Defects modeling in solid-state physics: coupling atomic scale with elasticity Emmanuel Clouet

Service de Recherche de Métallurgie Physique, CEA Saclay, France [email protected]

GdR ModMat, Istres last update: June 22, 2016 (SRMP lecture)

July 20-24, 2015

Introduction

Robert Hooke (1635 – 1703)

ceiiinosssttuv 1660 Ut tensio, sic vis As the extension, so the force 1678

source: wikipedia

Introduction

Years 2000 Atomistic simulations with several million atoms

Molecular-dynamics simulations of dynamic crack propagation in a fcc crystal using 100 million atoms with a Lennard-Jones potential V. V. Bulatov, F. F. Abraham, L. P. Kubin, B. Devincre and S. Yip Nature 391, 669 (1998).

Empirical potentials usually perfectly match elastic behaviour

Introduction Self-interstitial clusters in bcc iron : Cluster containing 8 SIAs ‹111› loop

GGA

ε=0 : ∆E = 5.6 eV σ=0 : ∆E = 0.6 eV

C15 aggregate1

elasticity can be used to understand (and withdraw) size effects in atomistic simulations

1 M.-C. Marinica, F. Willaime, and J.-P. Crocombette, Phys. Rev. Lett. 108, 025501 (2012).

Introduction Vacancy in bcc iron : interaction with an applied strain Simulation box 10x10x10 (2000 sites) Marinica 2007 EAM potential

 1 0 0 ε  0 1 0   0 0 1  

 0 1 1 ε  1 0 1   1 1 0  

elasticity can be used to model variations with the applied strain of the energies of defects (vacancies, self interstitials, solute, precipitates)

Introduction Dislocation dynamics simulations Coarsening kinetics of a distribution of prismatic loops 2 time steps - dislocation glide - dislocation climb (diffusion)

Al parameters T = 600K unfaulted {110} loops

elasticity can be used to model microstructure evolution at a mesoscopic scale (DD, phase field) D. Mordehai, E. Clouet, M. Fivel, and M. Verdier, Philos. Mag. 88, 899 (2008) B. Bakó, E. Clouet, L. Dupuy and M. Blétry, Philos. Mag. 91, 3173 (2011)

Introduction Nano devices and experiments mechanical tests on cantilevers (beams)

J. Gong, T. B. Britton, M. A. Cuddihy, F. P. Dunne and A. J. Wilkinson, Acta Mater. 96, 249 (2015) prismatic, basal, and slip strengths of commercially pure Zr by micro-cantilever tests

Introduction Nano devices and experiments

h

Four-point bending loading device pure shear stress but pure tensile stress in the irradiation zone

300 keV Zr ions 11.5x1014 ions/cm²

L. Tournadre, F. Onimus, J.-L. Béchade, D. Gilbon, J.-M. Cloué, J.-P. Mardon and X. Feaugas J. ASTM Int. 17, 853 (2014) Impact of hydrogen pick-up and applied stress on c-component loops: toward a better understanding of the radiation induced growth of recrystallized zirconium alloys

Introduction Creep: Irradiation vs Thermal Creep

σ

Creep: inelastic deformation at stress levels below the yield stress

-1

Strain rates (s )

σY

100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 4

long-term deformation relatively high T and applied loads needed Irradiation creep: sharp acceleration of creep deformation under the action of irradiation

ε

for structural materials in reactors irradiation creep dominates at T < 0.5 Tm mechanisms of thermal and irradiation creep are very different

Thermal Creep [18] Irrad. Creep under VHTR: 3x10-8 dpa/s

80 MPa 5.5x10-6 dpa/s

thermal creep: creation of point defects irradiation creep: diffusion and absorption

PM2000 6

8

10

12

14

16

1/T (10-4 K-1)

18

20

22

V. Borodin, J.-C. Chen, M. Sauzay, P. Vladimirov, Analysis and assessment of mechanisms of irradiation creep Matisse European project (2015)

Introduction Irradiation creep: Stress Induced Preferential Absorption (SIPA) No applied stress

Applied stress vcl

I

vcl

I

τ

τ

V

V vcl

The number of V and I absorbed by a dislocation does not depend on its orientation same climb velocity for all dislo (= 0 if dislo are the only sinks)

no macroscopic strain

vcl

Applied stress τ modifies V and I fluxes to dislo climb velocity depends on dislo orientation with respect to τ macroscopic strain

Outlines Defects modeling in solid-state physics: coupling atomic scale with elasticity I.

Elasticity Theory

II.

Inclusions, Inhomogeneities and Point Defects

III. Dislocations IV. Plane interfaces

I . Elasticity Theory

1. Deformation of an elastic body 2. Stresses in an elastic body 3. Thermodynamics of deformation 4. Hooke’s law 5. Equilibrium equation in homogeneous elasticity References: • L. Landau & E. Lifchitz, Theory of elasticity, Course of theoretical physics, vol. 7 (1967) • A. E. H. Love, The mathematical theory of elasticity (1927). • J.P. Hirth & J. Lothe, Theory of dislocations, chap. 2 (1982) • J. F. Nye, Physical Properties of Crystals - Their representation by tensors and matrices (1957). • D. J. Bacon, D. M. Barnett and R. O. Scattergood, Prog. Mater. Sci. 23, 51 (1980).

1. Deformation of an Elastic Body Strained state

Reference state

dx x dl

Deformation

dx ' d l ' x'

Displacement

u = x′ − x :

ui = xi '− xi

Distortion

F = ∇ ⊗u :

Fij = ∂ui / ∂x j

Strain

∂ui dxi ' = dxi + dx j = (δ ij + Fij ) dx j ∂x j

1 1  ∂ui ∂u j ∂un ∂un  t ε = ( Id + F ) ( Id + F ) − Id  : ε ij =  + +  2 2  ∂x j ∂xi ∂xi ∂x j  dl '2 = dl 2 + 2ε ij dxi dx j

1. Deformation of an Elastic Body Strained state

Reference state

dx x dl

Deformation

dx ' d l ' x'

Displacement

u = x′ − x :

ui = xi '− xi

Distortion

F = ∇ ⊗u :

Fij = ∂ui / ∂x j

Strain

∂ui dxi ' = dxi + dx j = (δ ij + Fij ) dx j ∂x j

1 ε = ( Id + F t ) ( Id + F ) − Id  2 1 −1 t −1 ε = Id − ( Id + F ) ( Id + F )   2

(Green-Lagrange) (Euler-Almansi)

symmetric 2nd rank tensors

1. Deformation of an Elastic Body Small deformation assumption • Strain

1 1  ∂ui ∂u j  t ε = ( F + F ) : ε ij =  +  2 2  ∂x j ∂xi 

• Rotation

1 1  ∂ui ∂u j  t Ω = ( F − F ) : Ωij =  −  2 2  ∂x j ∂xi  no contribution in linear elasticity because no internal torque

diagonal components: elongation / compression

dl '− dl ε jj = dl

dl

off-diagonal components: shear angle

dl '

γ

ε ij = γ

γ

symmetric tensors 3 eigenvalues = elongation / compression along principal axes volume change

dV ' = dV (1 + ε (1) )(1 + ε (2) )(1 + ε (3) ) = dV (1 + ε (1) + ε (2) + ε (3) )

dV '− dV Tr(ε) = dV

1. Deformation of an Elastic Body Small strain

1  ∂ui ∂u j  1 ε = ( ∇ ⊗ u + u ⊗ ∇ ) : ε ij =  +  2 2  ∂x j ∂xi 

Spherical coordinates

Cylindrical coordinates

1. Deformation of an Elastic Body Strain extraction from atomistic simulations • average strain: from vectors defining simulation cell (periodic or not)

a

3

a a

3

a '

Deformation

2

2

a '

1

H = ( a1 | a 2 | a 3 ) H′ = (Id + F)H

1

a '

H ' = ( a1 ' | a 2 ' | a 3 ' ) distortion:

F = H′ H −1 − Id

strain, rotation, …

1. Deformation of an Elastic Body Strain extraction from atomistic simulations • local strain (on each atom): from vectors defining neighbours positions Deformation

r

N

r '

N

r r

2

2

r ' 1

r '

1

R = (r1 | r 2 |⋯ | r N )

R ' = ( r 1 ' | r 2 ' | ⋯ | r N ')

r n ' = (Id + F) r n , ∀n ∈ 1: N over-determined system: least square fitting distortion

F = (R ' R

t

)( R R )

t −1

− Id

strain, rotation, … C. S. Hartley and Y. Mishin, Acta Mater. 53, 1313 (2005).

1. Deformation of an Elastic Body Strain extraction from atomistic simulations screw dislocation in hcp Zr (EAM) Trace of the deviatoric strain tensor ~ average shear

21%

l b

e y = [0001]

0% ez =  2 1 10

ex = 0 110

Visualization: atomeye (Ju Li, MIT) J. Li, Modelling Simul. Mater. Sci. Eng. 11, 173 (2003).

2. Stresses in an Elastic Body Force acting on a volume element of a strained body

∫

V

FdV

- external body force (gravity, distribution of point forces, …)

f dV - internal forces corresponding to interaction between particles cancellation of forces between particles inside V (action – reaction) only forces corresponding to interaction with outside particles proportional to the surface (no long range interaction)

σ dS

∫

V

σzy

z x

y

σxy

σyy

Fi dV = ∫ fi dV + V

Example: pressure

∫

S

σ ij dS j

1 0 0 σ = −P  0 1 0    0 0 1  

2. Stresses in an Elastic Body Equilibrium: no resultant inside Force

∫

V

Fi dV = ∫ fi dV + V

∫

S

σ ij dS j

 ∂σ ij  = ∫  fi + dV  V ∂x j   Torque

∫

V

fi +

= ∫ ( xk f i − xi f k V

S

V

= ∫ (σ ik − σ ki ) dV

T a dS − σ dS = 0

∂x j

Ta n

k

ij

i

k

ij

kj

j

i

kj

j

σ ik = σ ki

V

+ equilibrium with applied forces at the boundary

=0

∫ ( x σ − x σ ) dS ∂ x σ − x σ ) dV ) dV + ∫ ( ∂x

M ik dV = ∫ ( xk f i − xi f k ) dV + V

∂σ ij

−σn

σ ij n j = T ja

3. Thermodynamics of Deformation Infinitesimal deformation corresponding to a displacement change Work of the internal forces (δW > 0 when energy flux from elastic body to outside)

∫ δ W dV = − ∫ V

V

= −∫

V

δu

fi δ ui dV − ∫ σ ijδ ui dS j S

∂ fi δ ui dV − ∫ σ ijδ ui ) dV ( V ∂x j

∂δ ui = − ∫ σ ij dV V ∂x j Variation of the internal energy:

= − ∫ σ ijδε ij dV

(defined per volume unit of the unstrained body)

δ e = T δ s − δ w = T δ s + σ ijδε ij of the free energy:

δ f = δ e − δ (Ts ) = sδ T + σ ijδε ij

V

 ∂e  1  ∂E  σ ij =   =    ∂ε ij  S V0  ∂ε ij  S  ∂f  1  ∂F  σ ij =   =    ∂ε ij T V0  ∂ε ij T

3. Thermodynamics of Deformation Example: hydrostatic strain

1 0 0 δV  0 1 0 δε =  3V0   0 0 1  

δ e = T δ s + σ ijδε ij = Tδ s +

δV 3V0

= Tδ s − P

1 ∂E σ ij = V0 ∂ε ij

S

1 ∂F = V0 ∂ε ij

Constant pressure ensemble: Constant stress ensemble:

T

σ ijδ ij

δV V0

with

1 P = − Tr(σ ) 3

∂E  ∂F    P = −  = −  ∂ V ∂ V  S  T H = E + PV

H = E − V0σ ijε ij

⇒ δ H = Tδ S + V δ P ⇒ δ H = T δ S − V0ε ijδσ ij

3. Thermodynamics of Deformation Stress from atomistic simulations: average stress

N atoms in interaction E = ∑ α

α

α

p p int + E 2mα

NkT 1 ∂E int σ ij = − δ ij + V V ∂ε ij

when the potential energy Eint only depends on atom positions

Xα

NkT 1 ∂E int ∂X kα NkT 1 σ ij = − δ ij + ∑ α α =− δ ij − ∑ α Fiα X αj V V ∂X k ∂ε ij V V ! in PBC, Eint also depends on periodicity vectors

in quantum mechanics:

E int = φ H φ

stress from Hellmann-Feynman theorem:

1 ∂H σ ij = φ φ V ∂ε ij

O. H. Nielsen and R. M. Martin, Phys. Rev. B 32, 3780 (1985).

3. Thermodynamics of Deformation Stress from atomistic simulations: atomic stress it cannot be always defined (not in ab initio) one needs to be able - to partition the energy in atom contributions - to associate a volume Vα to each atom

it has to be meaningful (average, equilibrium)

1 σ = ∑ α V α σα V

not always an easy task !

3. Thermodynamics of Deformation Stress from atomistic simulations: atomic stress it cannot be always defined (not in ab initio) one needs to be able - to partition the energy in atom contributions - to associate a volume Vα to each atom

it has to be meaningful (average, equilibrium)

1 σ = ∑ α V α σα V

not always an easy task !

Example: potentials depending only on the distance between atoms (pair potentials, EAM, 2nd moment approximation)

E int = φ ({rαβ })

α β α β X − X X − X 1 ∂φ ( i i )( j j ) α σ ij = α ∑ β αβ αβ 2 V ∂r (r )

V. Vitek and T. Egami, Phys. Status Solidi B 144, 145 (1987).

3. Thermodynamics of Deformation Stress from atomistic simulations: atomic stress screw dislocation in hcp Zr (EAM) Trace of the deviatoric stress tensor = Von Misès stress

61 MPa

l b

e y = [0001]

1.7 MPa ez =  2 1 10

ex = 0 110

4. Hooke’s Law Thermodynamics:

δ F = Sδ T + V σ ijδε ij expression of F(T,ε) needed

Undeformed state (reference): elastic body at equilibrium without any external force (surface, body)

1  ∂F  σ ij =   V  ∂ε ij T

1  ∂F  σ ij =   =0 V  ∂ε ij T

1 Small deformation assumption: F (T , ε) = F0 (T ) + VCijkl ε ij ε kl 2 1  ∂2F  elastic constants: Cijkl =   V  ∂ε ij ∂ε kl T 1 σ ij = Cijkl ε kl ∆F = V σ ij ε ij Hooke’s law: 2 2 Hydrostatic strain ∆V ∆ V ∆ V ε ij = δ ij σ ij = − Pδ ij P = −B ∆F = B 3V V V

4. Hooke’s Law Properties of elastic constants

1 ∆F = VCijkl ε ij ε kl > 0 ∀ε 2 ∂2 F ∂2 F = ∂ε ij ∂ε kl ∂ε kl ∂ε ij

Cijkl : 4th rank tensor definite positive Cijkl = Cklij (major symmetry)

ε ij = ε ji

Cijkl = Cjikl = Cijlk (minor symmetries)

21 coefficients (instead of 81) Elastic compliance (inverse tensor)

σ ij = Cijkl ε kl

ε ij = Sijklσ kl

with

Cijkl S klmn

1 = (δ imδ jn + δ inδ jm ) 2

4. Hooke’s Law Voigt notation

σ ij = Cijkl ε kl

σ I = CIK ε K

sym. tensors in R3

vectors in R6

(6 components)

(6 components)

Indexes

Indexes

11 22 33 23 / 32 13 / 31 12 / 21

1 2 3 4 5 6

1 ∆F = V σ ijε ij 2

1 ∆F = V σ I ε I 2 CIK = Cijkl

 σ 11   C11 C12 σ   C22  22    σ 33    =  σ 23    σ 13      σ  12    ε11   S11 ε    22    ε 33    =  2ε 23    2ε13      2 ε  12  

C13

C14

C15

C23 C24 C33 C34

C25 C35

C44

C45

S12

S13

S14

S15

S 22

S23 S33

S24 S34

S 25 S35

S44

S 45 S55

S IK = Sijkl ∀I , K

C55

C16   ε11  C26   ε 22    C36   ε 33    C46   2ε 23  C56   2ε13    C66   2ε12  S16   σ 11  S 26   σ 22    S36   σ 33    S 46   σ 23  S56   σ 13    S66   σ 12 

if 1 ≤ I , K ≤ 3

= 2 Sijkl

if 1 ≤ I ≤ 3 and 4 ≤ K ≤ 6

= 2 Sijkl

if 4 ≤ I ≤ 6 and 1 ≤ K ≤ 3

= 4 Sijkl

if 4 ≤ I , K ≤ 6

4. Hooke’s Law Elastic constants and rotation rotation R:

u → u ′ : u 'i = Rij u j C → C′ : C 'ijkl = Rim R jn Rko Rlp Cmnop

Elastic constants and symmetry if R is a symmetry operation of the Bravais lattice

Cijkl = Rim R jn Rko Rlp Cmnop isotropic : 2 coefficients (Lamé coefficients)

Cijkl = λδ ijδ kl + µ (δ ik δ jl + δ ilδ jk )

cubic : 3 coefficients (C11, C12, C44) hexagonal : 5 coefficients (C11, C33, C12, C13, C44)

4. Hooke’s Law Cubic elasticity

 C11 C12 C12 0  C11 C12 0  C11 0   C44    

0 0 0 0 C44

0  0   0   0  0   C44 

2 B = C11 + C12 3

Isotropic elasticity

C11 = λ + 2 µ Bulk modulus: Young modulus: Poisson coefficient:

C12 = λ

C44 = µ

1 C44 = ( C11 − C12 ) 2

2 B=λ+ µ 3 µ (3λ + 2 µ ) 9 µ B E= = = 2 µ (1 + ν ) µ +λ 3B + µ 3B − 2 µ λ E − 2µ ν= = = 2(3B + µ ) 2( µ + λ ) 2µ

4. Hooke’s Law Hexagonal elasticity

 C11 C12 C13  C11 C13  C33      

0 0

0 0

0 C44

0 0 C44

0  0   0   0  0   C44 

with

B=

1 C66 = ( C11 − C12 ) 2 2C11 + C33 + 2C12 + 4C13 9

Example: ab initio calculations in Zr Pwscf: DFT GGA Ultrasoft pseudopotential (5s2 4d2+ 4s2 4p6) Valence electrons described by plane waves (20 / 28 Ry) HCP unit cell (2 atoms)

4. Hooke’s Law C44 in hcp Zr

0 0 1 ε = ε 0 0 0   1 0 0  

0 0 1 σ = 2C44ε  0 0 0    1 0 0  

∆E (ε ) = 2VC44ε 2

4. Hooke’s Law C66 in hcp Zr

0 1 0 ε = ε 1 0 0   0 0 0  

0 1 0 σ = 2C66ε  1 0 0    0 0 0  

∆E (ε ) = 2VC66ε 2 1 with C66 = ( C11 − C12 ) 2

atomic relaxations when the crystal symmetry is broken by the strain

4. Hooke’s Law B in hcp Zr

1 0 0 ε = ε 0 1 0   0 0 1  

1 P = − Tr(σ ) = −3Bε 3

9 ∆E (ε ) = − VBε 2 2

slower convergence of the stress (work with stress differences)

4. Hooke’s Law Hexagonal elasticity  C11

       

C12 C11

C13 C13

0 0

0 0

C33

0 C44

0 0

Example: ab initio calculations in Zr

C44

0  0   0   0  0   C44 

1 with C66 = ( C11 − C12 ) 2

5. Equilibrium Equation in Homogeneous Elasticity

∂σ ij ∂x j

+

+ fi = 0

with

σ ij = Cijkl ε kl

and

∂ 2 uk Cijkl + fi = 0 ∂x j ∂xl

σ ij n j = T ja

Isotropic elasticity

and / or

ui = uia

1  ∂ui ∂u j  + ε ij =   2  ∂x j ∂xi 

at the boundary

∂ 2u j

∂ 2ui (λ + µ ) +µ + fi = 0 ∂xi ∂x j ∂x j ∂x j (λ + µ )∇(∇·u ) + µ ∆u + f = 0

5. Equilibrium Equation in Homogeneous Elasticity

∂ 2 uk Cijkl + fi = 0 ∂x j ∂xl

+

σ ij n j = T ja

and / or

ui = uia

at the boundary

Superpostion principle linear equation (linear elasticity) solutions can be added 1 2 if u and u are solutions corresponding to forces f 1 and

