Multiscale Modelling of Materials and Radiation Damage: Theory and Examples

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1 Perlado 2014 Instituto Fusion Nuclear UPM Multiscale Modelling of Materials and Radiation Damage: Theory and Examples Prof. J. M. Perlado Director Institute Nuclear Fusion, UPM

2 Space scales in materials and living systems

3 The two large problems with materials in irradiation environments Activation consequence = Radioactivity, with effects in Safety and Environment Damage consequence = changes in expected design properties of the materials Affecting Fission, Fusion, Transmutation systems

4 Cascades in fcc metals 35 ps Cu 20keV > Cu Au 20keV > Cu F.C.C. materials (low stacking fault energy) result in the formation of both vacancy and interstitial clusters at the end of the collision cascade Vacancies Interstitials Perlado 2014 Instituto Fusion Nuclear UPM Pb 30KeV > Pb Interstitials Vacancies

5 Evolution of the Microstructure After the generation of defects, those follow annealing /thermal, recombination; diffuse, agglomerate among them, dissociate.and finally Complex secondary defects: Small defect clusters Interstitial dislocation loops Vacancy dislocation loops Stacking fault tetrahedra Precipitates Voids He bubbles Dislocation loops Voids Stacking fault tetrahedra 200 nm 10 nm 50 nm

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9 Mechanisms in Neutron Damage: Modeling and Diagnosis Nuclear Reaction: the first process Atom Probe

10 Atomistic Scale: MULTISCALE MODELLING: COMPUTACIONAL Pure Quantum-Mechanics Mehods ( ab initio ) Quantum-Mechanics Mehods Tight Binding (IFN code development) Classical Molecular Dynamics (IFN code development) Microscopic Scale: Rate Theory (different approaches) Kinetic MonteCarlo (different solutions with/without lattice, IFN code development PARALLEL Mesoscpic Scale: Dislocation Dynamics (present collaboration with LLNL/UCLA) Macroscopic: Finite Elements Working Area of IFN

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12 Perlado 2014 Instituto Fusion Nuclear UPM Ab Initio Simulations of Matter The problem in its basic and fundamental form from Quantum Mechanics considers N nuclei with coordinates R 1,...R n R, and momenta P 1,...P n P, and N e electrons with coordinates r 1,...r n R, and momenta p 1,...p n R, and spin variables s 1,...s n s, with Hamiltonian We want to obtain eigenfunctions and eigenvalues of Schrödinger s: with nuclear and electron kinetic energies, and electron-electron, nuclear-nuclear, and electron-nuclear interactions operators. AN EXACT SOLUTION OF THIS EQUATION IS EXTREMELY DIFFICULT!

13 Perlado 2014 Instituto Fusion Nuclear UPM Ab Initio Simulations of Matter: Born-Oppenheimer This approximation consists in the recognition that there is a strong separation between the times scales of the nuclear and electronic motion, since the electrons are lighter than nuclei (three orders of magnitude), then: with electronic and nuclear wave functions respectively, in which the electronic wave function depends parametrically on the nuclear position. The Born-Oppenheimer approximation indicates that neglected. can be The reason is that the nuclear wave function (R) is more localized than the electronic wave function; in consequence, I >> I, and operating:

14 Perlado 2014 Instituto Fusion Nuclear UPM DENSITY FUNCTIONAL THEORY from Hohenberg-Kohn theorem 1998 Nobel Prize in Chemistry to Walter Kohn and John Pople The ground state energy 0 (R) at a given nuclear configuration, R, is obtained by minimizing a certain functional, (n), which depends on the electron density. That functional assumes its minimum at the ground state electron density. Equation 3 [Hohenberg and Kohn, Phys.Rev. 136 (1964) 864]: A convenient form of this functional is given by the scheme of KOHN-SHAM where a doubly occupied single-states, i (R), i=1,...n e /2, each containing an spin up and down is introduced. [W. Kohn and L. J. Sham, Phys Rev. 140 (1965) 1133] The density and functional are expressed: Unknow n Quantum kinetic energy Direct Coulomb term from Hartree-Fock theory Exact exchange and correlation Interaction of electron density with external potentials

15 Perlado 2014 Instituto Fusion Nuclear UPM Examples of Results Phase transformations of Si, Ge from Yin and Cohen (1982) Needs and Mujica (1995)

16 A new phase of Nitrogen Published in Nature. Dense, metastable semiconductor Predicted by theory ~10 years ago! Molecular form Mailhiot, et al 1992 Cubic Gauche Polymeric form with 3 coordination

17 Ab initio calculations of Equation of States (Hydrogen at very high pressure ) C. Guerrero, J. M. Perlado, EPL G33392/B16589 (2014) Perlado 2014 Instituto Fusion Nuclear UPM

18 Ab initio Calculation of Migration Energy of <110> SIA in Fe

19 Characterization of defects (Si-O 2.15 Å) Good agreement with experiments

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21 Perlado 2014 Instituto Fusion Nuclear UPM Binary Collision Approximation Interatomic Potentials: Screened Coulomb Potentials: Moliere, Firsov. Universal Potential from Ziegler, Biersack and Littmark (ZBL) using Hartree-Fock-Slater (HFS) atomic charge distributions: u a u 4 c i a i exp( d 0 ( Z i r a u ) Z ) 1

22 Perlado 2014 Instituto Fusion Nuclear UPM TRIM and Marlowe These are binary collision codes that provide a good initial picture of the cascade In SRIM/TRIM (Ziegler and Biersack), the ion and target atom have a screened Coulomb interaction during the collisions, including exchange and correlation interactions between the overlapping electron shells MARLOWE (Robinson) simulates atomic collisions in crystalline targets using the binary collision approximation and follows all moving atoms until they reach E d

23 Fe-implanted Fe (300 MeV) Perlado 2014 Instituto Fusion Nuclear UPM

24 Fe-implanted Fe (300 KeV) Perlado 2014 Instituto Fusion Nuclear UPM

25 Perlado 2014 Instituto Fusion Nuclear UPM Using TRIM code to assess the recoil spectra and penetration in Simulation of Experiment VENUS-II (150 Fe + in Fe) J.Marian, J.M. Perlado

26 Seeger (1958) Perlado 2014 Instituto Fusion Nuclear UPM

27 a 0 = nm <1,0,0> Cascade formation by MARLOWE 200 kev Cu in Cu / J.M. Perlado, J. Sanz et al JNM (1992) A part of study for comparison of defect structures between Fusion and Spallation Neutron Spectra Perlado 2014 Instituto Fusion Nuclear UPM

28 Perlado 2014 Instituto Fusion Nuclear UPM MOLECULAR DYNAMICS Integration of the Newton s equations of motion for all the atoms in a computational cell Trajectories Interaction between the particles through an empirical potential...for a conservative potential: Fi V 2 d r Fi 2 dt m i ( r )

29 Perlado 2014 Instituto Fusion Nuclear UPM MOLECULAR DYNAMICS Electronic stopping implemented in MD (viscous force) dv i m Fi vi dt where is given by Lindhard by 0.857NZ ( Z Z 2/3 1 7/6 1 2/3 2 Z ) 2 3/ 2

30 Perlado 2014 Instituto Fusion Nuclear UPM Critical Part!!!!!! MOLECULAR DYNAMICS Interatomic Potentials For displacement cascades simulations a range from a few ev to several kev needs to be covered. Force models based on semi empirical many body potentials for interactions not far from equilibrium, and pair potentials (Universal) for high energy range. Attention to fitting...!!!!!

31 Perlado 2014 Instituto Fusion Nuclear UPM MOLECULAR DYNAMICS Interatomic Potentials Some more used: Isotropic Embedded Atom Model (EAM) and modifications (Metals) Finnis- Sinclair method (alternative to EAM) Johnson-Oh (EAM for BCC structures) Non-isotropic, useful for covalent bonding (highly directional) Stillinger-Weber Tersoff Pearson (Born-Meyer + Axilrod-Teller)

32 Perlado 2014 Instituto Fusion Nuclear UPM MOLECULAR DYNAMICS Interatomic Potentials (some references) EAM model: M.S. Daw and M.I. Baskes, Phys. Rev. B 29 (1984) S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) EAM model other methods / models: M.W. Finnis and J.E. Sinclair, Philos. Mag. A 50 (1984) 45. R.A. Johnson, D.J. Oh, J. Mater. Res. 4 (1989) For Covalent Bonding (highly directional) F. H. Stillinger, T.A. Weber, Phys. Rev. B 31(1985) 5262 J. Tersoff, Phys. Rev. B 39 (1989) 5566

