters are passed through various lengthscales nonelastic reactions between dislocations that may occur have to be included in the simulation setup as interaction rules. Thus, an important issue in these approaches is to identify proper coupling variables to transition between the different scales. This is typically achieved by hierarchically coupling to full atomistic methods. Multiscale methods have also been particularly useful for polymers and biological materials. Figure 5.3 depicts a schematic that shows a systematic coarse graining of the molecular structure of a tropocollagen molecule. The concept here is to divide a larger molecular structure into a representation of super-atoms, or beads, which as a whole represent the entire tropocollagen molecule. A similar approach will be discussed in Sect for modeling of large carbon nanotube systems. The concept of hierarchical integration of computational approaches has also been used in the design of new materials, from bottom up, as illustrated in the design of Cybersteel [18, 19]. Figure 5.4 illustrates the hierarchical integration of computational tools in the design process. Such new material design methods are based on the bottom-up multiscale design, using multiscale modeling as the basic engineering tool. In the study reviewed in Fig. 5.4, the design process begins at the quantum scale, where the particle matrix interface decohesion is analyzed by using first principles methods, in order to determine the appropriate traction-separation law. This traction-separation law is then embedded into the simulation of the submicron cell that contains secondary particles. The sequential information flow across multiple hierarchies is repeated several times until the required macroscopic behavior is achieved.

6 162 Atomistic Modeling of Materials Failure Fig. 5.4 Hierarchical modeling of Cybersteel [18]. Subplot (a) shows quantum mechanical calculations that provide the traction-separation law. Subplot (b)depicts concurrent modeling of the submicron cell based on the traction-separation law. Subplot (c) illustrates concurrent modeling of the microcell with the embedded constitutive law of the submicron cell. Subplot (d) shows results of modeling the fracture of the Cybersteel with embedded constitutive law of the microcell. Subplot (e) depicts the fracture toughness and the yield strength of the Cybersteel as a function of decohesion energy, determined by geometry of the nanostructures. Subplot (f) shows snap-shots of the localization induced debonding process. Subplot (g) summarizes experimental observations. Reprinted from [18], Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp , W.K. Liu, E.G. Karpov, S. Zhang, and H.S. Park, An introduction to computational nanomechanics and materials, copyright c 2004, with permission from Elsevier Concurrent Integration of Tight-Binding, Empirical Force Fields and Continuum Theory Rigorous multiscale approaches for mechanical properties of solids were first reported in the 1990s. One of the first approaches in this field included an integration of atomistic simulation with finite element models [183]. The authors discuss simulations performed with a hybrid atomistic-finite element (FEAt) model, and compared the results with the continuum-based Peirls Nabarro model for different crack orientations in a nickel crystal. The researchers demonstrated the basic assumptions of the continuum model for dislocation nucleation, that is, stable incipient slip configurations are formed prior

7 5 Multiscale Modeling and Simulation Methods 163 to dislocation nucleation, and found relatively good agreement of the FEAt model with the Peierls model for critical loading associated with dislocation nucleation. In the FEAt model, the region with atomistic detail is determined prior to computation and cannot be updated during the simulation. Another concurrent modeling approaches spanning scales from quantum methods to continuum was developed as a method to model fracture dynamics of brittle materials [144, 174, 184]. Fracture dynamics is governed by processes over a range of interconnected lengthscales, all of which are vital in deciding how a crack propagates in crystals. This multiscale methodology links lengthscales ranging from the atomistic scale, treated with a tight binding (TB) approximation, through microscale, treated by classical molecular dynamics, to the mesoscale, handled by finite element methods in the context of continuum elasticity. The method was applied to study dynamical fracture in silicon, a covalently bonded brittle material whose fracture mechanics is intrinsically complicated and cannot easily be captured by empirical approaches. The formation and breaking of covalent bonds at and in the vicinity of the crack tip is treated by a near quantum mechanical approach, a semiempirical nonorthogonal tight-binding formulation, which describes the bulk, amorphous, and surface properties of Si well. Somewhat further away from the crack tip region, where strains are large, the chemical bonds are not broken but deviate from their ideal bulk bonding arrangements. In this region, the Stillinger Weber [124] empirical interatomic potential is used to describe the material with classical molecular dynamics. A continuum finite element region is used in the far-field, where the atomic displacements from ideal positions and strain gradients are small. The use of three different regions requires two different hand-shaking zones: FE/MD and MD/TB. The total Hamiltonian for the system is written as H tot = H FE + H TB + H FE/MD + H MD/TB. (5.1) The degrees of freedom of the Hamiltonian are the atomic positions r and atomic velocities v for the TB and MD regions, and nodal displacements and their velocities for the FE region. To achieve FE/MD coupling, the FE mesh dimensions are brought down to atomic dimensions at the interface between the two regions. Moving away from the interface and into the continuum the mesh size is expanded. The FE and MD regions share atoms and nodes on either side of the interface where they are given half weights each to the Hamiltonian. Since the FE/MD interface is far from the crack and any plastic deformation zone, the mapping is unambiguous and gives correct forces near the interface. For the MD/TB coupling, the dangling bonds on the TB side are saturated by pseudohydrogen atoms. The Hamiltonian matrix of these pseudohydrogen atoms are carefully constructed to tie off a single Si bond and ensure absence of any charge transfer when the atoms are in a perfect Si lattice position. The TB terminating atoms called silogens are fictitious monovalent atoms

8 164 Atomistic Modeling of Materials Failure forming covalent bonds with the strength and length of bulk Si bonds. Thus, at the perimeter of the MD/TB region, there are silogens sitting directly on top of the atoms of the MD region. The MD atoms of the interface, on the other hand, have a full complement of neighbors, including neighbors whose positions are determined by the dynamics of atoms in the TB region. Major successes in the description of silicon fracture through this method have been the correct depiction of brittle fracture at low temperatures with experimental agreement with crack propagation speeds, and elucidation of possible mechanisms of brittle-to-ductile transition in fracture at higher temperatures. A general problem with such interface coupling methods is the spurious reflection of elastic waves (phonons) as the boundaries due to changes in system description. In a subsequent paper, the authors reported that there was no visible reflection of phonons at the FE/MD interface and no obvious discontinuities at the MD/TB interface. However, this particular scheme is very confined to covalently bonded crystalline materials. One of the reasons for this limitation is that transition regions for other systems (e.g., metals) are more difficult to implement. More recent ongoing efforts are exploring the possibility of applying it to metals and metallic alloys, where the MD/TB region has to be coupled very differently owing to nondirectional bonding in metals. Also, the use of the TB method at the crack tip to describe bond breaking and making, limits the size of the crack affected zone computationally, since TB is a much costlier method than empirical potentials and FE methods. Thus, in materials where cracks branch off and/or have large plastic zones or voids around them, the computational requirements for the problem can escalate drastically. Other methods in this area coupled DFT level methods with empirical potentials, as for instance done for the case of metals [185,186]. These methods also included an incorporation of the quasicontinuum method (see Sect for details). A new development in this field is the bridging scale technique, which enables a seamless integration of atomistic and continuum formulations throughout the entire computational domain [18, 19, 187]. A core feature of this method is that it is assumed that the continuum and atomistic-scale solutions exist simultaneously in the entire computational domain. Molecular dynamics calculations are only performed in the parts of the domain where this level of accuracy is required. This is possible by decomposing the displacement functions into a slowly varying and rapidly varying part (see also the schematic of the displacement time history as shown in Fig. 2.4). By subtracting the bridging scale from the total solution, the authors arrive at a coarse-fine decomposition that decouples the kinetic energy of the two simulations. A major advantage is two different time-step sizes can be used for the two different scales. This method is particularly suitable for finite temperature applications, enabling to connect atomistic and continuum domains seamlessly, even at finite temperature. This is achieved by the definition of

9 5 Multiscale Modeling and Simulation Methods 165 Fig. 5.5 This plot shows a multiscale analysis of a 15-walled CNT by a bridging scale method. Subplot (a) illustrates the multiscale simulation model. It consists of ten rings of carbon atoms (with 49,400 atoms each) and a meshfree continuum approximation of the 15-walled CNT by 27,450 nodes. Subplot (b) shows the global buckling pattern captured by meshfree method, whereas the detailed local buckling of the ten rings of atoms are captured by a concurrent bridging scale molecular dynamic simulation. Reprinted from [18], Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp , W.K. Liu, E.G. Karpov, S. Zhang, and H.S. Park, An introduction to computational nanomechanics and materials, copyright c 2004, with permission from Elsevier appropriate boundary conditions for the atomistic simulation domain that mimics the fine atomistic-scale vibrations of atoms. Figure 5.5 depicts a multiscale analysis of a 15-walled CNT by a bridging scale method. The method has found many other applications, for instance to model intersonic crack propagation as reported in [188] The Quasicontinuum Method and Related Approaches A quasicontinuum (often also referred to as QC) model for quasistatic atomistic-continuum simulations was developed by Tadmor, Miller, Ortiz, Phillips, Shenoy, and coworkers [20, 175, 189] beginning in the mid 1990s. The quasicontinuum method is based on the observation that in many large-scale atomic simulations a large section of atomic degrees of freedom can be described by effective continuum models and only a small subset of atomic degrees do something different. The chief objective is to systematically coarsen the atomistic description by judicious introduction of kinematic constraints on the full atomistic representation. These kinematic constraints allow full atomic resolution to be preserved where required, for example in the vicinity of defects and interfaces, and to treat large number of atoms collectively in regions where the deformation fields vary slowly on the scale of the lattice. For example, in the simulation of dislocation mechanisms, the method enables to describe the dislocation core region fully atomistically, while most of the bulk region surrounding the dislocation is treated as a continuum.

10 166 Atomistic Modeling of Materials Failure The method begins from a conventional atomistic approach that computes the energy as a function of atomic positions. The configuration space of the solid is then reduced to a subset of representative atoms. The position of the remaining atoms are obtained by piecewise linear interpolations of the representative atom positions. Effective equilibrium equations are then obtained by minimizing potential energy of the solid over the reduced configuration space. The energy of the system is defined as the weighted sum of the representative atom energies. N U tot n i U i, (5.2) i=1 where n i is the quadrature weight signifying how many atoms a given representative atom stands for and U i is the energy of the ith representative atom. The selection of representative atoms is based on the local variations of the deformation field. For example, near dislocation cores and on planes undergoing slip, full atomistic resolution is attained with adaptive meshing. Far away from defects, the density of representative atoms sharply decreases, and the collective motion of very large number of atoms is described by a small number of representative atoms. The quasicontinuum method has been extended to complex Bravais crystals and polycrystalline materials. It has been applied to many problems, including dislocation structures, interaction of cracks with grain boundaries, dislocation junctions, and other crystal defects. One drawback is that because of the particular expressions for energy in the quasicontinuum method, the actual atomistic methods that can be implemented are limited to ones that can easily be expressed in a suitable form. Also, finite temperature applications remain challenging, although some new developments have been proposed that combine coarse-grained dynamical simulations with the quasicontinuum method (see, for instance [190]). Further, a fully nonlocal three-dimensional version of the method has been introduced and applied to the study of nanoindentation. A recent thrust area in the quasicontinuum field has been on incorporating ab initio methods, such as orbital free DFT, for example, instead of relying entirely on empirical potentials. One potential pitfall of the quasicontinuum method is the so-called ghost force at the interface between the coarse-grained representative atoms and the atomically resolved regions [35]. The error arises because of the discontinuity between neighboring cells where the cell sizes are less than the range of the atomistic potential. Care must be taken to correct these ghost forces. Quasicontinuum approach also shares some features with hierarchical methods as the constitutive equations for FE nodes are drawn from atomistic calculations, and hence there is a message passing across scales. To exemplify the approach and to illustrate how simulation domains appear in these methods, we review two examples. The quasicontinuum method finds particularly useful applications in studies of fracture and deformation, as it is illustrated here in a simple example of a thin films constrained