σ ij = σ ij1 + σ ij 2 = Cijkl ( ε ij1 + ε ij 2 )

f2 ∂ 2 (uk1 + uk 2 ) Cijkl + f i1 + f i 2 = 0 ∂x j ∂xl

1 ∆F = V (σ ij1 + σ ij 2 )( ε ij1 + ε ij 2 ) 2 1 1 1 1 1 2 2 = V σ ij ε ij + V σ ij ε ij + V (σ ij1ε ij 2 + σ ij 2ε ij1 ) 2 2 2 = ∆F 1 + ∆F 2 + ∆F inter Elastic interaction energy ∆F inter = V σ ij1ε ij 2 = V σ ij 2ε ij1 (between defects, …)

5. Equilibrium Equation in Homogeneous Elasticity

∂ 2 uk Cijkl + fi = 0 ∂x j ∂xl

+

σ ij n j = T ja

and / or

ui = uia

at the boundary

Superpostion principle: Green’s function: solution to a unit point force (Dirac delta function)

∂ 2Gkn Cijkl + δ inδ ( x ) = 0 ∂x j ∂xl

with

 0 if x ≠ 0 δ (x) =  ∞ if x = 0

Gkn (r ) : tensor field giving the displacement along the xk axis of a unit point force applied along the xn at the origin Solution to the force distribution

∫ f ( x′)dx′

uk ( x ) = ∫ Gkn ( x − x′) f n ( x′)dx′

σ ij ( x ) = Cijkl ∫ Gkn ,l ( x − x′) f n ( x′)dx′

with

Gkn ,l

∂Gkn = ∂xl

5. Equilibrium Equation in Homogeneous Elasticity Green’s function • isotropic elasticity:

1 Gkn ( x ) = 16πµ (1 −ν ) x

 xk xn  (3 − 4ν )δ ij + 2  x  

• anisotropic elasticity: analytical expression only for transverse isotropy (hexagonal) numerical evaluation needed D. M. Barnett, Phys. Status Solidi B 49, 741 (1972). but the radial dependence does not vary

1 Gkn (r ) = g (θ , φ ) r 1 Gkn ,i (r ) = 2 h(θ , φ ) r 1 Gkn ,ij (r ) = 3 f (θ , φ ) r

5. Equilibrium Equation in Homogeneous Elasticity

∂ 2 uk Cijkl + fi = 0 ∂x j ∂xl

+

σ ij n j = T ja

and / or

ui = uia

at the boundary

Superpostion principle: problem decomposition to enforce boundary conditions

T a on ST

T1

T2

=

u a on Su

T a −T1 −T 2

+

u1

+

u2

u a − u1 − u 2

E. van der Giessen and A. Needleman, Modelling Simul. Mater. Sci. Eng. 3, 689 (1995).

5. Equilibrium Equation in Homogeneous Elasticity

∂ 2 uk Cijkl + fi = 0 ∂x j ∂xl

+

σ ij n j = T ja

Fourier transform method linear partial differential equation

u ( r ) = ∫ U ( q ) e − i q ·r dq

and / or

at the boundary

solutions in reciprocal space and

−Cijkl q j qlU k + Fi = 0 with U k = (qq ) −ki1 Fi

f ( r ) = ∫ F ( q ) e − i q ·r dq

(qq )ik = Cijkl q j ql

Fourier transform of Green’s function:

σ (r ) = ∫ Σ(q ) e − i q ·r dq

ui = uia

where

(qq ) −ki1

Σij = −iCijkl ql (qq ) −kn1 Fn

complex microstructure (phase field, polycrystal homogenization), but - periodic boundary conditions - spatial resolution d in a box of dimension L limited by the number N of nodes of the FFT grid: N = L/d T. Mura, Micromechanics of Defects in Solids (1987).

II. Inclusions, Inhomogeneities and Point Defects 1. Spherical inclusion 2. Eshelby’s inclusion 3. Inclusion and applied stress 4. Point defect 5. Carbon - dislocation interaction in iron 6. Isolated point-defect in ab initio calculations 7. Inhomogeneity and polarizability References: • J. D. Eshelby, Proc. Roy. Soc. Lond. A 241, 376 (1957); ibid 252, 561 (1959). • G. Leibfried and N. Breuer, Point Defects in Metals I 81 (1978). • D. J. Bacon, D. M. Barnett and R. O. Scattergood, Prog. Mater. Sci. 23, 51 (1980). • R. W. Balluffi, Introduction to Elasticity Theory for Crystal Defects (2012). • C. Weinberger, W. Cai and D. Barnett, Elasticity of Microscopic Structures, Standford Univ. lecture notes (2005).

1. Spherical Inclusion Isotropic elasticity in spherical symmetry spherical symmetry isotropic elasticity no body force

u (r ) = u (r )er (λ + µ )∇(∇·u ) + µ ∆u + f = 0

∂  ∂u 2u   + =0 ∂r  ∂r r 

ε = ∇⊗u

∂u B ε rr = = A − 2 3 ∂r r

εθθ = ε φφ

B u (r ) = A r + 2 r u B = = A+ 3 r r

∆V = 3A V

σ ij = λδ ijδ kl + µ (δ ik δ jl + δ ilδ jk )  ε kl B B σ rr = ( 3λ + 2µ ) A − 4µ 3 σ θθ = σ φφ = (3λ + 2µ ) A + 2 µ 3 r

Density of elastic energy:

2 3 B f = (3λ + 2 µ ) A2 + 6 µ 6 2 r

r P = −3(λ + 2 µ ) A

1. Spherical Inclusion Spherical inclusion: infinite system

λ, µ

Inside the inclusion

RI

R

λI, µI

δ R = RI − R Continuity at the interface

uI ( r ) = A r I

εI = A

1 0 0  I 0 1 0   0 0 1  

homogeneous compression

Outside the inclusion

B u (r ) = 2 r B  −2 0 ε = 30 1 r  0 0 pure shear

B  I I I R + A R = R+ 2  R I + u I ( R I ) = R + u ( R)  R   I I σ ( R ) = σ ( R)  ( 3λ I + 2 µ I ) AI = −4 µ B  R3 I I 4 µ δ R 3 λ + 2 µ 3δR AI = − I B= I R I I 3λ + 2 µ + 4µ R 3λ + 2 µ + 4 µ R

0  0 1 

1. Spherical Inclusion

B u (r ) = A r + 2 r

Spherical inclusion: infinite system

λ, µ

Inside the inclusion

RI

R

λI, µI

δ R = RI − R Elastic energy:

Outside the inclusion

B u (r ) = 2 r B  −2 0 ε = 30 1 r  0 0

uI ( r ) = A r I

εI = A

1 0 0  I 0 1 0   0 0 1  

homogeneous compression

0  0 1 

pure shear

2 ∞ 3 4 B F = (3λ I + 2 µ I ) AI2 π R 3 + ∫ 6 µ 6 4π r 2 dr R 2 3 r

6 µ (3λ I + 2 µ I ) I  δ R  F= Ω   I I (3λ + 2 µ + 4 µ )  R 

2

with

4 3 Ω = πR 3 I

1. Spherical Inclusion Homogeneous spherical inclusion : infinite system Inside the inclusion

λ, µ

uI ( r ) = A r I

RI

R

εI = A λ, µ

1 0 0  I  0 1 0 0 0 1  

4µ δR A =− 3(λ + 2 µ ) R

Outside the inclusion

B u (r ) = 2 r  −2 0 B ε = 30 1 r  0 0

3λ + 2 µ 3 δ R B= R 3(λ + 2 µ ) R

I

δ R = RI − R

0  0 1 

Eigenstrain: inclusion strain before matrix relaxation

4µ ε = ε* ε* = − 3(λ + 2 µ ) R Elastic energy: 2 2 µ (3λ + 2 µ ) I  δ R  3 3λ + 2 µ  I * 2  F= Ω   = (3λ + 2 µ )Ω ( ε ) 1 −  (λ + 2 µ ) R 2 3( λ + 2 µ )    

δ R  1

0 0  0 1 0   0 0 1  

1 (δΩ ) F= B 2 ΩI

I 2

3λ + 2 µ   1 − 3(λ + 2 µ )   

I

2. Eshelby’s Inclusion 1)

C

2)

T

3)

C

ε*

1) Strain the inclusion to fit the hole elastic field in the inclusion elastic field outside

εI = ε * ε=0

σI = C : ε * σ=0

2) Weld the inclusion in the hole need to apply body force along the surface to compensate tractions caused by inclusion stress

T = − ∫ C : ε * dS

3) Relax surface traction J. D. Eshelby, Proc. Roy. Soc. Lond. A 241, 376 (1957); 252, 561 (1959).

2. Eshelby’s Inclusion 1)

C

2)

3)

T

C

ε* Elastic field outside the inclusion

ui ( x ) =

∫

S

Gij ( x − x′)T j ( x′)dS '

= − ∫ Gij ( x − x′)σ ( x′)nk ( x′)dS ' S

* jk

with

∂ Gij ( x − x′)σ *jk ( x′) dV ' = −∫ I Ω ∂x k = − ∫ I Gij ,k ( x − x′)σ *jk ( x′)dV ' Ω

because

* σ *jk = C jkmnε mn

∂σ *jk ∂xk

=0

2. Eshelby’s Inclusion 1)

C

2)

3)

T

C

ε* Elastic field outside the inclusion

ui ( x ) = − ∫ I Gij ,k ( x − x′)σ ( x′)dV ' * jk

Ω

with

ε mn ( x ) = − ∫ Gmj ,nk ( x − x′)σ *jk ( x′)dV ' ΩI

σ pq ( x ) = −C pqmn ∫ Gmj ,nk ( x − x′)σ *jk ( x′)dV ' ΩI

Elastic field inside the inclusion: add the eigenstrain

* σ *jk = C jkmnε mn

2. Eshelby’s Inclusion Ellipsoidal inclusion If the eigenstrain is homogeneous, then the strain inside the inclusion is also homogeneous

C

E ε ijI = Sijkl ε kl*

C

ε*

Eshelby tensor:

S

E ijmn

1 = Cklmn ∫ I Gik , jl ( x ) + G jk ,il ( x )  dV Ω 2

(minor symmetries but no major symmetry)

Elastic energy:

1 I 1 I * * E * F = Ω Cijkl ε ijε ij − Ω Cijkl S klmn ε ij*ε mn 2 2 1 I E * * = Ω Cijkl (δ kmδ ln − S klmn ε ) ij ε mn 2

3. Inclusion and Applied Stress Homogeneous spherical inclusion : infinite system Inside the inclusion

λ, µ

uI ( r ) = A r I

RI

R

εI = A λ, µ

1 0 0  I  0 1 0 0 0 1  

4µ δR A =− 3(λ + 2 µ ) R I

Outside the inclusion

B u (r ) = 2 r  −2 0 B ε = 30 1 r  0 0

0  0 1 

3λ + 2 µ 3 δ R B= R 3(λ + 2 µ ) R

Homogeneous spherical inclusion : finite system the displacement cancel in Re (no external strain) superposition of a homogeneous strain ε0 1   B ε I =  AI − 3   0  R E   0

0 0  1 0 0 1 

variation of the elastic energy (image forces)

u(R ) = E

B R

B  −2 ε = 30 r  0

E2

+ ε 0 RE = 0

0 0  1 0 0 1 

−

1 0 0   0 1 0   E3   0 0 1

B R

3. Inclusion and Applied Stress Homogeneous spherical inclusion : finite system Inside the inclusion

λ, µ RI

R

λ, µ

Outside the inclusion

 I B  uI (r ) =  A − 3 r  RE   I B   1 0 0  I ε =  A − 3   0 1 0  R E   0 0 1 

Interaction with an applied strain : ε

1 0 0  A 0 1 0   0 0 1  

AI = −

r  1 u ( r ) = B  2 − E3  r R  B  −2 0 0  B  1 0 ε = 3  0 1 0 − 3 0 1 r  0 0 1  R E  0 0 4µ δR 3(λ + 2 µ ) R

B=

3λ + 2 µ 3 δ R R 3(λ + 2 µ ) R

F inter = V σ ij1ε ij 2 = V σ ij 2ε ij1 4 3 4 E3 B A inter I A F = π R 3(3λ + 2 µ ) A ε − π R 3(3λ + 2 µ ) 3 ε 3 3 RE δR A inter I F = −Ω 3(3λ + 2 µ ) ε R F inter = −3Ω Iε * P A with P A = −(3λ + 2 µ )ε A

Interaction between 2 elastic fields (superposition)

0  0 1 

3. Inclusion and Applied Stress Eshelby inclusion

C

Interaction with an applied strain

C

ε*

F inter = − ∫ I Cijkl ε ij* ( x )ε klA ( x )dV Ω

for an homogeneous eigenstrain

F inter = −Cijkl ε ij* ∫ I ε klA ( x )dV Ω

4. Point Defect Infinitesimal Inclusion : interest only in the far-field elastic range limit of small inclusion volume Ω I → 0 assumption: homogeneous inclusion (not necessary because average eigenstrain) * σ *jk = C jkmnε mn

Elastic field outside the inclusion

ui ( x ) = − ∫ I Gij ,k ( x − x′)σ *jk ( x′)dV ' Ω

ui ( x ) = −Gij ,k ( x )Ω Iσ *jk

σ pq ( x ) = −C pqmn ∫ Gmj ,nk ( x − x′)σ *jk ( x′)dV ' Ω σ pq ( x ) = −C pqmnGmj ,nk ( x )Ω Iσ *jk

∝

1 r2

I

Interaction with an applied strain

F inter = ∫ I Cijkl ε ij* ( x )ε klA ( x )dV Ω

1 ∝ 3 r

F inter = Ω Iσ ij*ε ijA (0)

Infinitesimal inclusion fully characterized by

Ω Iσ ij* = Ω ICijkl ε kl* (strain source)

4. Point Defect Dipole : Point defect modeled as an equilibrated distribution of point forces: n

force F acting in

an

∑ Net torque : ∑

Equilibrium

Displacement created by the point defect: Long range displacement:

r ≫ an

Resultant :

n

n

F =0

n

F ×a = 0

n

n

ui (r ) = ∑ n Gij (r − a n ) Fj n

ui (r ) = Gij (r )∑ n Fj − n

∂Gij (r ) ∂rk

n n F a ∑n j k +…

= 0 (equilibrium: no force resultant)

Pjk = ∑ n Fj n ak n : elastic dipole modeling the point defect

(first moment of the force distribution) symmetric tensor (equilibrium: no torque)

ui (r ) = −Gij ,k (r ) Pjk

and

σ ij (r ) = −Cijkl Gkm,nl (r ) Pmn

4. Point Defect Dipole : Point defect modeled as an equilibrated distribution of point forces: n

force F acting in

Pjk = ∑ n Fj n ak n

an

Interaction with an applied elastic field:

F inter = −∑ n Fkn ukA (a n ) Limited expansion:

F inter = ukA (0)∑ n Fkn − ukA,l (0)∑ n Fkn aln + … = −ε klA (0) Pkl + …

D. J. Bacon, D. M. Barnett and R. O. Scattergood, Prog. Mater. Sci. 23, 51 (1980).

Point defects: 2 equivalent modelds: - Eshelby infinitesimal inclusion - elastic dipole

−1 Ω Iε ij * = Cijkl Pkl

F inter = −ε klA (0) Pkl = −σ klA (0)Ω Iε kl*

4. Point Defect Point defect in an elastic body: 2 equivalent models - Eshelby infinitesimal inclusion - elastic dipole −1 Ω Iε ij * = Cijkl Pkl

F inter = −ε klA (0) Pkl = −σ klA (0)Ω Iε kl* + infinitesimal dislocation loop

Microscopic elasticity theory (Khachaturyan, Cook, De Fontaine) Point defect modeled by Kanzaki forces Fn Only the first moment in dipole approach

Crystal answers through its force-constant matrices corresponding to its whole phonon spectrum Elastic continuum corresponds to the long wavelength limit (Cijkl ~ slope of the phonon dispersion in Γ)

4. Point Defect Determination of the elastic dipole from atomistic simulations Volume V - containing nPD point defects Pij - submitted to a homogeneous strain εij Elastic energy:

E=n E PD

PD

+E

PD inter

1 + VCijklεijεkl − nPDPijεij 2

EPD : point defect self energy : point defect interaction energy EPD inter Homogeneous stress:

1 ∂E nPD σij = = Cijklεkl − Pij V ∂εij V

1) Conditions with no stress (experiments):

σij = 0

n PD −1 εij = Cijkl Pkl V

Pij deduced from measured strain 2) Conditions with no strain (atomistic simulations): nPD

εij = 0

σij = −

V

Pij

Pij deduced from measured stress

(Vegard’s law)

5. Carbon – Dislocation Interaction in Iron C atom in α iron: interstitial in octahedral site with 3 variants: [100], [010], [001] modeled as a point-force dipole

 Px 0 P = ( ij )  0 

[001] variant z

=

y x

Stress

Atomic simulations: • EAM potential1,2 • Atomic positions relaxed

• Periodicity vectors fixed: εij

Py 0

0 0  Pz 

0

(MPa)

σxx = σyy

-40

=0

1 σ ij = − Pij V  Px 0 0  1  (σ ij ) = − V  0 Px 0  0 0 P z  

σzz

-80

-120 0

2×10-5

4×10-5

6×10-5

8×10-5

1/V (Å-3)

M.I. Mendelev, S. Han, D.J. Srolovitz, G.J. Ackland, D.Y. Sun, M. Asta, Philos. Mag. 83, 3977 (2003) C. Becquart, J. Raulot, G. Bencteux, C. Domain, M. Perez, S. Garruchet, & H. Nguyen, Comp. Mater. Sci. 40 (2007), 119. 1

2

0

5. Carbon – Dislocation Interaction in Iron Atomistic simulations • Empirical potential (EAM) for Fe-C alloys1,2 • Molecular statics (MS) binding energies between - C atom in octahedral interstitial site - ½〈111〉{110} screw or edge dislocation Ebind = E(dislo) + E(C) – E(dislo+C) – E(bcc) Elasticity theory

Ebind = Pijεijd = Vεijσijd

(model first proposed by Cochardt et al.3)

εijd and σijd = Cijklεkld : strain and stress due to dislocation calculated with

- isotropic elasticity - anisotropic elasticity

When the point defect acts only as a dilatation center

E bind = δΩ ∑ i σ iid / 3 with δΩ = V ∑ i ε ii : point defect relaxation volume (“size interaction” model of Cottrell and Bilby4)

M.I. Mendelev et al, Philos. Mag. 83, 3977 (2003) 2 C. Becquart,et al., Comp. Mater. Sci. 40, 119 (2007) 3 A. W. Cochardt, G. Schoek, and H. Wiedersich, Acta Metall. 3, 533 (1955) 4 A. H. Cottrell and B. A. Bilby, Proc. Phys. Soc. A62, 49 (1949) 1

5. Carbon – Dislocation Interaction in Iron Screw dislocation

C interstitial

[1 10]

x

Screw dislo

[111]

h

Glide plane

[1 12]

[001] variant – h = 3.5d110

[100] variant – h = 4d110 Ebind (eV)

Ebind (eV)

0.04

0.2

0 -0.04

0.1

-0.08 0

-0.12 -40

-20

0

20

40

x (Å)

Anisotropic elasticity: all stress components only pressure Isotropic elasticity: all stress components

-40

-20

0

20

Atomic simulations

40

x (Å)

5. Carbon – Dislocation Interaction in Iron Screw dislocation

C interstitial

[1 10]

x

Screw dislo

[111]

0.4

h = 3.5 d110

2 d110

1.5 d110

0 0.2

-0.5 d110

0.2

- 2 d110

- 2.5 d110

- 4 d110

0

0

-0.2

-0.2 -20

[001] variant

Ebind (eV) h = 4 d110

0.4

Glide plane

[1 12]

[100] variant

Ebind (eV)

h

-10

0

10

20

-20

-10

0

10

x (Å) Anisotropic elasticity: all stress components

20

x (Å) Atomic simulations

5. Carbon – Dislocation Interaction in Iron Edge dislocation

b

[121]

Edge dislo

Ebind

0.04

(eV)

C interstitial

[110] x

h

Glide plane

[111]

[100] variant – h = -9d110

Ebind 0

[010] variant – h = -7.5d110

(eV)

-0.02

0

-0.04

-0.04

-0.06 -0.08 -40

-20

0

20

40

x (Å)

-0.08

Anisotropic elasticity: all stress components only pressure Isotropic elasticity: all stress components only pressure

-40

-20

0

20

Atomic simulations

40 x

(Å)

5. Carbon – Dislocation Interaction in Iron Edge dislocation

b

[121]

[110] x

Edge dislo

Ebind

C interstitial h

[111]

[100] variant

(eV)

Ebind

0.4

4 d110 2 d110 0.2 0 -2 d110 -4 d110

[010] variant

(eV)

0.2

0

0

-0.2

-20

Glide plane

-10

0

10

x (Å)

2.5 d110 -0.2 0.5 d110 -1.5 d110 -3.5 d110 -0.4 -20

Anisotropic elasticity: all stress components

-10

0

10

Atomic simulations

x (Å)

5. Carbon – Dislocation Interaction in Iron Comparison atomic simulations / elasticity theory1 Quantitative agreement for r > rc when considering in elastic modeling - dilatation and tetragonal distortion due to the C atom - anisotropy • screw: rc ~ 2 Å • edge: rc ~ 20 Å elastic model can be used in mesoscale simulations (DD, AKMC, …) Diffusion under stress2 Stress dependence of the activation energy of the point-defect jump

Eact = Esaddle − Estable

Esaddle Eact

(

)

= E0saddle − E0stable − Pijsaddle − Pijstable εij

Estable

1 E. Clouet, S. Garruchet, H. Nguyen, M. Perez, and C.S. Becquart, Acta Mater. 56, 3450 (2008) 2 R. Veiga, M. Perez, C. Becquart, E. Clouet and C. Domain, Acta Mater. 59, 6963 (2011)

5. Carbon – Dislocation Interaction in Iron Diffusion near the core C diffusion bias induced by dislocation:

edge 300K

screw 300K

d = ∑ Pi→ jδi→ j i

δi→ j : jump vector Pi→ j : jump probability - EAM + NEB - elasticity

AKMC simulations

R. Veiga, M. Perez, C. Becquart, E. Clouet and C. Domain, Acta Mater. 59, 6963 (2011)

6. Isolated Point-Defect in Ab Initio Calculations Atomistic simulations: valuable tool to study point defects (vacancies, self-interstitial atoms, solute, clusters, …) structure formation, migration, interaction energies accuracy and transferability

• empirical potentials • ab initio calculations

size of the system

− DFT − hybrid functionals

Point defects in a simulation cell with periodic boundary conditions artifact due to interaction with periodic images charged defect: Coulombian interaction (Einter ~ 1/L) neutral defect: elastic interaction (Einter ~ 1/L3)

L

6. Isolated Point-Defect in Ab Initio Calculations Self-interstitial clusters in bcc iron : Cluster containing 8 SIAs ‹111› loop

GGA

C15 aggregate1

ε=0 : ∆E = 5.6 eV σ=0 : ∆E = 0.6 eV 1 M.-C. Marinica, F. Willaime, and J.-P. Crocombette, Phys. Rev. Lett. 108, 025501 (2012).

6. Isolated Point-Defect in Ab Initio Calculations Atomistic simulations of point defects with periodic boundary conditions

EPD atomistic Isolated point defect:

PD PBC EPD = E − 1 2 E ∞ atomistic inter

Elasticity theory

EPBC inter

- withdraw artifact due to PBC - consistency of ε=0 and σ=0 calculations

′ εPBC = − ∑ Gik,jl (Rnmp )Pkl ij n,m,p

with

R nmp = na1 + ma 2 + pa3

summation on periodic images

Inputs needed to evaluate interaction energy • stress / strain of the supercell containing the point defect: Pij • elastic constants of the perfect crystal: Gik,jl Fast and simple post-treatment

6. Isolated Point-Defect in Ab Initio Calculations Self-interstitial clusters in bcc iron : Cluster containing 8 SIAs ‹111› loop

GGA

C15 aggregate1

ε=0 : ∆E = 5.6 eV σ=0 : ∆E = 0.6 eV 1 M.-C. Marinica, F. Willaime, and J.-P. Crocombette, Phys. Rev. Lett. 108, 025501 (2012).

6. Isolated Point-Defect in Ab Initio Calculations Cluster containing 8 self-interstitial atoms: formation energy ‹111› loop

GGA

Uncorrected ε=0 : ∆E = 5.6 eV σ=0 : ∆E = 0.6 eV

C15 aggregate1

With elastic correction ε=0 : ∆E = 3.3 eV σ=0 : ∆E = 3.7 eV

1 M.-C. Marinica, F. Willaime, and J.-P. Crocombette, Phys. Rev. Lett. 108, 025501 (2012).

6. Isolated Point-Defect in Ab Initio Calculations Neutral Vacancy in diamond Silicon Si = semi-conductors: good description of the band gap needed simple DFT (LDA/GGA) fails RPA or hybrid functionals needed1 Vacancy: Jahn-Teller distortion long range elastic field HSE06 hybrid functional 2x2x2 k-points mesh

Ef = 4.26 eV

1 F. Bruneval, Phys. Rev. Lett. 108, 256403 (2012).

6. Isolated Point-Defect in Ab Initio Calculations Conclusions Isolated point defect: PD PBC EPD = E − 1 2 E ∞ atomistic inter

Interaction energy evaluated within anisotropic elasticity theory Inputs: • elastic constants of the perfect crystal • stress / strain of the supercell containing the point defect Fast and simple post-treatment (Fortran source code available as supplemental material♠ )

Validation • Interstitial clusters in bcc Fe • Self-interstitial in hcp Zr (formation and migration) • Neutral vacancy in silicon Perspective Convergency could be improved by also correcting atomic forces but the evaluation of ∂2E / ∂X ∂ε is needed (no Hellmann-Feynamn theorem) ♠ C. Varvenne, F. Bruneval, M.-C. Marinica, and E. Clouet, Phys. Rev. B in press (2013).

7. Inhomogeneity and polarizability Eliispoidal inclusion Eshelby inhomogeneous inclusion

equivalent homogeneous inclusion

C

C C’

C

ε*’

ε*

Strain in the inclusion: εI’

Strain in the inclusion: εI

Displacement continuity at the interface: Stress continuity at the interface:

εI '=εI C '(ε I '− ε * ') = C (ε I − ε * ) −1

ε = (C '− C ) S + C  C ' ε * ' *

E

= SE ε*

7. Inhomogeneity and polarizability Ellipsoidal inclusion with an applied strain Eshelby inhomogeneous inclusion

equivalent homogeneous inclusion

C

C C’

C

ε*’

ε*

Strain in the inclusion: εI’ + εA

Strain in the inclusion: εI

Displacement continuity at the interface: Stress continuity at the interface:

ε I '+ ε A = ε I + ε A C '(ε I '− ε * '+ ε A ) = C (ε I − ε * + ε A )

−1

ε = (C '− C ) S + C  C ' ε *'− (C '− C )ε A  *

E

= SE ε* + εA

strain source depends on the applied strain

7. Inhomogeneity and polarizability Point defect modeled in elasticity theory by an elastic dipole with an amplitude depending on the applied strain first order: 0

Pij = Pij + α ijkl ε kl

dipole (para-elasticity)

polarizability (dia-elasticity)

Elastic interaction with an applied strain

E inter = − Pij0ε ij − 1/ 2 α ijkl ε ij ε kl

dependence of the elastic constants with the concentration of point defects (solute atoms) … two strain sources

E inter = − Pij0 ( ε ij(1) + ε ij(2) ) − α ijkl ε ij(1)ε kl(2) − 1/ 2 α ijkl ( ε ij(1)ε kl(1) + ε ij(2)ε kl(2) ) coupling (SIPA)

Under finite T, paraelastic contribution also leads to polarizability for a point defect with several variants average dipole:

Pij = ∑ n

exp ( − Pkl( n )ε kl ) S

Pij( n ) with S = ∑ m exp ( − Pkl( m )ε kl )

7. Inhomogeneity and polarizability Vacancy in bcc iron : interaction with an applied strain Simulation box 10x10x10 (2000 sites) Marinica 2007 EAM potential

 1 0 0 ε  0 1 0   0 0 1  

 0 1 1 ε  1 0 1   1 1 0  

elasticity can be used to model variations with the applied strain of the energies of defects (vacancies, self interstitials, solute, precipitates)

7. Inhomogeneity and polarizability Irradiation creep: Stress Induced Preferential Absorption (SIPA) vcl

Point defects diffusion equation to dislocation: Fick’s law

vcl

I

τ

τ V vcl

vcl

DX CX J X = − DX ∇CX − ∇E inter kT (X = V or I)

Source of coupling with applied stress τ: - drift: interaction energy (applied + dislo stress) - kinetic: stress dependence of DX

Origin of drift coupling: point defect polarization model developed from Eshelby inhomogeneity problem point defect = 2nd phase inclusion with different elastic constants

τ

τ

E inter = − Pij ε ij(1) − Pij ε ij(2) − α ijkl ε ij(1)ε kl(2)

Morning’s Conclusions I. Elasticity Theory 1. Deformation of an elastic body 2. Stresses in an elastic body 3. Thermodynamics of deformation 4. Hooke’s law 5. Equilibrium equation in homogeneous elasticity II. Inclusions, Inhomogeneities and Point Defects 1. Spherical inclusion 2. Eshelby’s inclusion 3. Inclusion and applied stress 4. Point defect 5. Carbon - dislocation interaction in iron 6. Isolated point-defect in ab initio calculations 7. Inhomogeneity and polarizability

III. Dislocations

1. Dislocation: lattice defect 2. Elasticity theory of dislocations 3. Atomistic simulations 4. Dislocation dynamics 5. Crystal plasticity References: • D. J. Bacon, D. M. Barnett and R. O. Scattergood, Prog. Mater. Sci. 23, 51 (1980). • J.P. Hirth & J. Lothe, Theory of dislocations (1982) • D. Hull and D. J. Bacon, Introduction to Dislocations, 5th edition (2011). • V. V. Bulatov and W. Cai, Computer Simulations of Dislocations (2006). • C. Weinberger, W. Cai and D. Barnett, Elasticity of Microscopic Structures, Standford Univ. lecture notes (2005). • D. Rodney and J. Bonneville, Dislocations, in Physical Metallurgy, Eds. D. Laughlin and K. Hono (5th ed., 2014).

1. Dislocation: lattice defect Crystal shear: homogeneous vs localized shear τ

τ

displacement discontinuity

border line = dislocation

1. Dislocation: lattice defect Dislocation: frontier line of a surface of displacement discontinuity - line direction l (varying but oriented) - Burgers vector

b = ∫ du (conserved)

Linear crystal defect propagating the plastic strain

F S edge

b ⊥ζ

screw

S F

b ζ SF/RH convention (see chap. 1 in Hirth and Lothe)

1. Dislocation: lattice defect The displacement discontinuity needs to be bounded close dislocation lines (loops)

b S

F

S

F

1. Dislocation: lattice defect The displacement discontinuity needs to be bounded close dislocation lines (loops) or intersection with other defects (dislocations, surfaces)

b2

b b b1

b3 = b1 + b2

The Burgers vector needs to be a vector of the Bravais lattice otherwise a stacking fault is generated (perfect vs partial dislocation)

1. Dislocation: lattice defect When a dislocation segment moves, the corresponding surface area is displaced by one Burgers vector.

δl × x

The volume involved in the transformation is

δΩ = (δ l × x )·b = (b × δ l )·x

δl x

Glide = conservative motion: δΩ = 0 displacement needs to be orthogonal to line and Burgers vectors Burgers vector and line direction belong to the glide plane (screw dislocation: several glide planes)

Climb = non conservative motion: δΩ ≠ 0 displacement perpendicular to the glide plane diffusion needed

1. Dislocation: lattice defect Orowan loop (plasticity)

Prismatic loop (irradiation)

b ⋅n ≠ 0

b ⋅n = 0

(pure prismatic if

b n)

n n

b

glide plane

b

glide cylinder (unfaulted loop)

motion out of the loop plane: climb

motion in the loop plane: (self) climb

1. Dislocation: lattice defect Dislocations in real life

Daniel Caillard

TEM in situ straining experiments

Ti @ 150 K

Zr @ 150 K

E. Clouet, D. Caillard, N. Chaari, F. Onimus and D. Rodney, Nature Materials in press (2015).

1. Dislocation: lattice defect Dislocations in real life Fe @ 120 K

High yield strength Strong temperature dependence

Fe @ ambiant T

D. Brunner and J. Diehl, Phys. Stat. Sol. A 160, 355 (1997). D. