A Study of Crack Propagation in Metals in the Presence of Defects

Transcription

1 Indiana University of Pennsylvania Knowledge IUP Theses and Dissertations (All) A Study of Crack Propagation in Metals in the Presence of Defects Justin M. Petucci Indiana University of Pennsylvania Follow this and additional works at: Recommended Citation Petucci, Justin M., "A Study of Crack Propagation in Metals in the Presence of Defects" (213). Theses and Dissertations (All) This Thesis is brought to you for free and open access by Knowledge IUP. It has been accepted for inclusion in Theses and Dissertations (All) by an authorized administrator of Knowledge IUP. For more information, please contact cclouser@iup.edu, sara.parme@iup.edu.

2 A STUDY OF CRACK PROPAGATION IN METALS IN THE PRESENCE OF DEFECTS A Thesis Submitted to the School of Graduate Studies and Research in Partial Fulfillment of the Requirements for the Degree Master of Science Justin M. Petucci Indiana University of Pennsylvania May 213

3 213 Justin M. Petucci All Rights Reserved ii

4 Indiana University of Pennsylvania School of Graduate Studies and Research Department of Physics We hereby approve of the thesis of Justin M. Petucci Candidate for the degree of Master of Science Majid Karimi, Ph.D. Professor of Physics Thesis Advisor Gregory Kenning, Ph.D. Professor of Physics Graduate Coordinator Devki Talwar, Ph.D. Professor of Physics Department Chair ACCEPTED Timothy P. Mack, Ph.D. Dean School of Graduate Studies and Research iii

5 Title: A Study of Crack Propagation in Metals in the Presence of Defects Author: Justin M. Petucci Thesis Chairman: Majid Karimi, Ph.D. Thesis Committee Members: Gregory Kenning, Ph.D. Devki Talwar, Ph.D. Molecular dynamics (MD) and molecular statics (MS) simulations of crack propagation in the presence of defects in crystalline FCC metals under mode I loading are carried out on the (1)[1] crack system using the embedded atom method (EAM) interatomic potential. Substitutional impurity point defects are introduced into a 3D thin-strip slab of 16 atoms at various distances from the crack tip. The critical load required for the initiation of crack propagation is obtained, along with the atomic level stress distribution near the crack tip. The results indicate that the critical load is dependent on the defect species, geometry, and position. When located directly at the crack tip, the defects reduce the peak internal stress, increasing the critical load relative to the defect-free system. As the defects are moved away from the crack tip the critical load goes through a minimum and approaches the value of the pure material asymptotically. iv

6 ACKNOWLEDGMENTS I would like to express my sincerest gratitude to my thesis advisor, Dr. Majid Karimi. His knowledge and expertise, patience, and availability have been paramount in the completion of this project. I would like to thank the IUP Physics department, in particular, Dr. Gregory Kenning and Dr. Devki Talwar for serving on my thesis committee. A special thanks goes to Dr. Carl LeBlond for providing computational resources in addition to his ongoing assistance during the building and maintenance of the Einstein computer cluster. I would also like to thank our dean, Dr. Deanne L. Snavely, who provided funding for the purchase of a new server. A resource that this project has benefited greatly from. Finally, I wish to thank my family and friends for their unquestioning love and support over the years. v

7 TABLE OF CONTENTS Chapter Page 1 INTRODUCTION ANALYTICAL RESULTS OF FRACTURE Origins of Fracture Analysis Energetic Approach for the Thin Strip Geometry Stress Field Distribution and the Stress Intensity Factor Atomic View of Crack Propagation COMPUTATIONAL SIMULATION Molecular Dynamics NVE ensemble Molecular Dynamics NVT ensemble Boundary Conditions The Interatomic Potential Two Body Pairwise Potentials Pair Functionals Embedded Atom Method (EAM) Cutoff Radius, Neighbor Lists, and Binning Parallel Computing Aspects Molecular Statics Elastic Constants Surface Energy Temperature Atomic Virial Stress...57 vi

8 3.15 Crack Tip Location and Velocity Crack System and Defect Configurations Material Loading and Simulation Methods RESULTS AND DISCUSSION EAM Potential Verification Fracture Initiation in a Defect Free Material Fracture Initiation in the Presence of Impurity Defects at the Crack Tip Fracture Initiation in the Presence of Impurity Defects Away From the Crack Tip CONCLUSIONS...92 REFERENCES...93 APPENDICES...95 Appendix A: Derivation of the approximate failure tensile stress of a uniform perfect material...95 Appendix B: Pseudocode for the implementation of the Velocity-Verlet algorithm for an NVE ensemble...97 Appendix C: Pseudocode for the implementation of the Velocity-Verlet algorithm for an NVT ensemble...98 Appendix D: Vacancy of formation to cohesive energy ratio for linear and square root dependence of the cohesive energy...99 Appendix E: Derivation of the atomic force expression for the Embedded Atom Method interatomic potential...11 Appendix F: Pseudocode for the implementation of a general energy minimization algorithm...14 vii

9 LIST OF TABLES Table Page 1 Limiting temperature for the classical approximation for some FCC metals Pair potential system property predictions with known values for some FCC metals Number of Independent Elastic/Compliance constants for various crystal systems Various material properties calculated using the EAM potential Calculated critical loads for crack initiation in some FCC metals Calculated critical loads for crack initiation in Nickel with various FCC defects present Calculated critical loads for crack initiation in Cu with Ni and Au FCC defects present...87 viii

10 LIST OF FIGURES Figure Page 1 A uniformly stressed plate in the presence of an elliptical cavity. The internal stress is magnified near the edges of the ellipse A thin strip of thickness y, height z, and length x is strained by displacing the top and bottom surfaces by a total distance Δz. The crack tip present in the center of the material releases the stored strain energy as it propagates The three loading modes that can be applied to a crack system (a) A plot of potential energy and force magnitude as a function of atom separation distance. (b) An atomistic view of crack extension via bond breaking A two dimensional square lattice with nearest neighbor interactions within a simulation cell. (a) The top image shows fixed boundary conditions and the bottom image shows periodic boundary conditions. (b) Alternate view of periodic boundary conditions through the replication or mirroring of the system in all directions The Verlet neighbor list region defined as the union of the gray and yellow areas. The red center atom only interacts with the atoms in the gray region The crack system geometry utilized for atomic simulations, with dimensions of 72.24Å 14.8Å Å for Ni Impurity defect configurations considered in atomistic simulations. The impurity atoms are shown in red Energy pathway for successive bond breaking during crack extension for differing loads (energy release rates) Lattice trapping barrier with individual energetic contributions; γs bond breaking energy and Eel elastic relaxation. This figure is reproduced with express written permission from the authors Crack tip displacement for 5ps of a simulation run in pure Ni with a load of Gc Snapshot in the vicinity of the crack tip during a simulation with atoms colored according to (a) atomistic temperature and (b) atomic stress in the z-direction σzz. The coloring is scaled from blue (smallest values) to red (largest values)...73 ix

11 13a Stress along the crack plane for defect configuration A in Ni with Cu impurities b Stress along the crack plane for defect configuration A in Ni with Pd impurities c Stress along the crack plane for defect configuration A in Ni with Pt impurities d Stress along the crack plane for defect configuration A in Ni with Ag impurities e Stress along the crack plane for defect configuration A in Ni with Au impurities a Stress along the crack plane for defect configuration B in Ni with Cu impurities b Stress along the crack plane for defect configuration B in Ni with Pd impurities c Stress along the crack plane for defect configuration B in Ni with Pt impurities d Stress along the crack plane for defect configuration B in Ni with Ag impurities e Stress along the crack plane for defect configuration B in Ni with Au impurities a Stress along the crack plane for defect configuration C in Ni with Cu impurities b Stress along the crack plane for defect configuration C in Ni with Pd impurities c Stress along the crack plane for defect configuration C in Ni with Pt impurities d Stress along the crack plane for defect configuration C in Ni with Ag impurities e Stress along the crack plane for defect configuration C in Ni with Au impurities...82 x

12 16 Atomic strain energy distribution along the crack plane for pure Ni and for Ni with Cu defect configuration A present (top). Energy path for the extension of a crack by one atomic layer for pure and defect cases (bottom) Critical loads for various defect configurations and impurity species placed at different distances from the crack tip Stress along the crack plane for defect configuration A in Ni with Cu impurities located 4 atomic layers from the crack tip...91 xi

13 CHAPTER 1 INTRODUCTION The main failure mechanism of brittle materials occurs through the creation and propagation of cracks. By gaining a deeper understanding of this process and the factors that affect it, the strength of materials can be more accurately predicted and methods to prevent material failure can be identified. Encompassing an atomic scale crack tip and extending over macroscopic distances, crack propagation is inherently multi-scale in its nature. Most of the strain energy due to the external loading on the material is concentrated at the crack tip. When this strain exceeds the failure strength of the material, a cascade of atomic bond breaking occurs creating new free surfaces. The presence of material defects such as impurities, vacancies, dislocations, microcracks, etc., are known to alter the propagation of a crack. Depending on the type and location of the defects the propagation can be aided or impeded. In order to fully analyze these defects and their impact on the local stress field and strain concentration at the crack tip, a method that can account for the atomic level inhomogeneities of the material must be used. While traditionally a continuum mechanics based approach has been implemented in the study of crack propagation, it is inadequate for atomic level details since it models matter as being infinitely divisible and continuous. With the multi-core evolution of processing units (CPUs & GPUs) and fast efficient parallel simulation codes, appropriate size systems can now be easily modeled from a classical atomistic standpoint. This project will focus on studying the effects of defects on crack propagation. The modeling of a system at the atomic level is fundamentally quantum mechanical. Thus 1

14 crack propagation would be ideally treated by solving the time-dependent Schrodinger equation. This computational task is impractical for any moderate size system and approximations must be implemented. Treating the system classically significantly reduces the computational load. The accuracy of the results, however, depends on the reliability of the interatomic potential that is employed. For this reason, crack propagation in Face-Centered-Cubic (FCC) metals will be studied since efficient and accurate interatomic potentials are readily available for these metals via the embedded atom method (EAM). Most of the metals are ductile at intermediate temperatures but at low temperatures they exhibit brittle behavior along certain crystal directions. Using molecular dynamics (MD) and molecular statics (MS), crack propagation will first be examined in defect free crystalline nickel, gold, and copper. Once the critical load required for the initiation of crack propagation in the pure materials is determined along with the atomic stress distribution near the crack tip, defects will be added. Specifically FCC metal substitutional impurity defects will be added near the crack tip and the critical load and stress distribution will be recalculated. By examining the change in these quantities after the introduction of defects, their effect on crack propagation can be characterized. With this, one can gain a deeper understanding of the fracturing process. This thesis is organized as follows. In Chapter 2, the origin and background of the basic analytical fracture concepts utilized in this study will be presented. The computational simulation methods, concepts, and techniques employed are given in Chapter 3. In Chapter 4, the results are summarized and finally in Chapter 5 concluding remarks are given. 2

15 CHAPTER 2 ANALYTICAL RESULTS OF FRACTURE The analytical study of fracture from both the continuum and atomistic view has been extensive and fruitful. From the results of basic continuum mechanics one can obtain a lower estimate of the critical failure stress and strain of a homogenous linear material along with the stress field distribution near the crack tip. From an atomistic treatment one can obtain equivalent failure criteria since the governing equations must converge to those of linear continuum mechanics as a limiting case. Along with these, there is also the prediction of lattice trapping and the velocity gap which arise due to the discrete nature of materials. Since the basic continuum based approach assumes a linear homogeneous continuous material and the atomistic approach is only tractable for simplistic 1D and 2D models, their results cannot describe crack propagation in true 3D discrete materials in the presence of defects where atoms near the crack tip are far from their equilibrium position. 2.1 Origins of Fracture Analysis The analytical study of crack propagation in brittle materials was initiated in the 192's by A. A. Griffith, in an attempt to resolve conflicting results between a material's experimentally measured and theoretically predicted strength. It was widely known at the time that the theoretical failure stress of a material, which can be approximated as E/1 where E is the Young's modulus of the material, was several orders (3-4) of magnitude larger than the observed experimental failure stress.1 The approximation for the failure stress is derived by considering the force needed to break all the atomic bonds in the 3

16 cross sectional area of a material as shown in Appendix A.1 Griffith hypothesized that since the material was failing at stresses below the theoretical failure stress, features in the material must be acting to locally concentrate and amplify the internal stress that is induced by the applied load. These features exist in all materials since no bulk material is truly perfect and defect free. Material defects can include cracks, voids, impurities, grain boundary interfaces, etc. An important result in the magnification of stress around material flaws was developed by Inglis less than a decade before Griffith's studies on fracture mechanics.2 In a uniformly stressed Figure 1. A uniformly stressed plate in the presence of an elliptical cavity. The internal stress is magnified near the edges of the ellipse. homogenous linear material, the internal stress is equal to the applied stress. Inglis introduced an elliptical cavity in a uniformly stressed plate and showed that the internal stress is no longer uniform and equal to the applied stress, but is magnified by several times at the corners of the elliptical hole as shown in Figure 1.2,3 The maximum stress at the edge of the ellipse is given by, 4