11 5 Multiscale Modeling and Simulation Methods 167 Fig. 5.6 Results of a simulation of a crack in a thin film constrained by a rigid substrate, exemplifying a study using a concurrent multiscale simulation method, the quasicontinuum approach [20] by a substrate. A set of results for this case is shown in Fig Here we investigate a thin copper film with a (111) surface on a rigid substrate (the film thickness is h f 30 nm). The interatomic interactions are modeled by Voter and Chen s EAM potential for copper [34,35]. We consider a crack orthogonal to the surface. Such a crack could for instance be created by grain boundary cracking or constrained grain boundary diffusion [46]. Figure 5.6a shows different snapshots as the lateral mode I opening loading of the film is increased (the black line indicates the interface of substrate and thin film). The atomic region adapts and expands, as dislocations gliding on glide planes parallel to the film surface are nucleated and flow into the film material. Figure 5.6b shows a zoom into the crack tip region. Figure 5.7 shows another example application of the quasicontinuum method, here illustrating the simulation of a nanoindentation experiment [191]. A coupled atomistic and discrete dislocations (CADD) method has been developed [192], exemplifying a multiscale approach aimed at coupling a fully atomistic region to a defected dislocation dynamics region. In the CADD method, dislocations in the continuum region are treated with a standard discrete dislocation method, and the atomistic region can have any kind of atomic scale defects. Key strengths are automatic detection and smooth passing of dislocations back and forth in the atomistic and continuum regions. So far, this approach is restricted to two-dimensional, quasistatic problems. Good agreement to fully atomistic simulations has been shown in atomic scale void growth and two-dimensional indentation problems. A major challenge to extension

12 168 Atomistic Modeling of Materials Failure Fig. 5.7 Application of the quasicontinuum method in the simulation of a nanoindentation experiment. Subplots (a) and(b) depicts a cross-sectional view of the test sample used in the nanoindentation simulations for increasing indenter penetration (part of the indenter is also shown). Subplot (c) plots the dislocation structure at the indenter penetration corresponding to the indentation depth shown in subplot (b). Subplot (d) shows a load vs. displacement curve predicted by full atomistic (LS) and quasicontinuum (QC) simulations, illustrating that the two methods show excellent agreement. Reprinted from Journal of the Mechanics and Physics of Solids, Vol. 49(9), J. Knap and M. Ortiz, An analysis of the quasicontinuum method, copyright c 2001, with permission from Elsevier to three-dimensional problems is the representation of three-dimensional dislocation loops that would extend across the atomistic/continuum interface Continuum Approaches Incorporating Atomistic Information Recently, a virtual internal bond (VIB) model has been proposed as a bridge of continuum models with cohesive surfaces and atomistic models with interatomic potentials [157]. The VIB method differs from an atomistic model

13 5 Multiscale Modeling and Simulation Methods 169 in a sense that a phenomenological cohesive force law is adapted to act between material particles, which are not necessarily atoms. A randomized network of cohesive bonds is statistically incorporated into the constitutive response of the material based on the Cauchy Born rule (see Sect ). This is achieved by equating the strain energy function on the continuum level to the potential energy stored in the cohesive bonds due to an imposed deformation. The basic idea of this method is to compute the constitutive equation directly from the interatomic potential. Other features of the VIB model [157] can be found elsewhere. The method has been used to study crack propagation in brittle materials, and is able to reproduce many experimental phenomena such as crack tip instabilities or branching of cracks at low velocities. An important implication of the VIB method is that it provides a direct link between the atomic microstructure and its elastic properties, for any given potential. The method was recently extended to model viscoelastic materials behavior [193]. The fact that this method is able to perform simulations on entirely different lengthscales makes it interesting for numerous applications particularly in engineering, where more complex situations have to be modeled Hybrid ReaxFF Model: Integration of Chemistry and Mechanics In this section, we review a method to integrate different force fields within a single computational domain. This method is not a true multiscale method, but it rather is a multiparadigm method. The integration of chemistry and mechanical properties remains a challenging issue, 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 combination of the ReaxFF force field, which is capable of treating chemically complex materials, with nonreactive potentials is a possible strategy to overcome these limitations. Basic Concepts The first principles based reactive force field ReaxFF (see also discussion in Sect ) 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. There have been other attempts of modeling charge transfer in metal/metal-oxide systems, such as the modified charge transfer-embedded atom method potentials [194, 195].

14 170 Atomistic Modeling of Materials Failure However, ReaxFF has a wider range of applicability to other types of atomic structures, such as hydrocarbons, proteins, and semiconductors. It has been demonstrated that ReaxFF reproduces quantum mechanical results for both reactive and nonreactive systems, including hydrocarbons, nitramines, ceramics, metal alloys, and metal oxides. Due to the complexity of the underlying mathematical expressions in ReaxFF, and the necessity to perform a charge equilibration (QEq) [132] at each iteration, ReaxFF is approximately times more expensive computationally than simple FFs such as CHARMM, DREIDING, or covalent force fields such as Tersoff s force field. However, ReaxFF is several orders of magnitude faster than quantum mechanics-based ab initio methods. For details about the ReaxFF methodology and development we refer the reader to Sect , where some of the important aspects are reviewed. EAM-based models, on the other hand, have emerged as a well-established methodology to study mechanical deformation of metals. Fitted to experimental and ab initio data, they reproduce a wide range of properties such as equations of state, surface energies, stacking fault energies, and others quite accurately, which are important in correct representation of deformation behavior. The computational expense of EAM models is on the same order as that of empirical potentials, and systems of sizes of multibillion atoms have been investigated using EAM potentials. Thus a multiparadigm integration of ReaxFF and empirical force fields such as EAM and Tersoff, to capture chemistry at the regions of interest, for example at crack surfaces, using ReaxFF, and bulk elastic and plastic deformation using the EAM or Tersoff model is a quite promising approach to take advantage of the best of both approaches. Formulation of the Hybrid ReaxFF Model The force and energy contribution from different simulation engines is weighted as shown in Fig Every computational engine i has a specific weight w i associated with it that describe how much the energy of this particular computational engine contributes to the total energy. Thus, for two computational engines, the total Hamiltonian of the system is written as (here done for a combination of ReaxFF and EAM, but a similar approach can be used to couple ReaxFF): where H tot = H ReaxFF + H EAM + H ReaxFF EAM, (5.3) H ReaxFF EAM = w ReaxFF (x)h ReaxFF +(1 w ReaxFF (x))h EAM (5.4) describes the hybrid Hamiltonian formulation in the transition region shown in Fig The assignment of weights is based on the concept to specify a

15 5 Multiscale Modeling and Simulation Methods 171 Fig. 5.8 The interpolation method for defining a mixed Hamiltonian in the transition region between two different paradigms. As an alternative to the linear interpolation we have also implemented smooth interpolation function based on a sinusoidal function. This enables using slightly smaller handshake regions thus increasing the computational efficiency relative contribution from the two force fields that are being connected in the transition region. In this equation, the parameter w ReaxFF (x) is the weight of the reactive force field in the handshaking region. Forces on individual atoms given by the negative of the partial derivatives of H tot with respect to the atom s coordinates (see the discussion in Sect and (2.54), noting also that the kinetic part of the Hamiltonian does not contribute to the forces as the velocities do not depend on the atomic coordinates). Using (5.3) to calculate the forces in the domains in which only a single force field is used (that is, either the ReaxFF or the EAM domain in this example) is straightforward to implement. The forces are calculated the same way as for the individual potentials, either ReaxFF, Tersoff, or EAM. In the handshaking region, however, due to a gradual change of weights with position, one obtains F ReaxFF EAM = w ReaxFF (x)f ReaxFF +(1 w ReaxFF (x))f EAM which reduces to w ReaxFF x (H ReaxFF H EAM ), (5.5) F ReaxFF EAM = w ReaxFF (x)f ReaxFF +(1 w ReaxFF (x))f EAM w ReaxFF (U ReaxFF U EAM ), (5.6) x since the difference in the kinetic part of the different force fields in the transition region is identical (it only depends on the particle linear momenta).

16 172 Atomistic Modeling of Materials Failure Fig. 5.9 Example of the energy landscape of two force fields, a ReaxFF reactive force field and a nonreactive force field. The plot illustrates that the two models yield a similar energy landscape for small deviations from the minimum potential well, the equilibrium position. An exemplification of this effect specifically for silicon is shown in Fig This equation can be simplified quite a bit based on two conditions. First, if w ReaxFF (x) 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 is small), the last term in the equation can be neglected. Further, if the potential energies for the reactive force field and the EAM method are almost the same (that is, if the difference U ReaxFF U EAM is negligible in the transition region), the last term in (5.6) can be neglected. Both conditions lead to a simplified expression that only involves the weighting of forces in the transition region and not the considerations of the potential energies. F ReaxFF EAM = w ReaxFF (x)f ReaxFF +(1 w ReaxFF (x))f EAM. (5.7) In summary, this force handshaking algorithm requires a large transition region and therefore a slowly varying interpolation function, and if the handshaking is done in a regime in which both potentials provide an identical or very similar energy landscape. It works well when a transition region of larger than 7 Å width with a linear or sinusoidal interpolation of the force contributions is used. Figure 5.9 depicts the energy landscape of two force fields, illustrated here for interactions around the equilibrium position of atoms. Typically, for small deviations from the equilibrium position both potentials provide a similar description and as long as the handshaking is done in this regime, (5.7) provides a suitable approximation. This hybrid model is in principle not limited to two methods, but it can be generalized to N C different computational methods:

17 5 Multiscale Modeling and Simulation Methods 173 N C H tot = w i H i, (5.8) where the weight of N C computational engines add up to unity, i=1 N C w i =1. (5.9) i=0 The forces and energies are weighted accordingly, and the force vector of an atom j is calculated as F j = F j,i w i, (5.10) i=0,...,n where F j,i is the force contribution on atom j due to computational engine i and F j is the resulting force vector on atoms j. 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 about 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. Numerical Implementation: The Computational Materials Design Facility (CMDF) Multiscale methods often feature great computational complexity, and the development, application, and modification of numerical implementations may require significant effort. Thus several techniques have been developed that aid in making it easier to integrate different computational tools by providing a modular structure. For example, the computational materials design facility (CMDF) is a Python [196]-based simulation framework allowing multiparadigm multiscale simulations of complex materials phenomena operating on disparate lengthscale and timescale [197]. Individual computational engines are wrapped using the Simplified Wrapper and Interface Generator (SWIG) for rapid integration of low-level codes with scripting languages [196]. This framework enables complex multiscale simulation tasks encompassing a variety of simulation paradigms, such as quantum mechanics, reactive force fields, nonreactive force fields, coarse grain mesoscale, and continuum descriptions of materials. The CMDF framework enables the combination of ReaxFF to capture the QM description of reactions with classical nonreactive potentials to describe nonreacting regions providing the means for describing many complex materials failure processes as reported herein.

18 174 Atomistic Modeling of Materials Failure Since all simulation tools and engines can be called from a Python scripting level, the scale agnostic combinations of various modeling approaches can easily be realized. 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. An excerpt of a CMDF script is shown in Fig CMDF is designed to: Provide a general, extensible approach of a simulation environment utilizing a library of a variety of computational tools spanning scales from quantum mechanics to continuum theories. Establish a reusable library of highly complex computational tools that can be used as black boxes for most applications, while being initialized with standard parameters for easy usage in standard cases. Enable atomistic applications to be used by engineers and experimental scientists, while retaining the possibility of building highly complex simulations and models. Close the gap in coupling fundamental, quantum mechanical methods such as DFT to the ReaxFF reactive force field, to nonreactive force field descriptions (e.g., DREIDING, UFF). Provide a test bed for developing new model and algorithms, making it simple to develop new communication channels between computational engines (e.g., developing a new force fields combining distinct methods as QEq, Morse potentials, ReaxFF, or M/EAM). The CMDF approach reviewed here is only one out of many other approaches. Many other software suites such as Konrad Hinsen s Molecular Modeling Toolkit (MMTK) or the CAMD Open Software project (CAMPOS) of the Center for Atomic-Scale Materials Design at Danmarks Tekniske Universitet provide similar approaches. In addition, codes like NAMD can also be driven by a Python script, providing further opportunity to integrate other codes. Example of CMDF Model of Oxidation Figure 5.11 depicts an example study of a nanoscale elliptical penny-shaped crack in nickel filled with O 2. The system is under 10% tensile strain loading in the x-direction (orthogonal to the long axis of the elliptical defect). Oxidative processes leading to formation of an oxide layer are competing with extension of the crack. In later stages of the simulations, the oxide layer still remains, keeping the Ni half spaces together, indicating that it involves strong Ni O bonds. The reactive region can expand or shrink during the simulation and is determined by the positions of the oxygen atoms. Failure initiates by formation of nanovoids in the Ni bulk phase. Classical modeling schemes, for example based on the EAM method, cannot describe such complex organic metallic systems.

19 5 Multiscale Modeling and Simulation Methods 175 Fig Example CMDF script (upper part) and schematic of the structure of CMDF (lower part) 5.5 Advanced Molecular Dynamics Techniques to Span Vast Timescales Not only multiscale methods are developed to bridge spatial dimensions, but also other methods are focused on bridging across vast timescales. In classical molecular dynamics schemes it is in principle possible to simulate arbitrarily large systems, provided sufficiently large computers are available. However, the timescale remains confined to several nanoseconds. Surprisingly, this is also true for very small systems (independent of how large computers we use). The reason is that very small systems cannot be effectively parallelized. Also, time cannot easily be parallelized. Therefore, surprisingly there exists little trade off

20 176 Atomistic Modeling of Materials Failure Fig Study of a nanoscale elliptical penny-shaped crack in nickel, filled with O 2, illustrating the hybrid ReaxFF-EAM approach (crystal is loaded in tension, in the horizontal direction) between the desired simulation time and desired simulation size. This problem is referred to as the time-scale dilemma of molecular dynamics [89, 134, 198]. Many systems of interest spend a lot of time in local free energy minima before a transition to another state occurs. In such cases, the free energy surface has several local minima separated by large barriers. This is computationally highly inefficient for simulations with classical molecular dynamics methods. An alternative to classical molecular dynamics schemes is using Monte- Carlo techniques such as the Metropolis algorithm. In such schemes, all events and their associated energy must be known in advance. Note that in kinetic Monte-Carlo schemes all events and associated activation energy that take place during the simulation should be known in advance. For that purpose, the state space for the atoms has to be discretized on a lattice. Besides having to know all events, another drawback of such methods is that no real dynamics is obtained. To overcome the time-scale dilemma and still obtain real dynamics while not knowing the events prior to the simulation, a number of different advanced simulation techniques have been developed in recent years (for a more

21 5 Multiscale Modeling and Simulation Methods 177 extensive list of references see [199]). They are based on a variety of ideas, such as flattening the free energy surface, parallel sampling for state transitions, and finding the saddle points or trajectory-based schemes. Such techniques could find useful applications in problems in nanodimensions. Time spans of microseconds, seconds, or even years may be possible with these methods. Examples of such techniques are the parallel-replica (PR) method [200, 201], the hyperdynamics method [202], and the temperature-accelerated dynamics (TAD) method [203]. These methods have been developed by the group around Voter [89] (further references could be found therein) and allow calculating the real time-trajectory of atomistic systems over long time spans. Other methods have been proposed by the group around Parrinello, who for instance developed a Non-Markovian coarse grain dynamics method [199]. The method finds fast ways out of local free energy minima by adding a bias potential wherever the system has been previously, thus quickly filling up local minima. The methods discussed in these paragraphs could be useful for modeling deformation of nanosized structures and materials over long time spans, such as biological structures (e.g., mechanical deformation of proteins and properties at surfaces). A drawback in many of these methods is that schemes to detect state transitions need to be known. Also, the methods are often only effective for a particular class of problems and conditions. We give an example of using the TAD method in calculating the surface diffusivity of copper (modeled by an EAM potential [34]). We briefly review the method. The simulation is speed up by simulating the system at a temperature higher than the actual temperature of interest. Therefore, in this method two temperatures are critical: The low temperature at which the dynamics of the system is studied, and a high temperature where the system is sampled for state transitions during a critical sampling time. This critical sampling time can be estimated based on theoretical considerations in transition state theory [89]. For every state transition, the time at low temperature is estimated based on the activation energy of the event. Among all state transitions detected during the critical sampling time, only the state transition that would have occurred at low temperature is selected to evolve the system and the process is repeated. To calculate the surface diffusivity of copper, we consider a single atom on top of a flat [100] surface as shown in Fig The atom is constrained to move at the surface. The total simulation time approaches t = s. This is a very long timescale compared to classical molecular dynamics timescales (see Fig. 2.11). The surface diffusivity is calculated according to xi (t) x i (t 0 ) 2 D s = lim. (5.11) t 6(t t 0 ) The simulation is carried out at a temperature of T 400 K with N = 385 atoms. The high temperature in the TAD method is chosen to be 950 K. The

22 178 Atomistic Modeling of Materials Failure Fig Atomistic model to study surface diffusion of a single adatom on a flat [100] copper surface Fig Study of atomic mechanisms near a surface step at a [100] copper surface. The living time (or temporal stability) of states A (perfect step) and B (single atom hopped away from step) as a function of temperature. The higher the temperature, the closer the living times of states A and B get integration time step is δt = s. The diffusivity is then calculated to be Ds MD = m 3 /s 1. (5.12) This value is comparable to experimental data Ds exp m 3 /s 1 [204]. The activation energy of all state transitions is found to be 0.57 ev. We further show an example of how the temperature accelerated method could be used. Here we consider the atomic activities near a surface step in a [100] copper surface. We find that atoms at the surface step tend to hop away from the perfect step. This defines two states (A), the perfect step, and (B), when the atom is hopped away from the step. The simulation suggests that over time, the two states A and B interchange. Figure 5.13 shows the time-averaged stability of the two states as a function of temperature. It can be observed that for low temperatures, the living time of state (B) is much smaller compared to that of state (A). State (A) is observed to be

23 5 Multiscale Modeling and Simulation Methods 179 State transition (from to) Activation energy (ev) A B B A Table 5.1 Activation energy for different state transitions Fig Snapshots of states A (perfect step) and B (single atom hopped away from step) stable up to several hundred seconds. Figure 5.14 shows the two states in a three-dimensional atomic plot. Table 5.5 summarizes the different activation energies is higher than that of the reverse process. The activation energy to get from state (A) to state (B) is higher than that of the reverse process. This immediately explains why state (B) is not as stable as state (A). Such methods have recently also been applied to better understand rate dependence effects in the deformation of metals [21]. In this work, the authors illustrated the rate dependence of twinning of metals across a large range of timescales, showing how simulation and experimental results can be connected. The authors used a combination of the CADD method to reduce the number of atomic degrees of freedom, together with the parallel replica method. This method enabled them to study dynamical materials failure mechanisms over many orders of magnitudes of timescales (see Fig. 5.15). The brief examples reviewed here illustrate the great appeal of these advanced simulation techniques. Experimental techniques are currently not able to provide the resolution in space and time to track the motion of single atoms. On the other hands, advanced molecular dynamics simulation techniques can track the motion of atoms on a surface on a relatively long timescale, with a very high resolution of time.

24 180 Atomistic Modeling of Materials Failure Fig Hybrid CADD-Parallel Replica study of mechanical twinning of a metal. Subplot (a) shows the simulation domain, illustrating the continuum/discrete dislocation regime and the full atomistic domain (blow-up in right part). Subplot (b) shows a comparison of atomistic simulation results with the predictions of an analytical model. The plot shows the time to nucleation of a trailing or twinning partial versus applied load in Al at a temperature of 300 K. The circles refer to the multiscale simulation results covering many orders of magnitudes in timescales. The dashed lines correspond to the predictions of the analytical. Reprinted with permission from Macmillan Publishers Ltd, Nature Materials [21] c Discussion Multiscale simulation methods have developed quite significantly over the past decades. In particular in the past 10 years, many new methods have been developed that contributed to an extensive database of available methods.