33 Perlado 2014 Instituto Fusion Nuclear UPM MOLECULAR DYNAMICS Embedded Atom Model (EAM) Metal as positively charged ions embedded in a local electron density. Energy of the system from an embedding energy and the ion core repulsion. Assumption from density functional theory... Total electronic energy is a unique function of the local electron density in which the ion is embedded Local density as superposition of atomic densities of surrounding atoms 1 E ( ) tot F i ( Ri, j ) 2 j i, j( i)

34 MOLECULAR DYNAMICS Integration of the equations of motion Fourth order Predictor- Corrector method. Given values at time t, we compute at time t+t Predicted values (not included the equations of motion): ) ( ) ( ) ( ) ( ) ( ) ( 2 1 ) ( ) ( ) ( ) ( 6 1 ) ( 2 1 ) ( ) ( ) ( t b t t b t tb t a t t a t b t t ta t v t t v t b t t a t t tv t r t t r p p p p Other very popular integrator for these problems is that of VERLET (1960) Perlado 2014 Instituto Fusion Nuclear UPM

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36 Consequences of the Interaction of Radiation in Materials Electrons = simple Frenkel pairs Ions/Neutrons/Gammas = Atomic displacement cascades 28.6 nm point structure defects: vacancies, intersitials clusters of vacancies, clusters of interstitials segregation of alloying elements Damage in fused silica: DENSIFICATION 5 kev recoil (from M.J. Caturla et al.) 0.10 ps 0.16 ps 2.67 ps Large production of Oxygen Deficient Centers produced along the cascade tracks during the cascade. Residual defects observed after the cascade.

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38 g(r) Example: 20 kev recoil of Cu in Cu Vacancy clusters generated by a cascade Vacancies Interstitials 20 kev Cu in Cu Vacancy cluster Interstitial clusters Temperatures (>> T m ) at short time (ps) Stress distribution 2.5ps y (a o ) x (a o ) Temperature (K) Cu 20keV on Cu at 1.5 ps (Maximum damage) Perfect crystal at 300K Box (6.4nm) r (Å) Compressive Stress Tensile Stress Perlado 2014 Instituto Fusion Nuclear UPM 89 ps y (a o ) x (a o ) Hydrostatic Stress (GPa)

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41 Alurralde et al JNM (1990) Perlado 2014 Instituto Fusion Nuclear UPM

42 Validation of cascade simulation: nanoscale melting Non equilibrium nanoprecipitates of ZrO 2 are observed in the ZrO 2 - SiO 2 due to the fast cooling of the liquid droplet (cascade). Zinkle et al., Nature 395 (1998) 56 Perlado 2014 Instituto Fusion Nuclear UPM

43 Validation of cascade simulation: subcascade behavior Fission (0.1-3 MeV) Equivalency of damage produced by fission and fusion neutrons due to subcascade formation (also valid for other high energy particles) Fusion (14 MeV) 590 MeV protons Perlado 2014 Instituto Fusion Nuclear UPM

44 Time [ps] Event Result Parameters < 1 PKA: transfer of recoil energy Formation of displacement cascade Spike formation and relaxation 3-10 Core solidification and cooling t> 10 Thermal escape of interstitials and vacancies Reactions of the moving defects Lattice local disorder T PKA T dam dσ/dt Depleted (vacancies) Interstitial ejection Molten region Shock front Stable SIAs Atomic mixing Vacancy collapse Disordered zone Amorphous zone zone Thermal escape of interstitials and vacancies Reactions of the moving defects Perlado 2014 Instituto Fusion Nuclear UPM N d n sc : avge. number of subcascades e-ph coupling Spike temperature Max. melt volume Max. melt lifetime Atomic mixing efficiency Irradiation temperature

45 Visualization of the objects produced in MD simulations Experiments e.g. Stacking fault tetrahedra in irradiated copper? Simulations 2 nm 50 nm Cu 0.01 dpa RT weak beam g(6g) g = (200) Molecular dynamics simulation Pair potential method, atoms Schaublin Perlado 2014 Instituto Fusion Nuclear UPM

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47 Perlado 2014 Instituto Fusion Nuclear UPM Large loops in ferritic steels MD simulation. J Marian LLNL USA, J. M Perlado Interstitial loop in pure Fe 937 interstitial <100> loop on (100) plane Weak beam g(4.1g) g=(200) 200 kv thickness 18 nm Image width = 22.9 nm

48 Cluster density [m -3 ] Perlado 2014 Instituto Fusion Nuclear UPM open symbol: p + irradiation solid symbol: n irradiation + and - centered: ion irradiation Cu Fe Pd Stainless Steel Ni Ferritic Steel NiAl Ni 3 Al Dose [dpa]

49 defect number density [m -3 ] Perlado 2014 Instituto Fusion Nuclear UPM Defect accumulation in technical alloys 6x10 23 Defect accumulation in 304L St.St. compared to that in F82H 5x x x x10 23 SSCLp SSSCl SSFl SSSFT SStotal F82Hp 1x dose [dpa]

50 Perlado 2014 Instituto Fusion Nuclear UPM DIFFUSION OF DEFECTS Spatial and Temporal scale (Problems in MD) Spatial problems in MD perhaps solved through periodic boundary solutions. Temporal problems... actually... Very difficult to solve. Combination of MD with other methods (/models) Rate Equations MonteCarlo (Kinetic MonteCarlo, MC)

51 DIFFUSION How do atoms move through solids? Substitutional diffusion Interstitial diffusion High diffusivity paths, diffusion along grain boundaries, free surfaces, dislocations Factors that influence diffusion: Diffusing species and host solid (size, bonding) Temperature Microstructure Perlado 2014 Instituto Fusion Nuclear UPM

52 Perlado 2014 Instituto Fusion Nuclear UPM DIFFUSION OF DEFECTS: MonteCarlo Static Metropolis Dynamics Lattice Kinetic Object Event Hybrid Lattice-Kinetic

53 Perlado 2014 Instituto Fusion Nuclear UPM Kinetic Monte Carlo Simulation Methodology Start Read Input: dose, dose rate diffusivities, binding energies Do while time < final dose or total time Do while time < time/cascade (dose rate) Read a cascade Select a particle Calculate time Adjust status of the particle picked and the old and new neighbours End

54 Until final time or dose Perlado 2014 Instituto Fusion Nuclear UPM Kinetic Monte Carlo Simulation Methodology Defect Distribution Rate for each Event R i Input Data From MD Cascade distribution Defect Jump: Migration Energy Cluster Generation and Dissolution: Binding Energy New Cascade: Dose Rate Total Rate, R: R = i R i N i with N i number of particles for Event i Select a particle from all the possible events: Random x R Update Time: Time = -log (random)/r Do Event: find neighbors of atoms that moved

55 Perlado 2014 Instituto Fusion Nuclear UPM Determination of Event frequency Cluster Average Lifetime - For a determined cluster is reached that: where: D = D 0 exp (-(E msi + E b ) /kt) E msi = E SIA migration + E binding of the cluster D = D 0 exp (- E cluster migration /kt) = 2 / t = first nearest neighbors / lifetime of the cluster

56 Perlado 2014 Instituto Fusion Nuclear UPM KINETIC MONTECARLO (KMC) Probabilities To choose the particle to jump... Distributions: P P if Defined the particle/defect to jump... Identify the jump length, such that: x 2 +y 2 +z 2 = 2. Finally... Potential reactions (recombinations...) N if P J N J i i if, ic, vf, vc if i

57 Perlado 2014 Instituto Fusion Nuclear UPM Determination of Event frequency We have already calculated the Diffusion coefficient Relation between jump frequency and diffusion: D[cm 2 /s] = 2 / t = = [first nearest neighbors] / [time between jumps] 1/t = = jump frequency What happen is determined...extracting a random number, which represent the time... It s faulted to calculate the lifetime of the cluster

58 Molecular Dynamics calculations of D cu en FeCu0.9 % in the case of high temperatures and high Vacancy concentrations. These results with identification of new mechanisms modified previous results that underestimates the coefficient by a factor of 10 or 100. Perlado 2014 Instituto Fusion Nuclear UPM D Cu (cm 2 s 1 ) J. Marian Thesis and PRB 65 (2002) / (a) experiment D i (T)= exp( 2.43/kT) D Cu (T) = exp( 2.31/kT) Cu vacantes (cm 2 s 1 ) D Cu /kT (ev 1 ) experiment (b) D Cu (T) = exp( 2.42/kT) D Cu (T) = exp( 2.31/kT) data point at 600 K D Cu = exp( 2.31/kT) /kT (ev 1 )