17 c σ edge=σ applied (1+ 2 ρ ) (2.1) where c is the semi major axis of the ellipse and ρ is the radius of curvature at the ellipse edge defined by ρ=b2/c.3 This equation demonstrates that the internal stress in a material around long narrow voids (large c, small b) with a very sharp tip (small ρ) is significantly amplified over the base applied stress value. Griffith discovered numerous micro-crack defects in the materials he was studying that had this perfect geometry to magnify the internal stress locally near the sharp tip. When this local stress exceeds the predicted failure stress of the material, bonds are broken and sheared causing the material to fail even though the measured external or applied stress is below the theoretically predicted failure stress. This simple model qualitatively explained why materials were failing at applied stresses well below the theoretical failure value. Furthermore, according to equation 2.1 an infinitely sharp crack tip generates a singularity in the stress at the tip. Griffith approached fracture mechanics energetically in order to circumvent this singularity predicted by continuum mechanics and to obtain a failure criterion for fracture in brittle materials. 2.2 Energetic Approach for the Thin Strip Geometry A crack will propagate in a material if the energy available for crack extension is larger than the material's resistance to crack growth. For a perfectly brittle material under fixed displacement loading, the energy available for crack growth is equal to the decrease in the system's elastic energy caused by the extension of the crack. This quantity is formally defined as the energy release rate, G, which is the energy released per unit thickness and per unit length of crack tip advance. The resistance to crack growth in a 5

18 Figure 2. A thin strip of thickness y, height z, and length x is strained by displacing the top and bottom surfaces by a total distance Δz. The crack tip present in the center of the material releases the stored strain energy as it propagates. brittle material is the energy necessary to create the new free surfaces that are formed when the crack advances. The crack resistance term acts to increase the total system energy since new free surfaces are formed while the crack driving term acts to decrease the energy of the system by relieving the strain in the portion of the material that the crack has passed through. The energy released goes into breaking bonds and forming the new free surfaces. Expressions for the critical stress and strain necessary to make a crack propagate in a material can be obtained by analyzing a specific crack system energetically. Figure 2 depicts the thin strip crack geometry that will be utilized in molecular dynamics simulations. The system is strained by displacing the upper and lower surfaces from their equilibrium position and holding them fixed. This fixed-grips constraint prevents the external load from performing any additional work on the system during crack propagation. When the crack extends a length, a, in a linear (Hookean) isotropic material the elastic energy dissipated or released is equal to, 6

19 1 2 U = E ' ϵ a y z 2 (2.2) where ϵ is the engineering strain defined as the ratio of change in height to original height, z is the unstrained height of the material, and the area of crack extension is A=a y. E' is the elastic constant in the direction of the applied strain and is often referred to as the effective Young's modulus of the three dimensional isotropic material. In section 11 of the following chapter the elasticity matrix will be presented and with this the elastic constant for a given direction can be obtained. Typically two special cases that reduce this matrix are considered. Plane strain occurs when a single dimension of a material is very large compared to the others. Due to the large size, the principal strain in that direction along with the corresponding shear strains are negligibly small and can be approximated as zero. In contrast, when a single dimension is very small compared to the others the principal stress in that direction is negligibly small and along with the corresponding shear stresses, can be assumed zero. This is known as the plane stress condition. The energy required to create the new free surfaces as a crack extends is given by the Griffith condition, U =2 γ s a y (2.3) where γs is the surface energy. The energy release rate and crack resistance terms are found by differentiating 2.2 and 2.3 with respect to the area of crack advance giving, G= du du da E ' ϵ2 z = = da da da 2 R= du du da = =2 γ s da da da (2.4) where G is the energy release rate and R is the crack resistance. When a material is strained the magnitude of G increases until a critical strain in which G=R is reached. At a strain infinitesimally larger than the critical value, the amount of energy available for 7

20 crack growth exceeds the resistance to crack growth. At this point it is energetically favorable for the crack to continually extend through the material since it lowers the overall system energy. As long as G remains larger than the resistance term R, enough energy will be continually funneled to the crack tip to break bonds and allow propagation through the entire material. Equating G and R, the critical strain and stress values required for a crack to propagate in a system with the geometry of Figure 2 are given as, ϵ c= 4 γs E' z σ c= 4 γs E ' z (2.5) An important result for the critical stress and strain expressions for the strip geometry is that they are independent of the initial crack length. In general this is not true for other crack system geometries, such as the thin plate. The identification of the crack resistance term with twice the free surface energy, γs, by Griffith is widely known to underestimate the actual crack resistance term of a material.4,5 Thus the equations of 2.5 are known to provide a lower bound for the critical applied strain and stress required to initiate crack propagation in a material. One way to improve on this result is to include an additional resistance term due to the fact that a perfectly brittle material does not exist. This term is due to the work required for the plastic or non-reversible deformation that occurs in the material around the crack tip. It can also include energy dissipated into the system through surface waves that emanate from the crack tip and travel through the material. Additionally there is evidence that the Young's modulus, E, and the free surface energy, γs, depend on the strain state of the material and the energy release rate, G, and the crack resistance, R, depend on the crack tip velocity.4,6 Ultimately however there are atomic level phenomena that arise due to the discreetness of matter that continuum mechanics can never predict. 8

21 2.3 Stress Field Distribution and the Stress Intensity Factor In the 195's G. R. Irwin et al. developed the stress intensity approach, which describes the stress field distribution near the crack tip in a loaded homogeneous linear elastic material.6 The three independent loading modes a crack system can experience are shown in Figure 3. Figure 3. The three loading modes that can be applied to a crack system. Under Mode I loading the applied stress is normal to the crack plane and direction. Mode II loading is also referred to as the sliding mode and occurs when the stress is applied parallel to the crack plane and crack direction. The third mode occurs when the loading condition is parallel to the crack plane but perpendicular to the crack direction. For the thin strip crack geometry analyzed earlier, which is an example of Mode I uniaxial loading in the z direction, the general equations of Irving for the stress around the crack tip are given by, σ xx= [ ( ) ( )] KI 5θ 3 cos θ +cos π r [ { σ yy= ν(σ xx +σ zz ), plane strain, plane stress σ xz = ( )] KI 5θ σ zz = 5 cos θ cos π r () KI 3θ sin (θ)cos 2 2 2π r ( ) } (2.6) σ xy =σ zy = where r and θ are polar coordinates defined from the crack tip, and ν is Poisson's ratio.6 9

22 The equations of 2.6 are valid near the crack tip where the distance from the crack tip is much less than the crack length, r << a. The stress field along the crack plane is given when θ= which significantly simplifies the equations in 2.6, KI σ xx =σ zz = 2 π r { 2 KI ν( ), plane strain σ yy = 2 π r, plane stress } σ xz =σ xy=σ zy= (2.7) where KI is the stress intensity factor which gives the magnitude of the stress field and depends on the external loading conditions and on the geometries of the material and crack.1,6 The stress is maximum at the crack tip and falls off with a (1/ r) dependence. As with the analysis of Inglis, there is still a singularity in the stress at r=. 2.4 Atomic View of Crack Propagation At the atomic level, when an external load is applied to a material the atoms are displaced from their original equilibrium position straining their bonds. In order for a bond to be severed and an atom to be permanently displaced from its equilibrium position, the external load on the material must translate to an acting atomic level force that is greater than the cohesive force binding the atoms. Additionally, this force must perform a minimum amount of work equal to the bond energy. Figure 4a shows an idealized interaction between two atoms as a function of their separation distance along with an atomistic view of crack extension shown in Figure 4b. The equilibrium atomic separation or bond length occurs at the minimum of the potential energy where the force is zero. The magnitude of the force (slope of the potential) on the atoms is shown for distances beyond equilibrium. Here the binding/bond energy is shown to be the minimum of the potential energy graph and the cohesive force is the maximum of the 1

23 force graph. As shown on the force graph, as the bond is strained and the separation distance increases a restoring force acts to counteract the straining force due to the external loading. If the straining force is larger than the maximum restoring force, the atoms will continue to separate and the bond between them will be severed. As shown in Figure 4. (a) A plot of potential energy and force magnitude as a function of atom separation distance. (b) An atomistic view of crack extension via bond breaking. the Figure 4b, a crack occurs when the local force experienced by atoms due to the external loading is enough to drive a cascade of bond breaking. This illustrates the necessity of modeling crack propagation atomistically rather than from a continuum standpoint as a crack initiates at the atomic level. In this simplistic atomic view of fracture, known as the Orowan and Gilman Model or the Cohesive Strength Model, a prediction for the critical stress and strain can be obtained by requiring that the bond energy or cohesive energy is equal to twice the energy of the surface created in the material once the bond breaks, as per the Griffith condition. This yields a result equivalent to that derived in equation 2.5 and follows from Appendix A.2 More sophisticated models such as the point-atom/spring-bond models or simply 11

24 lattice theory predict phenomena that arise purely due to the discreteness of matter.2,7-1 Lattice trapping is a term coined to describe the phenomenon of when a crack tip becomes stuck or trapped between atoms, which can arrest a crack in motion and increase the load necessary to initiate crack propagation to well above the Griffith load. The crack essentially encounters an energy barrier, while there is enough energy available for crack extension stored in the strained material ahead of the crack tip, the bonds directly in front of the crack tip are not strained enough to break.7,9,1 Another phenomenon that arises solely due to the discrete nature of matter is known as the velocity gap. There exists a band of forbidden velocities for the steady state motion of crack propagation from zero to around 3% of shear wave speed.7,9,1 This gap is a consequence of the existence of an upper bound for the time between the breaking of successive bonds as a crack tip travels. In addition to this a comprehensive study performed by Marder concluded that macroscopically identical elastic samples can exhibit differing fracture properties, which depend explicitly upon the atomic level structure of the material.7,9,1 From the results of continuum mechanics, it is established that the internal stresses around the tip of a crack are no longer uniform and equal to the applied stress, but are considerably magnified. Assuming that the strain energy released during the extension of a crack equals the energy required to create the new free surfaces, a lower estimate for the applied stress and strain required for the initiation of crack propagation can be obtained. Expressions can be found for a wide variety of crack geometries including the thin strip configuration previously analyzed. While correction terms can be added to improve the critical load estimate, ultimately a continuum based approach fails to completely describe the fracturing process since the atomic level details cannot be accounted for. The atomic or lattice 1-D and 2-D models approximate materials as atoms 12

25 connected by springs that obey Hooke's law up to a certain cutoff criterion where the interaction ceases, simulating the process of bond breaking.9 While these models are able to predict phenomena due to the discreetness of the atomic lattice such as lattice trapping and the velocity gap, they do not represent real materials in their dimensionality or their atomic interaction. Full 3-D atomic models with more sophisticated interactions that include the anharmonicity, which accompanies the large deviations of the system from equilibrium that occur during fracture, are analytically intractable.2,9 One must therefore approach the problem numerically using computer simulations. 13

26 CHAPTER 3 COMPUTATIONAL SIMULATION Computational simulations via numerical analysis and approximation methods act as a bridge between theory and experiment. When theoretical models are only analytically tractable for simplistic idealized cases there is usually a need for solutions with additional levels of complexity that more closely resemble the real physical system that one desires to model. In this respect, numerical simulation complements theory but it can also be used to augment experiment. Accurate computational models can probe physical aspects of systems under conditions that are too difficult or expensive to perform in an actual experimental setting. Of course in any case the computational results are only as reliable and physical as the underlying model that produces them. The fracturing process begins at the atomic level with the breaking of bonds at the crack tip. A continuum based approach fails to take into account the atomic level details due to the discreetness of matter and the inhomogeneities that occur with material defects present. Additionally the extreme conditions in the process zone near the crack tip diverge strongly from linear elastic behavior.1 Ideally a model of crack propagation in a macroscopic system will, therefore, include an accurate treatment of each atom in the solid. The modeling of a system at the atomistic level is inherently quantum mechanical in nature and therefore should involve solving the Schrodinger equation. However, solving the Schrodinger equation for systems beyond the hydrogen atom require numerical and approximation methods. These methods are used in so called ab initio computer simulation algorithms and are by far the most accurate and reliable methods in modeling materials. This accuracy comes with a huge computational cost such that only 14

27 small systems can be treated. The next step is to attempt to model the system classically treating the atoms as point particles within a Newtonian or equivalent (Lagrangian/Hamiltonian) framework. While simplistic low dimensional atomic models have been pursued assuming a Hookean interaction with an appropriate cutoff to simulate bond breaking with some success, ultimately three dimensional models with realistic interactions remain analytically intractable.1 Using molecular dynamics one can numerically solve the equations of motion for realistic interatomic potentials. These potentials are typically empirical in nature and fit to known experimental values or first principle calculations with ad hoc functional forms that have some basis from quantum theory. Treating the system classically significantly reduces the computational task resulting in the ability to model larger systems (up to tens of billions of atoms) for larger time spans (up to tens of nanoseconds) when augmented with efficient parallel algorithms and numerous computational cores. This chapter will outline the basics of the classical computational methods used in this study to model the fracturing process in FCC metals. A review of molecular dynamics and molecular statics will be presented. The embedded-atom method potential that is used to model the materials will be outlined. The equations for the calculation of surface energy, elastic constants, virial stress, crack tip location and velocity, and temperature are given. The parallel computation methods (software and hardware) will be over-viewed along with the visualization tools/techniques. 3.1 Molecular Dynamics NVE ensemble In molecular dynamics the atoms are treated as point masses and their trajectories 15

28 are found by the numerical integration of Newton's equations of motion (or an equivalent formalism) which are given by,11 F i = r V (r 1,..., r N )= i V (r 1,..., r N ) d vi d2 x =mi a i=mi =mi 2 i ri dt dt (3.1) The forces are derived from an energy function, V, that contains all of the physics of the atomic interactions. For a system of N unconstrained atoms there are a total of 6N equations for positions and velocities of the atoms at each time step. A common algorithm based on the finite difference method to numerically solve the classical equations of motion is the Velocity-Verlet method. In this scheme the position and velocity equations at each time step for an atom, i, are given as,11 ai (t )Δ t 2 r i (t +Δ t )=r i ( t)+ v i (t)δ t+ 2 v i (t+ Δ t)=v i (t )+ (a i (t )+ a i (t +Δ t ))Δ t 2 (3.2) Using these equations the system can be evolved along its phase space trajectory. One of the costs of numerically solving Newtons equations is that time is now discretized. The position and velocity of each atom can be found at any time t = nδt where n is integer and Δt is the time step value. The pseudocode for a typical implementation of the Velocity-Verlet integrator MD algorithm is given in Appendix B. With any numerical method it is important to have an estimate of how the error scales. The Velocity-Verlet algorithm is a second order integrator meaning the global error scales as O(Δt)2.Careful consideration must be taken when choosing a Δt time-step value. If too small of a value is chosen, the system will only evolve for a small portion of its phase space trajectory. A large time-step runs the risk of high error with the O(Δt)2 error scaling. Additionally for large time-step values the integrator algorithm can become unstable resulting in a wildly inaccurate evolution of the system. When modeling a solid 16

29 the time-step is usually of the order 1-15s, as the general practice is to choose a time-step such that there is enough resolution to see the highest frequency vibrations.12 Solving Newton's equations of motion for a system generates a microcanonical ensemble if all the forces are conservative. This is commonly referred to as an NVE ensemble, in which the number of atoms, volume, and energy of the system are constant. The conversion of energy from potential energy to kinetic energy in a closed system raises the system temperature. When modeling brittle fracture in metals it is imperative that the temperature of the system is kept very low near the limit of where classical physics is applicable in lieu of quantum mechanics. This is because metals are naturally ductile materials and only exhibit brittle behavior at low temperatures and along certain crystal planes, where the blunting of the crack tip is suppressed. The applicability of classical mechanics can be roughly approximated by the de Broglie thermal wavelength formula given by, Λ= h2 2π mkbt (3.3) where h is Planck's constant, kb is Boltzmann's constant, T is the temperature of the system, and m is the atomic mass.12 In a similar argument as with the diffraction limit, if this wavelength is much smaller than the interatomic spacing, the system can be treated as classical particles. Setting this wavelength equal to the nearest neighbor distance and solving for the temperature will give a lower bound for the temperature at which classical mechanics is no longer applicable, T= h2 2πmkBd2 (3.4) where d is the nearest neighbor distance, equal to c/ 2 for FCC structures, and c is the 17

30 lattice constant. Table 1 gives this lower bound temperature for the FCC metals that will be used in this study. For temperatures around 1 Kelvin, the metals modeled in this study are expected to behave classically and the quantum effects can be neglected. Table 1. Limiting temperature for the classical approximation for some FCC metals. Element Mass (g/mol) Lattice Constant T (K) Ni Cu Pd Ag Pt Au Molecular Dynamics NVT ensemble Numerous thermostatting methods have been developed for use in molecular dynamics to control the system temperature and keep it constant (on average) for the duration of the simulation.13 Methods range from simple velocity rescaling and the addition of drag terms in Newton's equations to coupling the system to a heat bath. While each of these many methods can ensure that the average of the instantaneous temperature is a constant, few guarantee that the system samples a true NVT canonical ensemble with the necessary temperature fluctuations.13 In addition, it is imperative that the thermostat coupling is as minimal as possible, to ensure the dynamics of the system are not significantly altered by it. The Nose-Hoover thermostat method samples a true NVT ensemble and will be utilized in the molecular dynamics simulations of crack 18

31 propagation. It is also important to mention that the Nose-Hoover method is deterministic and thus conserves the time reversibility of classical mechanics as opposed to stochastic based thermostats, which are not. The heat bath is incorporated by extending the system and introducing a new artificial dynamical variable ζ', with an associated mass Q.13 The new variable also acts as a time scaling factor such that in the extended system the total timescale is stretched by ζ', such that dt'=ζ' dt.13 The positions of the atoms in the extended system are equal to those in the real system, however the velocities are altered due to the time scaling. r =r ' r = d r d r dt ' d r ' dt ' = = = r ' ζ ' dt dt ' dt dt ' dt (3.5) The Lagrangian of the system in terms of the time stretched extended system coordinates is given by, ext (r ', r ', ζ ', ζ ' )= N 1 m ζ ' 2 r i ' 2 V (r ' )+ 12 Q ζ ' 2 3N k B T o ln ζ ' 2 i=1 i (3.6) where the initial two terms are the Lagrangian of the real system in the external system coordinates and the last two terms are kinetic and potential energy associated with ζ'.13 This potential energy term is selected such that the method produces an NVT ensemble. Applying the Euler Lagrange equations and converting back to the real system variables gives the new equations of motion as, r i= Fi γ r i mi γ = 3N kb ( T (t ) T o ) Q (3.7) where To is the target temperature of the system, T(t) is the instantaneous temperature, and Q is the mass associated with the introduced dynamical variable.13 The importance of the Q parameter is evident in equation 3.7, as it determines the coupling between the heat bath and the real system. The extended system (heat bath and real system) sample an 19

32 NVE ensemble of constant energy while the real system samples a canonical ensemble of constant temperature. Energy is transferred between the heat bath and the real system regulating the temperature with appropriate fluctuations for a true NVT ensemble. According to equation 3.7, if T(t)<To, γ will decrease which lowers the amount of damping to the system. The opposite is true if the desired temperature of the system is below the instantaneous temperature, γ will increase resulting in increased system damping. When γ <, energy is being added to the system to increase the temperature and energy is being drained from the system when γ >. A small value of Q corresponds to strong coupling to the heat bath which equilibrates the system to the desired temperature quickly. Although too small of a value can result in high frequency fluctuations of the system temperature.13 Large Q values produce weak coupling to the heat bath and thus take a much longer time span to converge to the desired temperature. In order to implement this new temperature control during each time-step, the NVE Velocity-Verlet in Appendix B must be modified as shown in Appendix C. The novel feature of the Nose-Hoover thermostat is that the effective velocity damping term, γ, is not proportional to difference between the desired and instantaneous system temperature, rather its time derivative is. This produces the necessary energy and temperature fluctuations in accordance with the variance equations, σ 2E = 2 2=k B C v T 2 σt2 = 2 2=6 N T 2 (3.8) of a true canonical ensemble.13 In equation 3.8, H and T are the Hamiltonian (instantaneous internal energy) and instantaneous temperature of the system respectively, Cv is the heat capacity, and T is the macroscopic system temperature. In molecular dynamics a single system is simulated rather than an ensemble of a very large number of 2

33 identical replica systems. Therefore the averages in equation 3.8 are taken as time averages rather than ensemble averages which is valid due to the ergodic hypothesis of statistical mechanics. 3.3 Boundary Conditions One of the first steps of any atomistic computer simulation involves initializing the system by defining and arranging the atoms into the desired configuration. These atoms are placed within a simulation cell of finite length, width, and height for a three dimensional system. A rectangular simulation cell is the most common although in general non-orthogonal parallelepiped cells can be used.12 The treatment of the simulation cell boundaries is defined by the choice of boundary conditions which are typically limited to fixed and periodic. For fixed boundary conditions, atoms do not interact across the given simulation box boundary. Additionally the position of the boundary is held constant for the duration of the simulation run. If an atom moves beyond this fixed simulation boundary, it is permanently removed from the simulation. Fixed boundary conditions simply terminate the system, resulting in the creation of a surface since the boundary atoms have less neighboring atoms than those towards the center of the material. This is illustrated in the top portion of Figure 5a, where a two dimensional square lattice is depicted with nearest neighbor interactions. The red atom is along a boundary and only interacts with its three nearest neighbors, since there is no atom above to interact with. The blue atom is located away from the boundary and therefore experiences bulk conditions interacting with its four nearest neighbors. 21

34 Figure 5. A two dimensional square lattice with nearest neighbor interactions within a simulation cell. (a) The top image shows fixed boundary conditions and the bottom image shows periodic boundary conditions. (b) Alternate view of periodic boundary conditions through the replication or mirroring of the system in all directions. The surface area to volume ratio of a cubic material scales as 6/a, where a is the length of one side of the cube. This ratio therefore increases as the system size decreases, meaning for smaller systems the ratio of surface atoms to total atoms also increases. In turn this makes the surface effects more dominant in simulations of these smaller systems. If the goal is to interpolate the simulation results to a macroscopic size system, where the surface effects are less pronounced, periodic boundary conditions must be utilized. They allow for interactions across boundaries which eliminates surfaces and therefore allows the system to mimic the behavior of a much larger system. Generally with periodic boundary conditions the size of the simulation cell is still held fixed during the simulation although it can be permitted to adjust in response to applied stresses. If an atom exits through one boundary it will reappear in the system through the opposite 22

35 boundary thus conserving linear momentum but not angular momentum. The bottom portion of Figure 5a shows the application of periodic boundaries. The red atom now acts through the boundary with its fourth nearest neighbor and thus is no longer considered a surface atom. This is one of the main reasons for taking advantage of periodic boundaries, it eliminates surface effects so the system is under bulk conditions. Another way to view this type of boundary condition is shown in Figure 5b where the original simulation cell is replicated in all directions. In reality this replication is effectively infinite in all directions. We can see that the red atom still interacts with four nearest neighbors as in Figure 5a, resulting in the elimination of surface effects. 3.4 The Interatomic Potential In the classical approximation of modeling materials, one forgoes any attempt at treating the system via the Schrodinger equation and instead must choose a potential energy function from which the interatomic forces are derived. This classical approach is validated on the Born-Oppenheimer approximation, which allows for the total system wavefunction to be separated into an electronic contribution and a contribution for the more massive nuclei. The electron wavefunction is a function of both the electron and nuclei coordinates. However, the nuclei wavefunction is solely a function of the nuclei coordinates based on the assumption that the much lighter electrons relax quickly to their instantaneous ground state configuration upon a change in position of the nuclei. This leads to the concept of the classical interatomic potential, which is a function of the nuclei coordinates which are treated as point particles. This interatomic potential must somehow account for the effects of the now absent electrons and thus the choice of potential depends on the type of bonding present in the system, the desired accuracy and 23

36 transferability, the system size and time scale, and the computational resources available. Any choice of classical interatomic potential is an approximation and one that is more accurate, reliable, and transferable comes at a higher computational cost. For example, the computation task or cost of simple limited range potentials generally scale linearly, O(N), with system size. Meaning simulating a system of size 2N would ideally take twice the number of calculations as simulating a system of size N. More complicated potentials can scale as O(N2), which is still preferable over the O(N3)-O(N5) scaling of ab initio computational methods. In order to reduce the computational cost, interatomic potentials are assumed to have a limited range and only allow each atom to interact locally with neighbors within a predefined cutoff radius. Limiting the interaction range can turn a simple pair potential, where the computational task scales as O(N2) if every atom is allowed to interact with the other (N-1) atoms of the system, into a O(N) scaling potential. These classical potentials are developed by choosing a functional form for the model with a set of adjustable parameters such that the internal energy is a function only of the atomic positions. The functional form is normally based on a mix of theory, ad hoc assumptions, and physical intuition. The model's parameters are fit to system properties that are known from experiment or first principle calculations such as cohesive energy, lattice constants, bulk and shear moduli, vacancy formation energy, etc. The idea is then to apply the model, with the expectation that the form and fitting is general enough, to predict system properties away from the equilibrium conditions that the potential was parametrized with. 24

37 3.5 Two Body Pairwise Potentials The earliest computer simulations of the 6s and 7s utilized the simplest of interatomic potentials, the pairwise or two body potential almost exclusively.14 Examples of some commonly used pair potentials include the Morse, Lennard-Jones, harmonic, and Coulomb potentials. An example of the Lennard-Jones potential is plotted in the top portion of Figure 4a in Chapter 2 which has the functional form of, V i j =4 ε [( σ ri j ) ( )] 12 σ ri j 6 (3.9) where ε is the potential well depth, σ is the zero of the potential, rij is the separation between the pair of atoms, and Vij is the energy of the pair of atoms. Typically these twobody interactions are strongly repulsive (Pauli exclusion repulsion) at short atomic separation and weakly attractive (interacting instantaneous fluctuating dipoles) for large separation distances. Pair potentials are solely functions of the separation between two atoms. The total potential energy of a system of N atoms interacting via a pair potential can be found by a summing Vij over all atom pairs, N V total = 1 V (r ) 2 i j i pair i j (3.1) where the leading 1/2 is for the double counting of the summation. As discussed previously, the range of the potential can be altered so atoms only interact with other atoms locally within a cutoff rather than with all of the other atoms of the system. This typically requires adding a cutoff function along with a smoothing function to the potential form in order to prevent abrupt jumps in the interatomic potential which can lead to ill defined derivatives of the potential. The applicability of pair potentials is generally limited to the inert noble gases 25

38 where atoms only interact via the weak Van der Waals forces or in qualitative studies where accurate material properties are not needed.12 As Table 2 indicates, applying pair potentials to metallic systems can lead to erroneous results for some critical system properties. The ratio of two of the three independent elastic constants for cubic FCC metals, C12/C44 is predicted to be 1 by pair potentials as per the Cauchy relation. This prediction is not valid for metals and can lead to an underestimation by more than a factor Table 2. Pair potential system property predictions with known values for some FCC metals. 15 System C12/C44 EV/EC EC/(kBTmelt) Ni Cu Pd Ag Pt Au Pair Potential of 3 for some metals such as gold and platinum. Pair potentials overestimate the ratio of the vacancy formation energy (energy to change the coordination of Z atoms from Z to Z1) to the cohesive energy (energy to change the coordination of 1 atom from Z to ) and underestimate the ratio of cohesive energy to the melting point temperature. These discrepancies are a result of the pair potential's inability to account for the local bond environment. In a pair potential the strength of a bond between two atoms is completely independent of the configuration of atoms around the bond. The cohesive energy per atom grows linearly with coordination number (number of neighbors) in a system 26

39 modeled by a nearest neighbor pair potential. Consider for example the blue atom in Figure 5a interacting with its four nearest neighbors. The cohesive energy of the blue atom is given by, 1 E c = V pair Z 2 (3.11) where Z is the coordination number (here Z=4) of the atom, Vpair is the bond energy between the blue atom and one of its neighbors, and the 1/2 indicates that half of the bond energy goes to each atom in the particular bond. Equation 3.11 clearly shows that the cohesive energy scales linearly with coordination for pair potentials. Experimental results, ab initio simulations such as density functional theory (DFT), and theoretical models such as the second moment approximation of tight binding theory indicate that the cohesive energy of an atom scales closer to Z1/2.12 With this square root relationship an atom's bonds get weaker as more and more are formed, which adds an environmental dependence. For example, dividing equation 3.11 by the number of bonds and multiplying by 2, gives the bond energy which as expected gives Vpair. If, however, the cohesive energy per atom scales as Z1/2 then this example yields a bond energy of 1/2 Vpair. If two more bonds are added, the energy per bond decreases to Vpair( 6)/6.41 Vpair. This can help explain the discrepancy for the pair potential's prediction of the vacancy formation energy to cohesive energy ratio. As adopting the square root dependency for the atomic cohesive energy reduces this ratio to.51 as shown in Appendix D. Table 2 indicates that the actual ratio is around 1/3 but this exemplifies that the pair potential's lack of environmental dependance is its major limitation. Another inconsistency predicted by pair potentials is the outward relaxation of surfaces. In most metals, the surfaces relax inwards due to the under-coordination of the atoms present on the surface which 27

40 increases the strength of the remaining bonds between the first and second atomic layers. 3.6 Pair Functionals A class of potentials informally known as pair functionals were first developed in the 198's and include models such as the embedded-atom method (EAM), Finnis and Sinclair potentials, quasiatom, effective medium theory (EMT), and glue model potentials. These potentials attempt to include the dependance of a bond's strength on the local environment around it, since an atom's bonds weaken when more are formed. While all of these different pair functional forms are based on different physical rationales, i.e. EAM has its roots in density functional theory approximations and the Finnis and Sinclair potentials are based on approximations to tight binding theory, they eventually arrive at the same general analytical form for the interatomic potential which is given by, N V= N 1 ϕ (r )+ U i (ρi ) 2 i j i i j i j i ρi = g j ( r i j ) (3.12) j i where φij is a pair potential and Ui is a functional that gives the energy required to place an atom at a point whose environment is defined by the function ρi.12,15 This second term accounts for the local environment where gj(rij) is a linear combination of contributions from the neighboring j atoms of atom i.[12][15] As gj(rij) has no angular dependance it does not account for any directionality and is therefore spherically symmetric depending only on the distance from an atom to its nearest neighbor. The pair functional approach marks significant improvement over simple pair potentials while at the same time continuing to scale as O(N), requiring only 2-5 times the computational work, allowing for the simulation of large atomistic systems.16 These potentials are most successful when applied to modeling metallic bonding in metals with filled or nearly filled d-orbits (l=2) 28

41 such as the FCC metals Ni, Cu, Pd, Ag, Pt and Au.12 For these materials the pair functionals considerably improve the pair potential's deficiencies previously mentioned such as the environmental dependance of bond energy, the Cauchy relation, and surface relaxation. In metallic bonding the outer shell valance electrons delocalize resulting in an attractive force between the positively charged metal ions and the electron cloud of the conduction electrons. The filled d-band allows for the assumed spherical nature gj(rij) to be a more reasonable approximation.12 In covalently bonded materials where bonding is highly directional this assumption fails leading to a poor description of these materials by the pair functional approach. While all the various pair functional models have the same analytical form of equation 3.2, they differ in their definition and interpretation of U, ρ, and g along with their parametrization and fitting schemes. The EAM potential is one of the most popular implementations in open source molecular dynamics packages and has been met with wide success in modeling material properties. 3.7 Embedded Atom Method (EAM) The Embedded Atom Method (EAM) was originally proposed by M.S. Daw and M.I. Baskes in ,18 In a later paper published in 1986, with the addition of S.M. Foiles, the functional fitting and parametrization procedures were generalized and potentials for the FCC metals Cu, Ag, Au, Ni, Pd, and Pt were presented.19 This 1986 version of the EAM potential, referred to as u3 parametrization, will be utilized for the atomistic modeling of crack propagation. EAM is grounded on the fundamental basis of density functional theory in which Hohenberg and Kohn proved that the ground state energy of a system is given by a functional of its electron density. Within EAM, each 29

42 atom is viewed as an impurity, with an energy equal to the energy required to embed it into the local electron density of the other host atoms of the material plus an electrostatic pair contribution due to the repulsion between the atom cores.18,19 The embedding energy is a functional of the local electron density which is approximated as a linear superposition of atomic charge density contributions from the host atoms. Comparing this implementation to the generic pair functional outlined in the previous section, the local environment is then defined by the electron density contributed by the neighboring atoms. The functional form of the EAM interatomic potential is given by, E pot = F i (ρhi )+ i 1 ϕi j (r i j ) 2 i j i h a ρi = ρ j (r i j ) (3.13) j i where ρhi is the electron density at atom i due to host atoms, rij is the separation distance between atoms i and j, Fi(ρhi) is energy to embed atom i into the host electron density ρhi, φij(rij) is an electrostatic pair interaction between atoms i and j, and ρaj is the atomic electron charge density of atom j that is contributed to the host electron density.18,19 The force on an atom k for a system modeled via the EAM potential is given by, Fk= E pot = [ F 'l (ρhl ) ρ'ka ( r l k )+ F 'k (ρhk )ρ'l a (r l k )+ ϕ'l k (r l k ) ] r l k rk l k (3.14) as derived in Appendix E, where the summation is over the neighboring atoms within the interaction range of atom k. To implement the embedded atom method the embedding function Fi(ρ), pair potential φ(r), and the atomic electron densities ρa(r) must be determined.19 The embedding function is independent of the electron density source and, therefore, only needs to be determined for each embedded atomic species while the pair potential must be determined for each species along with each combination of atomic species. Using first principle calculations the qualitative behavior of the embedded function and pair 3

43 potential can be determined. The embedding energy should be concave up and have a negative slope for the electron density values found in metals while converging to zero for zero electron density.19 Along with the monotonic decrease of ρa(r) with increasing separation r and the interpretation of ρhi as some continuous measure of coordination or bond order, this prescribed behavior of the embedding function ensures that as the number of bonds of an atom increases the individual bonds weaken. The pair potential is strictly repulsive and interactions between unique atomic species can be approximated by the geometric mean of the individual species pair interaction.19 In the u3 parametrization, the embedding function is determined by utilizing the universal binding energy relation (UBER) developed by Rose et al.19 This relationship displays the universality of the cohesive energy curves of metallic and covalently bonded materials where only two parameters are needed to reasonably approximate the cohesive energy versus lattice constant curve of the material.12 The cohesive energy as a function of lattice constant is given by the universal function, * E (a)= E coh (1+ a * )e a a * =(a /a )/( E coh / 9BΩ)1/ 2 (3.15) where Ecoh is the magnitude of the equilibrium cohesive energy, ao is the equilibrium lattice constant, B is the bulk modulus of the material, and Ω is the equilibrium atomic volume. The required input to obtain the equation of state for particular material is the equilibrium lattice constant, cohesive energy, bulk modulus, and crystal structure. The embedding function can then be determined through equations 3.13 and 3.15, provided that the pair interaction and atomic electron densities are known. The host electron density in equation 3.13 must be inverted to give rij as a function of ρ which is then substituted in for rij in the result from the previous step yielding Fi(ρ). Typically this 31

44 inversion and algebra must be done numerically resulting in the use of tabulated embedding functions for computer simulations. With the assumption that the atomic electron density of an individual host atom is well represented by the spherically averaged free-atom densities, ρa(r) can be found from the wavefunctions calculated via the Hartree-Fock method, which are available in the data tables of Clementi et al.19 In order to use the single free-atom densities calculated from the Hatree-Fock wavefunctions as the single bulk atom densities ρa(r), the following equation is applied, ρa (r )=n s ρs (r )+n d ρd (r ) (3.16) where ns and nd are the number of electrons in the outer s and d orbitals and ρs and ρd are the densities calculated from the s and d orbital wavefunctions.19 In order to eliminate a free parameter the sum of ns and nd is fixed to 1 for Ni, Pd, and Pt and 11 for Cu, Ag, and Au, leaving the parameter ns to be fit to give the correct heats of solution of alloys.19 The pair potential is assumed to be electrostatic in nature due to the repulsion of the positively charged screened atom cores (due to electron delocalization). Thus the pair potential is Coulombic and given by, ϕ i j ( r)= Z i (r ) Z j (r ) r (3.17) where Zi and Zj are the effective charges of the atom types i and j.19 First principle calculations predict that the effective charges are strictly positive and decreases with increasing distance, r. The form of Z(r) is assumed to be, Z (r )=Z (1+β R ν )e α r (3.18) where Zo is the number of outer electrons of the atom type, Zo =1 for Ni, Pd, and Pt, Zo =11 for Cu, Ag and Au.19 Of the three remaining parameters, α and β are determined by 32

45 fitting to the shear moduli of the different material types and the vacancy formation energy.19 The final parameter ν, is empirically fit to reproduce the material's elastic constants with a chosen value of ν =1 for Ni, Pd, and Pt, and ν =2 for Cu, Ag and Au.19 The cohesive energy, bulk modulus, and equilibrium lattice constant are guaranteed to be predicted correctly for pure materials by this implementation of EAM due to the definition of the embedding function in terms of Rose's universal binding energy relation. One of the strengths of the embedded atom method is the ease in which it can handle modeling interactions between different atomic species. This allows for the simulation of alloy materials and materials with impurity and interstitial defects as long as all the alloy and defect elements involved are parametrized for the potential. 3.8 Cutoff Radius, Neighbor Lists, and Binning The potential energy expressions for the general pairwise and pair functional interatomic potentials given in equations 3.1 and 3.12 are sums of the individual energetic contributions from each atom that makes up the system. These atomic energies are found by summing the interactions each atom has with its surrounding neighbor atoms. For example the EAM potential energy can be rewritten as, N E pot = E i i E i=f i a j 1 ( ρ ( r ))+ 2 ϕ ( r j i ij ij ij ) (3.19) j i where the individual atom energy Ei is obtained by first calculating the total electron density at the site of atom i by summing contributions from its j surrounding neighbor atoms. The embedding energy is then calculated for this electron density and added to the pair function contribution which is also found by summing contributions from the neighbor atoms of atom i. Defining a specific neighbor region around an atom therefore 33

46 determines the set of atoms it interacts with. For an infinite range interaction in a system of size N, each atom interacts with the other N-1 atoms of the system. This is considered an infinite neighbor region and is computationally costly, as the force and energy calculations scale as O(N2). To lessen the computational task molecular dynamics simulations make use of short range interactions where the effective range of a potential is limited by a defined cutoff radius rcut, which reduces the scaling to O(N). The neighbor region is thus defined as a sphere of radius rcut. Interactions between atoms that are separated by a distance greater than rcut are disregarded. This cutoff radius is determined such that the disregarded interactions are negligible. In order to ensure the cutoff radius does not introduce any discontinuities in the energy or its derivatives, the potential energy is either shifted so that it converges to zero at the cutoff distance or a smoothing function is used. The summations in the individual atom energy equation can be calculated as long as the neighboring atoms within the cutoff distance of each atom are known for each step of the simulation. A simplistic approach for each time step is to first compute the separation distance, rij, between all the atom pairs in the system. Then for the energy calculation of atom i, only allow interactions with atoms for which rij rcut, i.e. those atoms within the neighbor region sphere of atom i. This method is highly inefficient as this N choose 2 combination requires N(N-1)/2 distance calculations and can therefore spoil the O(N) scaling because many of the distance calculations are unnecessary. The goal for efficiency is then to find a method that checks as few atoms as possible to see if they are within the defined neighbor region of a particular atom, minimizing the amount of wasteful distance calculations. One approach to optimize this neighbor checking process is the Verlet neighbor list method which was first proposed by Loup Verlet in the late 196s.12 As the name 34

47 suggests the strategy is to store and maintain, for each atom, a list of the nearby atoms. The initial creation of the neighbor list follows the inefficient O(N2) approach outlined above. However, once the list is formed the subsequent energy and force calculations for a particular atom only need to check the atoms of its neighbor list. This list must be maintained throughout the simulation as atoms could enter and exit an atom's neighbor region. With the forethought that the lists need to be actively updated, the Verlet method does not store neighbors within rcut to an atom's list, but stores those within a distance of rlist = rcut + δ as depicted in Figure 6. The value of δ depends on the dynamics of the system being modeled, but is typically around 2% of rcut and δ > to extend the neighbor region. Some time is wasted in checking the atoms of the list in the region between rcut and rlist,but this is substantially less than if every atom of the system was checked.12 The neighbor lists need to be rebuilt before any atom can move a distance greater than or equal to δ, with respect to any other another atom of the system. Figure 6. The Verlet neighbor list region defined as the union of the gray and yellow areas. The red center atom only interacts with atoms in the gray region. As an atom traversing the distance δ could enter another atom's interaction range defined by rcut. If an atom moves a distance less than δ, it could only enter the region between rcut 35

48 and rlist where it is ultimately neglected and makes no contribution. A fail-safe criterion to check against δ to determine if the neighbor lists need rebuilt is the sum of magnitudes of the two largest atom displacements, d1 + d2, since the last neighbor list update.12 If d1 + d2 < δ, the lists do not need rebuilt, but if d1 + d2 δ the neighbor lists need to be rebuilt as it is possible an atom moved from the white region of Figure 6 into the gray interaction region. Once the neighbor lists are created and actively maintained the advantage of this method is clear, as the main goal was to minimize the number of atoms that had to be checked when determining the set of atoms that another atom interacts with. With Verlet lists only the atoms contained in one's neighbor list need to be checked, which is a significant reduction from checking all the atoms of the system. A disadvantage of this method is its extra storage requirements. The complete neighbor list for each atom of the system must be stored along with a snapshot of the atomic configuration from the last neighbor update. For large values values of N, such as tens of thousands, the occasional rebuilding of the neighbor lists (which is O(N2)) becomes too costly and the O(N) scaling of this method begins to be spoiled.12 An alternate approach is the link-cell method which is commonly referred to as binning. For this method, during each time-step the simulation cell region is divided into cubic bins of length rcut. Each atom of the system is then a member of one of the bins that the simulation area is now partitioned into. At the most, each cubic bin is at the center of a 3bin 3bin 3bin cube where it is surrounded by 26 other bins. The neighbors within an atom's interaction range rcut, must be contained in the atom's own bin or one of its 26 neighboring bins. Thus with this binning method, only the atoms contained in 27 bins need to be checked for interactions versus checking all the atoms of the system. This routine involves two overhead operations that take it away from perfect efficiency. One is 36

49 the splitting of the simulation box and assignment of each atom to a bin needs to be repeated each simulation step. This can be compared to the overhead of rebuilding neighbor lists in Verlet method. While these neighbor lists updates are performed less frequently, they require O(N2) computational work as opposed to the binning process which is only O(N). The other is that extra atoms in the 27 bins that are not within the interaction range of an atom will still be checked, similar to the checking of atoms in the padded region between rcut and rlist of the list method. However since the region checked is local and not the entire system, this portion of each routine scales as O(N), making the binning method O(N) overall. By finding the volumes of the regions checked by the two methods discussed, the total number of extra atoms that they check can be compared. For the binning method, the atoms contained in a volume of (3rcut)3 are checked and for the list method the volume is 4π(rcut+δ )3/3. As long as δ is a reasonable value the binning method will always check more unnecessary atoms than the neighbor list method, since the volume it checks is larger. Naturally a hybrid method that eliminates the weaknesses of the two approaches would produce optimal performance. The weakness of using lists is that when they are updated, it requires an O(N2) calculation and as just discussed the main weakness of binning is the large neighbor region volume that is checked for interacting atoms. The best approach is then to incorporate binning into the neighbor list rebuilding and then continue as normal in the Verlet list routine, using the neighbor lists to check atoms and performing updates when d1 + d2 δ. Whenever an update of the neighbor lists is required, the system is divided into cubic bins of length rlist assigning each atom to one of the bins. The neighbor list of a given atom is compiled by searching for atoms that are within a distance rlist in its own bin and the 26 adjacent bins. This new method now has 37

50 O(N) neighbor list rebuilding and only checks a small number of atoms within a volume of 4π(rcut+δ )3/3 for interactions in between neighbor list rebuilds. Each unique atomic force calculation executed in the algorithms of Appendices B and C require taking the derivative of the interatomic potential as per equation 3.1. In order to form the complete interatomic potential expression with a computational cost of O(N), some routine must be implemented to make the potential short-ranged and to minimize the number of neighbors that are checked to be in an atom's interaction range. With the hybrid method, the sums over j in equation 3.19 only include atoms within a distance of rcut of a given atom. In order to find this group of atoms, a small number of extra atoms need to be checked in a neighbor list that is built with O(N) scaling. One final modification to reduce the amount of work is to take advantage of Newton's third law for all pairwise force calculations. Using this, only half of the pairwise interactions need to be computed since fij = -fji. 3.9 Parallel Computing Aspects The goal of parallel computation is to achieve increased performance by breaking a given job into smaller pieces and distributing them to separate processing cores for concurrent completion. In an ideal setting the total problem is evenly divided amongst all the cores and no additional work is required in the parallel implementation. For this special case, a job that takes a time T to be completed in serial (1 core) will take a time of T/P to be completed by P cores. The overall speedup is then defined as, S= Ts Tp (3.2) where Ts is serial run time and Tp is the parallel run time.2 For the ideal case where 38

51 Tp=Ts/P, the speedup is linear with the number of cores used, S=P. Unfortunately in reality few jobs are perfectly parallel or embarrassingly parallel, as some level of overhead must be introduced when converting serial code to parallel code. For shared memory systems a mutual exclusion mechanism will need to be used to ensure that no cores access a memory element at the same time. For distributed memory systems, where each core has its own private memory, the overhead is the inter-node communication. Additionally if the job is unbalanced across the P cores such as if one core is given 75% of the work, the overall speedup can suffer. In general, a problem is parallelizable if the order that its various constituents are completed does not matter. In high performance computing environments two forms of parallel approaches are typically utilized, task and data parallelism. In task parallelism all the tasks that are required to complete a given problem are partitioned across the available cores of the computing system. For data-parallelism, the data necessary to solve the given problem is divided amongst the cores and the same set of operations or tasks are performed by each core using the local data. As an example of the two approaches, consider the job of sorting a standard deck of 52 playing cards into piles according to their suit by four people. One approach would be to locate the cards in a central location and have each person search for and gather a pile for a single suit, i.e. person 1 is responsible for hearts, person 2 for diamonds, etc. Another approach would be to give each person 13 cards and have them sort the cards by suit into small piles. When each person is finished, their smaller common suit piles are combined. In the first approach the total job is broken into four separate tasks and one is given to each person for completion, an example of task parallelism. In the second approach which is an example of data parallelism, the data is split among the four people and they each perform the four tasks 39

52 on their own data. Most jobs are comprised of a mix of parallel and serial parts, as some tasks are well suited for parallelism and others are not. A large majority of the molecular dynamics algorithm is parallelizable such as the force calculations along with the position and velocity updates. The data parallelism approach is best suited for molecular dynamics since each task requires the results produced by the previous one. For a fixed system size, an estimate of the speedup can be obtained using Amdahl's law given by, S= Ts T 1 = s = T p t p +t s ( 1 λ)+ λ P (3.21) where P is the number of cores used, λ is the percentage of the program that is parallelized, tp is the run-time of the parallel portion of the code, and ts is the run-time of the serial portion of the code.2 If zero parallel overhead is assumed and no portion of the program is serial λ =1, linear speed up with core number is obtained as with the perfect case of equation 3.2. If the limit of equation 3.21 is taken as P, we obtain an expression for the maximum possible speedup 1/(1-λ). Even with an infinite number of cores, code that is 1% serial has a maximum possible speed up of 1. The problem is worsened if a non-perfect parallel process with overhead is considered since this effectively increases the serial portion of the code. When writing parallel code the goal for maximum speedup is then to minimize the parallel overhead and maximize the percentage of the code that is parallel. The molecular dynamics code utilized in this study is the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) developed by Steve Plimpton et al. at Sandia National Labs.21 LAMMPS uses a spatial-decomposition (SD) algorithm to parallelize the atomic force calculations along with the velocity and position 4

53 updates.21 With SD, the simulation region is divided into 3-D boxes, where the atoms within each box are assigned to a single processing core. The number of boxes along with their size and shape depend on the number of cores used and the size, shape, and number of atoms of the system being modeled. For this method the communication cost, which is proportional to the surface area of the boxes, is minimized by keeping each box as cubic as possible.21 As the atoms move throughout the simulation region they are reassigned to the processor responsible for their new location. The processor that currently owns a particular atom calculates the force on it and updates its position and velocity during each time-step. Each processor locally stores the position, velocities, neighbor lists, etc. for the atoms in its box and also stores the position of the atoms in it's neighboring boxes for the force calculations. Data must be exchanged between processors in order to obtain the position of atoms not currently in their box. This communication must be done until each processor has local access to all the positions of atoms within a distance rlist of its box. The main benefit of the SD algorithm is that communication is local, as processors only need information from a few processors that own the atoms in the surrounding boxes. This significantly reduces the overhead as compared to other parallel schemes such as the Atom-Decomposition and Force-Decomposition algorithms which require global communication, where each processor needs to communicate with every other one.21 Depending on the size of the processor grid, the required atom positions can be obtained in as little as 6 data exchanges, all of which are with the 6 nearest neighbor processors.21 The communication is handled by the message-passing interface known as MPI which can support both shared and distributed memory systems. MPI is an application programming interface which consists of a library of functions that can be directly called in programs written in C, C++, FORTAN, etc.2 MPI allows for inter-process 41

54 communication via simple send and receive functions along with global collective communication functions. Two distributed memory Beowulf computer clusters are utilized in this study, the Einstein and Penrose clusters of IUP's physics and chemistry departments. A Beowulf cluster consists of locally networked consumer grade personal computers and servers running a Unix-derived free and open source operating system. Using libraries such as MPI and specialized programs, the computers of the cluster can communicate and therefore share in the processing of a particular workload. The layout consists of a frontend node and compute or slave nodes. The frontend node which is commonly referred to as the master or head node, controls the cluster and serves the jobs to the slave nodes using a batch-queuing system such as Sun Grid Engine (SGE). Queue systems such as SGE handle the allocation of the cluster's resources and also schedule and manage the jobs distributed to the compute nodes. The head node is connected to an external network and serves as the access console for users of the cluster. It contains post processing software as well as a large storage capacity to store the results of the parallel computation. In contrast, the compute nodes are purely for computation and only execute the processes served to them by the scheduler. The Einstein cluster consists of 13 nodes with a total of 4 physical processing cores connected via a gigabit Ethernet network, 69.2GB of ram, and over 2TB of storage capacity. 3.1 Molecular Statics Any system in which dissipative forces are present will reach a minimum of the potential energy in its steady state configuration. This ensures that the entropy is maximized according to the second law of thermodynamics. In a classically modeled 42

55 system, the physics of the atomic interactions is contained in the interatomic potential energy function. This potential energy function is solely dependent on the atomic position and defines a 3N-dimensional potential energy surface. The stable equilibrium configurations occur at the minimums of the potential energy surface, where the net force on each atom is zero. Locating these minimum energy configurations is of great importance as they correspond to physically realizable stable states and can be used to probe material properties of the system. One obvious approach to finding the stationary points of a function is to set its gradient equal to zero, V(r1,...,rN)=, and solve for the satisfying configurations. Unfortunately, for a realistic 3N dimensional potential energy function, this method is tantamount to solving a system of 3N nonlinear equations, which typically is not analytically tractable. The standard numerical approach to this geometric optimization problem is to repeatedly vary the atomic coordinates from some starting configuration until a local or global minimum in the potential energy is found. The direction and distance that every atom's position is varied during each iteration in the search for a minimum depends on the specific minimization method being used. After each iteration a criterion is checked to determine if the system configuration has converged to a minimum of the potential energy. If the criterion is satisfied, the minimization routine is terminated and the current configuration is stored, if not the routine continues with the next iteration. At a minimum of the potential energy the net force on each atom is zero and a small deviation of the configuration should result in a correspondingly small change in the potential energy. This gives two criteria that can be used to determine if a minimum is reached, 43

56 F i < force tolerance value OR (3.22) V i V i 1 < energy tolerance value where Fi is the global force vector for the system configuration of the ith iteration, Vi is the potential energy of the system configuration of the ith iteration, and the energy and force tolerance values are chosen to be small enough to ensure convergence while at the same time large enough to minimize the number of required iterations. One method to determine the search direction for each iteration is to utilize the information provided by the gradient of the potential energy energy function. The negative gradient of any function points in the direction of greatest decrease. For the potential energy function, this is the direction of the net force. A reasonable choice for varying the system coordinates in the search of a minimum is then to move in the direction of the net force, i.e. each atom of the system moves in the direction of the net force acting on it. This is the basis of the Steepest Descent minimization algorithm. For each iteration the force on every atom is calculated and then they are displaced by some step size in the direction of their net force. This process continues until a convergence criterion is reached. Within this method the current system configuration Ri=(r1,...,rN) is then updated using the following equation, Ri +1= Ri γ V ( Ri ) (3.23) where γ determines the step size in the direction of the greatest decrease. This quantity is either defined to be a small constant to ensure that a step does not increase the total energy or it is determined through a series of one dimensional line searches. In these line searches, the length of the step size that each atom moves in the direction of its local force is chosen to minimize the energy for that particular direction. The system trajectory 44

57 traced by the successive configurations found by the Steepest Descent method is physical and corresponds to an overdamped or highly quenched dynamical system with a governing equation of dr/dt=- V.12 The motion of such a system is continually in the direction of the net force until the force becomes zero, at the local minimum, ceasing the motion. While the Steepest Descent approach is simple to implement and robust in its applicability, it is inefficient as many iterations are required to reach convergence. Other methods such as the Conjugate Gradient (CG) and Newton-Raphson (NR) employ more sophisticated search direction and step size routines to decrease the number of iterations required to locate a minimum. This increased efficiency comes at the cost of sacrificing the realistic system trajectory, as those produced by CG and NR are unphysical and no longer correspond to a dynamical system. Fortunately, the goal of energy minimization is simply to find the minimum energy configuration associated with an initial starting configuration, not the transition path connecting the two. Methods such as Nudged Elastic Band (NEB) can be used to find the transition path or minimum energy trajectory between two stable configurations. Pseudocode for a general energy minimization algorithm is given in Appendix F Elastic Constants In the ground state configuration of a perfect monatomic crystalline material all atoms are equivalent. In accordance with equation 3.19 for the EAM pair functional, each atom, in the bulk, contributes Ec=Ei=Epot/N, to the total potential energy of the system. If a small homogeneous strain is applied along the x-direction, the new per atom energy can be approximated by a Taylor series expansion of the cohesive energy with respect to the applied strain, 45

58 E= E c (r ij ) r + E c (r ij ) e,ij r 1ij 1 ij ( r r 1 e, ij re,ij 2 1 E c (r ij ) )+ (r 1ij r 1e,ij ) r ij r ij r (3.24) e,ij where re,ij is the equilibrium atomic separation distance, rij is the atomic separation distance after the small strain is applied, and the x components of the vectors rij and re,ij are given as r1ij and r1e,ij respectively for a coordinate system fixed to the material. In general, the strain on a plane can be decomposed into parallel and perpendicular components. The parallel components are called shear strains and those normal to the plane are called normal strains. In three dimensions (x=1, y=2, z=3) a general strain can be represented by a second rank tensor, ϵ11 ϵ12 ϵ13 ϵ= ϵ21 ϵ 22 ϵ23 ϵ31 ϵ 32 ϵ33 [ ] (3.25) where ϵij indicates a strain in the i-direction on a plane normal to the j-axis. As a notational example, ϵ11 indicates a normal strain in the x direction on a plane normal to the x-axis and ϵ21 indicates a shear strain in the y direction on a plane normal to the xaxis. Equation 3.24 can be rewritten using the ϵ11 strain tensor component, E= E c (r ij ) r + e,ij E c (r ij ) r 1ij (r re,ij 1 e, ij 2 1 E c ( r ij ) 1 2 ϵ 11)+ (r e,ij ϵ11) r 1ij r 1ij r (3.26) e,ij and the approximate atomic energy for a generalized internal strain state is given by, E= E c (r ij ) r + e,ij a,b E c (r ij ) r bij re,ij a r e,ij ϵ ab+ 2 E c (r ij ) 1 a c r e,ij ϵab r e,ij ϵcd +... b d 2 a,b, c, d r ij r ij r (3.27) e,ij At equilibrium each atom is at a local minima of the potential energy surface where the net force is zero. This equilibrium condition eliminates the second term of equation The atomic energy per unit volume (energy density) is given as, 46

59 1 U=Ω a [ E c (r ij ) r + e,ij ] 2 E c (r ij ) 1 r a ϵ r c ϵ a,b,c, d r bij r ijd r e,ij ab e,ij cd e,ij (3.28) where Ωa is the volume per atom. Equivalently the energy density can be expressed as, 2 1 E c (r ij ) C abcd = r ae,ij r ce, ij b d Ωa r r ij ij r 1 1 U= E c (r ij ) r + C abcd ϵ ab ϵcd +... Ωa 2 e,ij (3.29) e,ij where summations are implied by repeated indices from 1 to n, where n is the system dimensionality. The elastic energy, due to the distortion of the material, is the second term in the energy density, U, of equation Taking the derivative of the energy density with respect to the strain ϵab gives the generalized form of Hooke's law, U =σab=c abcd ϵ cd ϵ ab σ=c :ϵ (3.3) where σ is the second rank stress tensor, C is the fourth rank elasticity (or stiffness) tensor, and summations are again implied by repeated indices. The elasticity tensor has 34=81 components known as the elastic constants. The generalized Hooke's law can be inverted to give the strain as a function of stress, ϵ=s :σ ϵ ab=s abcd σcd (3.31) where S is the fourth rank compliance tensor comprised of 81 compliance constants in its general form. The elements of the elasticity and compliance tensors are the constants of proportionality for the linear relationship that exists between the components of the stress and strain tensors, as given by Hooke's law. These constants are fundamental quantities for a material that determine its strength and stability through its response to a small externally applied stress or strain. For a general material the number of independent components of the elasticity and compliance tensors can be significantly reduced by considering various symmetry 47

60 arguments. The first reduction which reduces the number of independent components from 81 to 36 is due to the required symmetry of the stress and strain tensors. These so called minor symmetries arise since a material in static equilibrium experiences no net force or torque. This condition reduces the number of independent off diagonal or shear terms of the stress and strain tensor from six to three. Using equation 3.3 and the symmetry condition of the stress and strain tensors, the minor symmetries of the elasticity tensor are given as, σ ab=σba σ 12=σ 21, σ13=σ 31, σ23=σ 32 C abcd =C bacd ϵ cd =ϵ dc ϵ12=ϵ 21, ϵ13 =ϵ31,ϵ 23=ϵ 32 C abcd =C abdc (3.32) C abcd =C bacd =C abdc These minor symmetries reduce the stress and strain tensors from nine to six independent components which in turn reduce the elasticity tensor from 92=81 to 62=36 components. This number can be further reduced using the definition of the energy density in terms of the elasticity tensor components given in equation Taking the second derivative of the energy density with respect to the strain components yields the so called major symmetry of the elasticity tensor, 2 U 2 U C abcd = ϵ ϵ = ϵ ϵ =C cdab ab cd cd ab (3.33) which arises since partial differentiation is commutative. This reduces the number of independent elastic constants to An analogous argument employing the major and minor symmetries of the compliance tensor can be applied to reduce the number of independent compliance constants to 21 for a general material. Voigt notation allows for the reduction of tensor rank due to symmetry.12 The tensor forms of Hooke's law given by equations 3.3 and 3.31 can be contracted into 48

61 matrix form where the elasticity and compliance tensors are 6 6 symmetric matrices.12, 23 Voigt notation summary tensor indices ab, cd : ,32 matrix index i, j : ,31 12, c ij =C abcd σi =σ ab ϵi =ϵ cd for i =1,2,3 (3.34) ϵi =2 ϵ cd for i =4,5,6 6 Hooke ' s Law : σ i=cij ϵ j = cij ϵ j c ij = j=1 σ i ϵj Equation 3.34 summaries the rules of Voigt notation for the elasticity tensor. Hooke's law then takes the elastic or compliance form presented in equation [ ][ [ ][ c 11 σ1 c 12 σ2 σ 3 = c 13 σ4 c 14 σ5 c 15 σ6 c 16 s 11 ϵ1 s 12 ϵ2 ϵ 3 = s 13 ϵ4 s 14 ϵ5 s 15 ϵ6 s 16 c12 c 22 c 23 c 24 c 25 c 26 s12 s 22 s 23 s24 s 25 s26 c13 c 23 c33 c34 c35 c36 s13 s 23 s33 s 34 s35 s 36 c 14 c 24 c 34 c 44 c 45 c 46 s 14 s 24 s 34 s 44 s 45 s 46 c 15 c 25 c 35 c 45 c 55 c 56 s 15 s 25 s 35 s 45 s 55 s 56 ][ ] ][ ] c16 c26 c36 c 46 c56 c66 s 16 s 26 s 36 s 46 s 56 s 66 ϵ1 ϵ2 ϵ3 ϵ4 ϵ5 ϵ6 (3.35) σ1 σ2 σ3 σ4 σ5 σ6 The number of independent elastic and compliance constants can still be reduced further by considering the rotational symmetry of the material. In general, as the rotational symmetry of the material increases, the number of independent constants decreases. For isotropic materials in which the properties are homogenous in all directions, there are only two independent constants. In cubic crystal materials, such as the FCC materials studied here, the four threefold axes of rotational symmetry reduce the 49

62 number of independent constants to three.22 The number of independent constants for the various crystal classes are summarized in Table 3. Table 3. Number of Independent Elastic/Compliance constants for various cyrstal systems.24,25 Crystal System Rotational Symmetry No. of Independent constants Triclinic none Monoclinic 1 twofold Orthorhombic 2 perpendicular twofold Tetragonal 1 fourfold Trigonal 1 threefold Hexagonal 1 sixfold Cubic 4 threefold Isotropic infinite or 7 6 or For cubic materials the rotational and translational symmetries imply, c 11=c 22=c 33 c 12=c 13=c 23 c 44=c55=c 66 s11=s 22=s 33 s12=s 13=s 23 s 44=s55=s 66 (3.36) where all other constants are zero This simplifies the matrix form of Hooke's law in equation 3.35 to, [ ][ [ ][ ][ ] ][ ] c 11 c 12 c 12 σ1 c 12 c 11 c 12 σ2 σ 3 = c 12 c 12 c 11 σ4 c 44 σ5 c 44 σ6 c 44 s 11 s12 s 12 ϵ1 s 12 s11 s 12 ϵ2 ϵ 3 = s 12 s12 s 11 ϵ4 s 44 ϵ5 s 44 ϵ6 s 44 ϵ1 ϵ2 ϵ3 ϵ4 ϵ5 ϵ6 (3.37) σ1 σ2 σ3 σ4 σ5 σ6 The relationship between the elastic and compliance constants can be found through the identity, c = s -1. Inverting the compliance matrix and equating its elements to those of the 5

63 elasticity matrix yields the relations, c 11= s11 + s12 (s11 s 12)(s 11+ 2s12) c 12= s12 ( s11 s 12)(s 11+ 2s12 ) c 44= 1 s 44 (3.38) For isotropic materials an additional elastic and compliance constant can be eliminated. This reduction arises since c44 and s44 are no longer independent from the other constants still remaining, c 44= c 11 c 12 2 s 44=2(s 11 s 12) (3.39) The two surviving independent elastic and compliance constants are typically replaced by the well known engineering elastic constants, the Young's modulus E and Poisson s ratio ν, through the relations, s11 = 1 E E (1 ν) c 11= (1+ ν)(1 2 ν) 2(1+ ν) E s12= ν E 2(s 11 s12 )= Eν c 12= (1+ν )(1 2 ν) c11 c12 E = 2 2(1+ ν) (3.4) with this the matrix form of Hooke's law for an isotropic material takes the form, [] σ1 σ2 E σ3 = (1+ ν)(1 2 ν) σ4 σ5 σ6 [ 1 ν ν ν ν 1 ν ν ν ν 1 ν 1 2 ν ν ν 2 51 ][ ] ϵ1 ϵ2 ϵ3 ϵ4 ϵ5 ϵ6 (3.41a)

64 [] [ ϵ1 1 ν ν ϵ2 ν 1 ν ϵ 3 = 1 ν ν 1 E 2( 1+ ν) ϵ4 2(1+ ν) ϵ5 2(1+ ν) ϵ6 ][ ] σ1 σ2 σ3 σ4 σ5 σ6 (3.41b) The special cases of plane stress and strain briefly detailed in Chapter 2 eliminate stress and strain elements and thus reduce the elastic and compliance matrices. Consider a material in which the y-dimension is large as compared to the others, a regime in which plane strain is applicable as the displacement vector in the y-direction throughout the material can be assumed as zero. With this assumption the strain elements ϵ2, ϵ4, and ϵ6 along with the shear stress elements σ4 and σ6 are also zero. It is important to note that the stress element σ2 is nonzero due to the Poisson effect of maintaining a constant material length in the y-direction.25 Applying this reduction to 3.41a, Hooke's law takes the form, [ 1 ν ν σ1 ν ν E σ2 = ν 1 ν (1+ν )(1 2 ν) σ3 1 2 ν σ5 2 [] ][ ] ϵ1 ϵ3 ϵ5 (3.42) To obtain the compliance form of Hooke's law for an isotropic material in the plane strain regime this equation along with the square matrix formed by rows 1, 3 and 4 must be inverted which gives, [] [ ϵ1 1+ ν 1 ν ν = ν 1 ν ϵ3 E 2 ϵ5 ][ ] σ1 σ3 σ5 (3.43) For the case of plane stress, a material in which the y-dimension is small as compared to the others is considered. Due to this small length only a negligibly small stress is able to form in this direction of the material.25 With this, the stress elements σ2, 52

65 σ4, and σ6 along with the shear strain elements ϵ4 and ϵ6 can be assumed as zero. Similar to the case of plane strain, even though the stress element σ2 is zero, ϵ2 remains non-zero due to the Poisson's effect. Applying the plane stress conditions to the compliance form of Hooke's law for an isotropic material given in equation 3.41b yields, [] [ ϵ1 1 ν ϵ 2 = 1 ν ν E ν 1 ϵ3 2(1+ ν) ϵ5 ][ ] σ1 σ3 σ5 (3.44) Inverting this equation with the square matrix comprised of rows 1, 3, and 4 gives Hooke's law in elastic form for plane stress, [ ][ ] 1 ν σ1 E ν 1 σ3 = 2 1 ν 1 ν σ5 2 [] ϵ1 ϵ3 ϵ5 (3.45) The effective Young's modulus, E', that enters into the critical strain and stress formulas for the thin strip geometry discussed in section 2 of chapter 2 can be obtained from the elasticity matrix for an isotropic material. For the cases of plane strain and stress, if a uniform strain is applied in the z-direction the effective Young's modulus in that direction is the 32 and 22 matrix elements of equations 3.42 and 3.45 respectively, E'= E (1 ν ) E (1 ν)2 = (1+ ν)(1 2 ν) (1 ν 2 )(1 2 ν) (plane strain) (3.46) E'= E 1 ν 2 (plane stress) which can be used in the critical stress and strain expressions of equation 2.5. It is common practice to use the effective Young's modulus given in 3.46 in the critical stress and strain expressions of equation 2.5 despite the crystalline structure of the material for 53

66 the special cases of plane strain and stress.5,26 The independent elastic and compliance constants for the crystal are averaged in such a way as to obtain a reasonable approximation for the Young's modulus and Poisson's ratio. This essentially approximates the material as polycrystalline. In general there is no unique way to determine the two engineering elastic constants from the three or more independent constants of the crystal material due to the overdetermined nature of the problem, however well known averaging methods exist.23,25 Among them, the Voigt and Reuss methods are the most widely used and theoretically justifiable as they provide an upper and lower bound respectively for the elastic and shear moduli. In the Voigt method the average is performed under constant strain conditions while the Reuss method is performed under constant stress. Given that these two methods establish the maximum and minimum possible values of the various elastic moduli, Hill proposed taking the arithmetic mean of the two methods which has been shown to provide a more reliable estimate.23 The equations that follow from the combined Voigt-Reuss-Hill averaging method for a cubic material are given as, EV = (c 11+ 2c12)( c11 c c44 ) 2c11+ 3c12 +c 44 GV = c11 c12 +3c 44 5 G R= E R= 5 3s11+ 2s12 + s ( s11 s 12)+3s 44 νvrh = E VRH = GVRH = EV + E R 2 G V +G R 2 (3.47) E VRH 1 2G VRH where EVRH and νvrh are the Voigt-Reuss-Hill approximation values for the Young's modulus and Poisson's ratio to be used in the determination of the effective Young's modulus in equation For a cubic crystal material subject to fixed displacement loading (constant strain) in the z direction only, one term of the elastic energy density (equation 3.29) is nonzero, 54

67 1 2 U elastic= c 11 ϵ 3 2 (3.48) since no other external forces act along any other direction to do work and add energy to the system. This is precisely the energy density available in the thin-strip geometry system ahead of the crack tip. Therefore, the entire averaging method and polycrystalline assumption to obtain an estimate of two elastic constants from three can be circumvented simply by using c11 as the effective Young's modulus in equation 2.2, E'=c11. For a thinstrip system uniformly strained in the z-direction, the energy release rate (strain energy released per unit area of crack advance) is obtained by multiplying equation 3.48 by the unstrained height of the material. The calculation of the elastic constants of a material is most simply performed using a molecular statics approach which is effectively performed at zero temperature. The differential form of Hooke's law given in equation 3.34 can be utilized in its difference quotient form, Δσ c ij = Δ ϵ i j (3.49) where Voigt notation is used. The general procedure for the calculation of the elastic constants is as follows. The system is first minimized and the initial stress tensor comprised of the 6 independent elements is calculated and stored. The material is then strained in a direction corresponding to an element of the strain tensor, such as the xdirection for the ϵ1 element, while all other strain elements remain zero. The system is then minimized again and the stress tensor is recalculated and stored. Repeating this procedure for each of the six independent strain elements and applying equation 3.49, the elements of the elasticity matrix can be found. 55

68 3.12 Surface Energy As a crack propagates through a material new free surfaces are created in the wake of the crack tip. The crack resistance term of Chapter 2 is the energetic cost per unit area of crack advance. This term expectedly depends on the surface energy (energy per unit area) of the material along the direction that the crack is traveling. The surface energy can be calculated by dividing the difference in the system energy before and after the creation of a surface by the area of the surface. Typically in atomistic simulations a system is created with the surface plane in question oriented along two of the boundaries of the simulation box region. The system is minimized with periodic boundaries in all directions. The periodicity is then removed along the boundaries adjacent to the surface plane under consideration and the system is minimized again and the energy recalculated. The surface energy can then be found by applying, γ s= E f E i 2A (3.5) where the factor of 2 accounts for the two surfaces created when the periodicity is removed in a given direction Temperature The macroscopic temperature of a three dimensional system is related to the average kinetic energy through the relation, 3 K = κ = N k B T 2 (3.51) where K is the average kinetic energy of the system, κ is the instantaneous kinetic energy, and T is the macroscopic average temperature of this system. This equation follows from 56

69 the equipartition theorem and the ergodic hypothesis of statistical mechanics. The instantaneous kinetic energy for a given simulation time-step is given by, N κ= 1 m r 2 2 i i i (3.52) Combining equations 3.51 and 3.52, the average and instantaneous system temperature can be calculated through the relation, T = = K= κ = 3N kb 3 N kb 3N kb N i mi r i 2 (3.53) where T is the instantaneous system temperature for a given time-step Atomic Virial Stress In order to analyze the stress field near the crack tip, a local measure of atomic stress is necessary. Employing the virial theorem which relates the average kinetic and potential energy of the system, an instantaneous atomic virial stress tensor can be defined as, [ r l ki rl k j 1 i j ' h 'a ' h 'a ' σij = k mk r k r k + [ F l (ρl )ρk (r l k )+ F k (ρk )ρl ( r l k )+ ϕl k (r l k ) ] rl k V l k k ] (3.54) for the EAM interaction where rlki is the ith component of rlk, r k i is the ith component of r k, mk and Vk are the atomic mass and volume of atom k, and the summation is over the interaction neighbors of atom k.28 The applicability, validly, and interpretation of equation 3.54 has recently come into question.12,28-31 Firstly the atomic virial stress tensor defined in equation 3.54 is not equivalent to the continuum based macroscopic Cauchy stress tensor defined in section 11 of this chapter. This would be akin to interpreting the instantaneous temperature of a 57

70 single atom as the macroscopic system temperature. The local atomic virial stress tensor converges to the Cauchy stress tensor when averaged spatially over the system and temporally over the simulation duration.28,32 Zhou et al. recently claimed that the virial stress could not be averaged to correctly represent the Cauchy stress and questioned the validity of the kinetic term.12,3,31 These claims were later found to be erroneous due to a discrepancy in the definition of the coordinate system which lead to a confusion between absolute and relative velocity terms.12,31 Aside from this, the atomic volume is an ill-defined quantity especially near material voids and surfaces. Typically it is calculated indirectly by dividing the volume of a perfect unit cell of the material by the number of atoms in the unit cell. This method does not account for the change in atomic volume that occurs when the material is deformed, which is especially important in the high stress/strain region in the vicinity of the crack tip. A voronoi cell tessellation approach has also been used for defining an atomic volume. In this method planes are drawn that perpendicularly bisect the lines connecting each atom to its nearest neighbors. This creates a closed volume region around each atom which is essentially closer to it than any other atom of the system. Since this atomic volume quantity is recalculated each simulation time-step, changes due to material deformation are accounted for, however inaccuracies still arise near surfaces and material voids. Using the atomic virial stress tensor, an expression for the Cauchy stress tensor can be obtained by averaging over the volume of the system, σij = 1 V ( N k m k r k i r k j + [ F 'l (ρhl )ρ'k a (r l k )+ F 'k (ρhk )ρ'l a (r l k )+ϕ'l k ( r l k ) ] l k r l ki r l k j rl k ) (3.55) where the ensemble or phase average can be replaced by a time average over a molecular 58

71 dynamics simulation interval as per the ergodic hypothesis. The total pressure is found by summing the diagonal elements of the Cauchy stress tensor, P= = 1 1 tr(σ)= (σ11 +σ 22+σ 33) V N ( mk r k 2+ [ F 'l (ρhl )ρ'ka (r l k )+ F 'k (ρhk )ρ'l a (r l k )+ϕ'l k ( r l k )] r l k) k l k N 1 = 3 N k B T + [ F 'l (ρlh)ρ'ka (r l k )+ F 'k (ρhk )ρ'l a (r l k )+ ϕ'l k (r l k ) ] r l k 3V k l k (3.56) where the factor of 3 accounts for the dimensionality of the system Crack Tip Location and Velocity The crack tip is located by monitoring the distance between the atoms in the upper and lower crack planes. A separation criterion is established and the material is searched along the crack plane coming from the intact material for the first atomic separation that exceeds the criterion. Once the separation criterion is calibrated, this method locates the crack tip position to within a single nearest neighbor distance. Using the position data, the crack tip velocity is found by, v i +1= xi +1 xi t i +1 t i (3.57) With this method the position of the crack tip evolves in discrete jumps from one atomic layer to the next. This in turn can cause the calculated velocity to become partially oscillatory in the steady state Crack System and Defect Configurations The (1)[1] crack system is studied, in which the crack propagates in the [1] 59

72 Figure 7. The crack system geometry utilized for atomic simulations, with dimensions of 72.24Å 14.8Å Å for Ni. direction along the (1) plane. In several atomistic studies, FCC metals have been shown to exhibit a brittle nature for this particular crack system at low temperatures with periodic boundaries along the thickness of the system (y-direction).5,33 To ensure that the functional form of the energy release rate for the thin strip geometry is applicable, a system length to height ratio of at least 4 (L 4h) is used and the crack tip of the initial starter crack is located a distance of L/4 from the system's edge.33 For the study of impurity defects, a base nickel lattice comprised of 16 atoms with dimensions of 72.24Å 14.8Å Å is used, as shown in Figure 7. The simulation box is periodic in the y-direction [1]. This constrains the strain in the y-direction to be zero, a regime in which plane strain conditions apply. The starter crack region is comprised of 1 atomic layers above and below the crack plane as depicted in blue and bluish green in Figure 7. The interaction between these two groups of atoms is turned off, severing the atomic bonds and allowing for the formation of a seed crack in the material. The atoms in the last layer perpendicular to the x-direction (right edge) are constrained to move only in y and z-directions. This prevents the edge from bowing inwards due to the material's nonzero Poisson s ratio minimizing its interaction with the crack until the crack tip is 23 atomic layers from the edge. Two Stokes' damping regions are present in the system to absorb the acoustic/shock waves emitted from the crack tip. These regions consist of the first and 6

73 last 1 atomic layers perpendicular to the x-direction in which a drag force proportional to the atom's velocity, Fd=-bv where b=.1 (ev*ps/å2), acts to drain energy out of the system. If the excess energy emitted by the crack tip that does not go into the formation of new free surfaces is not dissipated, it can return to the crack tip and severely alter the stress field, cause blunting, produce instantaneous voids, etc, that annihilates the steady state motion of the crack and the brittle nature of the material.7,9,1 In addition, temperature fluctuations can cause a crack to be trapped or can knock a crack out of a trapped state by instantaneously increasing or decreasing the energy release rate in the vicinity of the crack tip by producing material voids, kinks, dislocations, and blunting the crack tip.7,9,1 The crack tip is not only a concentrator of stress and strain, but also is a region of high temperature. For this reason an NVT ensemble is implemented with a target temperature of 1K and damping parameter of.1ps, in order to minimize the aforementioned undesired effects of temperature fluctuations. Three impurity defect configurations are investigated in Nickel in which the native atomic species at a lattice point is replaced by a differing species, as shown in red in Figure 8. In configurations A and B, four defect lines comprised of 4 impurity atoms each are placed throughout the thickness of the material. In configuration A, the defect lines are grouped together along the upper and lower crack planes. The pairs of defect lines in configuration B are separated by 4 atomic layers in the z-direction and lie off the upper and lower crack planes. In configuration C, two defect lines are again grouped together on the upper and lower crack planes along with two pairs both above and below separated each by 3 atomic layers in the z-direction. The critical load required to initiate crack propagation is first found for the defect free material. This is then recalculated for 61

74 Figure 8. Impurity defect configurations considered in atomistic simulations. The impurity atoms are shown in red. the three defect configurations with varying impurity defect species (Cu, Au, Ag, Pd and/or Pt) and distances from the initial crack tip. The stress fields and strain energy of the atoms along the crack plane are analyzed to interpret the resulting changes in the critical loads along with the energy path of the system Material Loading and Simulation Methods In atomistic simulations of the thin strip geometry, two fixed grip loading methods are commonly utilized to apply a strain to a system. In the first approach, to apply a strain of δ/z in the z-direction of the system in Figure 7, the top-most and bottom-most layers are displaced upwards and downwards by a distance δ/2 respectively. The atoms in these 62

75 two layers are then held rigidly fixed in all directions. The atomic layers in-between subsequently respond and after a relaxation period the system reaches an equilibrium configuration in which the internal stress and strain fields are static and far away from any material defects the strain is uniform and equal to the applied strain. While this method closely resembles the experimental method of loading a material, the abrupt displacement of two atomic layers generates undesired shock waves that travel through the material.7,1 In lab experiments a relaxation time of minutes is allowed in order for the waves to dissipate and the internal fields and thus atomic positions to reach their steady state configurations.34 This timescale is completely inaccessible for even classical molecular dynamics simulations of small systems. In order to decrease this relaxation time to a duration accessible to simulation, damping regions are used to absorb the generated waves and the strain is applied in very small increments. For example applying a total strain of δ/z, can be implemented in 1 smaller steps each with total displacements of δ/1. With the second method the strain is applied by directly scaling the atomic positions in the direction of the strain. The atoms of the top-most and bottom-most layers are then held fixed (in all directions) to maintain the strain. When a strain of δ/z is desired in the z-direction, the z coordinate of each atom is simply scaled and multiplied by 1+δ/z. This scaling method has the benefit of not producing shock waves or altering the general shape of the strain field and thus requires minimal time to reach a steady state configuration.7,1,33 It should however be noted that the shape of the strain and stress fields are only simply scaled if there are no instantaneous defects present in the material due to the thermal fluctuations, as previously discussed, on the time step the scaling is applied. For example if a material void is present in front of the crack tip, up scaling the 63

76 coordinates to apply a strain creates a larger void, which may not have occurred in the normal time evolution of the system. The impact on the internal fields near the void is unexpected to behave linearly and they therefore could change in overall shape. This in turn could result in either trapping or detrapping a crack, thus altering the value of the measured critical load required for the initiation of crack propagation. For this reason before scaling a minimization must be completed to eliminate any temperature related defects in the material. For the atomistic simulations performed in this study both loading methods are utilized. The material is first uniformly strained in the z-direction by scaling the atomic coordinates. This initial strain value corresponds to the energy release rate or load equal to the Griffith value, Go=2γ1 (equation 2.4), of the material. The Griffith load is the theoretical minimum value required to initiate crack propagation in a perfectly brittle material, equal to twice the energetic cost per unit area of new free surface creation along crack plane. When loaded to Go there is sufficient energy stored ahead of the crack tip to sever the material in half, breaking all the bonds along the crack plane. However, the Griffith load is widely known to underestimate the actual critical load for initiation of a static crack, Gc, even for a brittle material due to trapping of the crack tip. With trapping, a periodic energy barrier exists for the incremental extension of the crack from one atomic layer to the next. While trapping is attributed to the discreteness of the crystal lattice of the material (lattice trapping) other aspects affect the shape and height of the barrier such as material defects.7,1 Using multiple constrained energy minimizations in succession, the energy paths for the extension of a crack for three differing applied loads were mapped and are depicted in Figure 9. The horizontal axis is the reaction coordinate, which is a measure of 64

77 Figure 9. Energy pathway for successive bond breaking during crack extension for differing loads (energy release rates). 65

78 how far the system is along the reaction pathway of the crack extending by multiple layers in the horizontal x-direction. The energy or trapping barrier for the extension of the crack by one atomic layer can be identified as the path between two adjacent local minima. Two competing energetic processes are at play during crack extension, as one acts to increase the system energy while the other acts to decrease it. The breaking of bonds creates new free surfaces which increases the total energy of the system. The extension also allows the portion of the previously strained material to relax which in turn decreases the total energy. It is the interplay of these two energetic quantities that leads to the energy barrier.35 When the applied load is equal to the Griffith load, G=Go, the stable configurations (minima) are nearly equal in energy on either side of the energy barrier indicating that the competing energetic processes eventually completely cancel. However, the presence of the barrier implies that initially during the extension the bond breaking energy was dominate, leading to an increase in total energy. After the peak of the barrier the rate of change of the relaxation energy became larger in magnitude than that of the bond energy leading to the decrease in energy down the barrier. Bernstein and Hess explicitly calculated the two energetic contributions to the energy barrier, their result which illustrates this interplay, is reproduced in Figure 1.35 It is important to note that the presence of defects can locally alter the relaxation energy and bond breaking energy in the vicinity of the crack tip, thus impacting the trapping barrier. Two additional observations can be made from Figure 9. Firstly the lattice trapping barrier decreases with load, G. This decrease will continue until the critical value Gc is reached where the barrier disappears and crack propagation initiates in the material (if thermal excitations do not cause initiation beforehand). At this critical load, there is enough energy being funneled to the crack tip to not only break bonds in the creation of the crack surfaces, but there is 66

79 Figure 1. Lattice trapping barrier with individual energetic contributions; γs bond breaking energy and Eel elastic relaxation. This figure is reproduced with express written permission from the authors.35 an excess which is dissipated via the emission of phonons by the crack tip.7,1 Secondly, stable configurations exist on both sides of the energy barriers for all three loads depicted. For the case that satisfies G > Go, the extension of the crack is energetically favorable if the system can overcome the energy barrier since the total energy is decreased (forward barrier < reverse barrier). Through a process known as creeping, the crack can move forward via thermal fluctuations that excite the system over the energy barrier. At any non-zero temperature there is a finite forward creep rate of the crack proportional to exp[eb/kbt], where Eb is trapping barrier, when the energy release rate is above the Griffith value.7,1 After the system is uniformly strained to a load corresponding to Go via scaling, the system is relaxed through energy minimization with a force tolerance convergence criterion of (ev/ Å). The load is then increased by.5go using the top and bottom layer displacement method. Equation 2.4 is used to find the associated strain at the new load, ϵ=(2g/e'zo)1/2. The total displacement is calculated using the strain values, 67

80 δ=zo(ϵf - ϵi). In this first displacement step, Go and 1.5Go are used for G in the calculations of ϵi and ϵf respectively. A 15ps molecular dynamics simulation is then performed using a true NVT ensemble via a Nose-Hoover thermostat with a damping parameter of.1ps, a target temperature of 1K, and time-step of 1fs. The position, stress tensor, voronoi cell volume, potential energy, and coordination of each atom along with the system temperature and potential energy is recorded every 1 time-steps. The system trajectory and various physical quantities are then analyzed using a combination of visualization software (Xmovie and Ovito), custom shell scripts, and Mathematica code. If crack propagation was not initiated by the end of the run, the load is subsequently increased again by another.5go and a new MD run is performed. This process is repeated until a load is reached in which the crack begins to propagate. This value is defined as the critical load for the initiation of crack propagation in the material, Gc. 68

81 CHAPTER 4 RESULTS AND DISCUSSION The EAM interatomic potential is first verified by calculating the cohesive energy, lattice constant, surface energy, and elastic constants for six FCC metals Ag, Au, Cu, Ni, Pd, and Pt using molecular statics. The critical load, Gc, is then calculated for pure (no defects) Ni, Cu, and Au for the 16 atom (1)[1] thin strip crack system (Figure 7). Defect impurities of differing species, configuration, and distance from the crack tip are placed in the pure materials and the critical loads recalculated. By comparing the stress field and strain energy along with the trapping energy barrier in the vicinity of the crack tip for the pure and defect cases, the resulting changes in critical loads can be explained and interpreted. 4.1 EAM Potential Verification The cohesive energy and lattice constants are determined by minimizing a bulk system (periodic boundaries in all directions). The cohesive energy is determined by dividing the minimized potential energy by the number of atoms in the system. Calculating the nearest neighbor distance and multiplying by 2 yields the lattice constant for a material with FCC crystal structure. The elastic constants and surface energy are calculated following the methods outlined in Chapter 3. Table 4 summarizes the results, which are in good agreement with the values reported in the original implementation of the u3 EAM potential.19 The values of the elastic constants along with the surface energy are used to calculate the effective Young's modulus and Griffith load. 69

82 Table 4. Various material properties calculated using the EAM potential. Element Ni Au Ag Cu Pd Pt Lattice Atomic Radii Constant (Å) (Å) Ecoh (ev) c11 (GPa) c12 (GPa) c44 (GPa) γ1 (J/m2) Fracture Initiation in a Defect Free Material The critical loads calculated for pure nickel, copper, and gold are presented in Table 5, where Go=2γ1. As expected, for each material the critical load exceeds the Table 5. Calculated critical loads for crack initiation in some FCC metals. Element Critical Load Ni 1.9Go Cu 1.8Go Au 1.5Go Griffith value. This discrepancy is attributed to the trapping phenomena as previously discussed. In two previous atomistic studies, the critical energy release rate for Nickel has been calculated and reported as 1.2Go, by Karimi et al., and 1.4Go, by Gumbsch et al...5,33 Both papers utilized a different temperature control and material loading method than the study presented here. An NVT approximant ensemble was implemented in which the temperature is controlled by a Berendsen thermostat with regions of differing target temperatures. In the region containing the crack tip the desired temperature was set to 1K. In addition, the scaling method was utilized to strain the material throughout the simulations including the application of additional strain. The 7

83 study by Gumbsch et al. used the same EAM potential, by Foiles et al., as in this study. Karimi et al. utilized a different parametrization of the EAM potential developed Mishin et al. Since these two studies used exactly the same simulation setup, the discrepancy in their calculated critical load was attributed to their only difference, the EAM implementations.5 Many of the physical quantities predicted by the potential such as the trapping barrier can be altered by the parametrization potential. Comparing this present study to Gumbsch et al., the differing temperature control can account for the increase in the calculated critical load for Ni. Using a lower system temperature along with a different temperature control results in altered temperature fluctuations. Generally a lower temperature will result in smaller temperature fluctuations. With the order of magnitude decrease in temperature, the thermal excitation (or partial excitation) of the system over the lattice trapping barrier is less likely to occur. It will therefore require a larger applied critical load to overcome the equally sized lattice trapping barrier at a lower temperature. This increase in critical load over the result of Gumbsch et al., is not enough to reach the value of 1.2Go by Karimi et al. due to the different EAM potential. It is however expected, that if the present simulation was implemented utilizing the EAM potential of Mishin et al., a critical load greater than that of Karimi et al. would be calculated due to the lower simulation temperature. A graph of the crack tip position vs. time for a run in pure Ni loaded to the critical value is shown in Figure 11. This plot represents the typical behavior of the crack tip displacement in all simulations, once the crack begins to propagate. After an initial acceleration phase, the crack tip reaches a steady state energy flux and propagates at a constant velocity. In this case the slope yields a velocity equal to m/s. 71

84 Figure 11. Crack tip displacement for 5ps of a simulation run in pure Ni with a load of Gc. As previously discussed, the crack tip acts to amplify and concentrate the internal stress that is present in a material due to an externally applied stress. The strain energy released during the relaxation is funneled to the crack tip where a fraction is consumed in the process of breaking bonds and the rest is emitted and eventually dissipated in the material. The high strain and subsequent bond breaking in this area makes it a region of high temperature. Figure 12 depicts two images in the vicinity of the crack tip with atoms colored according to their temperature and σzz stress. This atomic coloring is scaled from the smallest (blue) to largest (red) values. These images serve to confirm that the atoms highest in temperature and stress are located around the crack tip. 72

85 Figure 12. Snapshot in the vicinity of the crack tip during a simulation with atoms colored according to (a) atomistic temperature and (b) atomic stress in the z-direction σzz. The coloring is scaled from blue (smallest values) to red (largest values). 4.3 Fracture Initiation in the Presence of Impurity Defects at the Crack Tip The presence of defects can disrupt the existing homogeneity and underlying symmetry of a material and thus directly alter the internal stress and strain fields. When defects are in the vicinity of an atomically sharp crack tip, they can affect the peak stress and alter the trapping barrier due to their direct affect on its two energetic contributions. Each of these in turn have an impact on the critical load required for the initiation of crack propagation. Impurity defects consisting of Cu, Pd, Pt, Ag, and Au were first placed in Ni in the three configurations depicted in Figure 8. The critical loads were recalculated with the various impurities present and are reported in Table 6. It is observed that the critical load is increased for each defect configuration and impurity species as compared to the result for pure Ni. Loosely, the critical load increases with the atomic radius of the defect species, although exceptions to this exist for each configuration particularly for 73