59 Experiments Y.Dai, M. Victoria (MRS) This is a clear example of successful use of KMC in comparison with experiments M.J. Caturla, N. Soneda, E. Alonso, B. Wirth, T. Díaz de la Rubia, J.M. Perlado, JNM 276 (2000) Perlado 2014 Instituto Fusion Nuclear UPM

60 Number of He desorbed Derivative of #He Results of OKMC simulations under the conditions of Edwards et. al He 400 ev -> Ni damage distribution obtained with SRIM Implantation at 300K defects followed until steady state is reached Annealing and measurement of desorption of He from 300K until 1200K Ramp rate 18.4 K/s Two peaks are also obtained from the simulations corresponding to the dissociation of He 2 V and HeV Temperature (K) Temperature (K)

61 Visible defect concentration at 600K in Zr (Arevalo, Caturla, Perlado, Nucl.Inst.Meth (2006) Visible: clusters with more than 50 defects ~ 2nm According to the model at 600K most of the defects are Vacancy type The concentration of self-interstitials is very low due to the 1D migration

62 Pure Iron Ultra High Pure Iron An example of simulation up to Diffusion KMC (VENUS-II) trying to be validated by experiments in the line of other international such as REVE 150 kev Fe + Ions 2 x 10-4 dpa/s 1 dpa T= 300 ºC Perlado 2014 Instituto Fusion Nuclear UPM

63 Nº de iones por ion incidente Perlado 2014 Instituto Fusion Nuclear UPM y 100 nm Incoming Ion 5 4 Distribuciòn Spacial de los atomos de retroceso Irradiaciòn por una cara Defects distribution x Rango (nm) PBC applied to x and y direction 100 nm z An example of simulation up to Diffusion KMC (VENUS-II) trying to be validated by experiments in the line of other international such as REVE

64 Migration prefactor D o =A n -1/2 A= 7.43 E -3 Migration Energy E m =A+B/n 1/3 A=0.059; B=0.067 Binding Energy Interstitial - Impurity 0.65 ev Visibility Conditions The minimum dimension of a TEM visible cluster is 2 nm. Considering interstitial clusters as Plane dislocations loops with <111> orientation R 2 b=n V Minimum size for a visible clusters is of 67 Interstitial Conditioning Results Perlado 2014 Instituto Fusion Nuclear UPM An example of simulation up to Diffusion KMC (VENUS-II) trying to be validated by experiments in the line of other international such as REVE

65 Modeling thin film Fe irradiation with 150keV Fe Additional Simulations KMC on the same experiments different parameters and physics of defects 100nm 150 kev Fe Surfaces 100% recombination Input data for kmc Cascade database from Roger Stoller Self-interstitial cluster evolution rules from Jaime Marian s MD (PRL 88,25 (2002) /4) Ø Loops formed in cascade as <111> mobile Ø Interaction between two loops: If resulting BV is <100> and both > 15 defects <100> loop If resulting BV is <100> <111> loop Ø Interaction between loop and Impurities: <111> loop + Impurity <111> loop (trapped) <111> trapped loop with other loops same rules as in (2) Binding energy <111> loop and Impurity between 0.4 and 1 ev Perlado 2014 Instituto Fusion Nuclear UPM

66 Perlado 2014 Instituto Fusion Nuclear UPM Modeling thin film Fe irradiation with 150keV Fe Density of defects in <100> and <111> loops Additional Simulations KMC on the same experiments different parameters and physics of defects Density of Interstitials in clusters (cm -3 ) <100> loops 20 appm 80 appm 400 appm DPA Density of Interstitials in clusters (cm -3 ) <111> loops 20 appm 80 appm 400 appm DPA <100> loops are formed during defect evolution at 300 C If no impurities all <111> loops at surfaces and no <100> loops If binding energy with impurities = 0.4 ev all <111> loops at surfaces

67 Perlado 2014 Instituto Fusion Nuclear UPM Physics Modeling: Experimental Validation in very simple samples Simplified and well controlled irradiation experiments to determine the influence of impurities, temperature and fluence on the damage characteristics Previous CIEMAT results from UHP Fe pure Fe loops larger than observed in Microstructure of UHP-Fe Irradiated with Fe ions TEM examination of pure and UHP Fe irradiated with Fe ions at different temperatures and fluences Characterisation of damage is giving size distribution, density, morphology and character of irradiation induced defects, that will be directly comparable with results from computer simulation

68 Perlado 2014 Instituto Fusion Nuclear UPM Dose Rate dependence of void swelling can be extracted from kmc simulations (courtesy of M.J. Caturla) V/V (%) dpa/s 10-8 dpa/s 10-9 dpa/s Temperature (K) Change in Volume due to He-V clusters at 0.03 dpa as a function of dose rate and temperature V/V()) voids = C V void(n) ()x r (n)) C voids V = concentration of Vacancies in Voids r = relaxation volume of a Vacancy in a Void n = number of defects in Void = Dose Characteristic swelling curve for FCC metals is obtained from kmc Swelling peak shift with dose rate

69 Perlado 2014 Instituto Fusion Nuclear UPM MESOSCOPIC scale of Multiscale Modeling: Simulating modifications of mechanical responses still NOT continuum methods. Dislocations: Long-range (1/r) elastic strain field (90% of the energy) + inelastic core (~10% of the energy). Long-range (1/r) interactions + large number of local reactions = complex non-linear behavior. Dislocation Dynamics: Solve the dynamics of dislocation lines in elastic continuum and include information about reactions. It was first developed for 2D simulations of infinite parallel dislocations [Lépinoux and Kubin, Scripta Met. 21, 833 (1987)] and then expanded to 3D systems [Kubin et al. Solid State Phenomena 23-24, 455 (1992)].

70 Dislocation Dynamics. Calculation of forces External Force. Constant Peach - Koehler force due to the applied external stress. Dislocation dislocation interaction. Calculated as sum of all Peach Koehler forces between the segments of all other dislocations in the system. Self force. Interaction between segments of the same dislocation, line tension force. Peierls force. Constant force in the glide plane that is always directed against the dislocation motion. It is small for FCC metals but can be significant for BCC metals, semiconductors, ionic materials. Image force due to the finite size of the simulated system. This contribution result from the stress relaxation in the vicinity of free surface or internal surface. Obstacle force due to the interaction with lattice defects other than dislocations. Perlado 2014 Instituto Fusion Nuclear UPM

71 Perlado 2014 Instituto Fusion Nuclear UPM A clear example of Multiscale Modeling in Radiation Damage using the different tools: Multiscale Modelling of plastic flow localization in irradiated materials. T.Díaz de la Rubia et al., Nature Vol 406, 24 August 2000, It is known that in metals the main features of neutron or ion irradiation is to induce changes in mechanical properties of different types: A sharp increase in yield stress The appearance of yield point followed by yield drop in fcc metals An instability resulting in a Plastic Flow localization within dislocation channels that leads to loss of ductility and premature failure. This last effect is fully simulated and compared with experiments. Defect free channel in irradiated Cu. Thisi is a TEM image after 600 MeV protons irradiation.

72 Perlado 2014 Instituto Fusion Nuclear UPM A clear example of Multiscale Modeling in Radiation Damage using the different tools: Multiscale Modelling of plastic flow localization in irradiated materials. T.Díaz de la Rubia et al., Nature Vol 406, 24 August 2000, Agreement with Experimental Observations. Cu. Channel width of 200nm with a channel spacing of 1000nm.

73 Perlado 2014 Instituto Fusion Nuclear UPM A clear example of Multiscale Modeling in Radiation Damage using the different tools: Multiscale Modelling of plastic flow localization in irradiated materials. T.Díaz de la Rubia et al., Nature Vol 406, 24 August 2000, Vacancy SFT are predominant defects in materials such as Cu, and selfinterstitial atom Frank sessile loops in such as Pd.

74 Dislocation Dynamics for Partial Dislocations Strength of SFT to the Screw Passage Yield curve for a 70 nm dislocation length and a 4.7 nm SFT Stress vs. Height of a BA(d) screw dislocation face-on depending on the SFT s size

75 Following material is taken from very descriptive ATOMS AND THE CONTINUUM Coupling Atomistic Simulations and Continuum Elasticity for Multiscale Simulations from Noam Bernstein Center for Computational Materials Science Naval Research Laboratory Washington, DC Perlado 2014 Instituto Fusion Nuclear UPM

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78 Perlado 2014 Instituto Fusion Nuclear UPM Connection between atomistic and continuum modeling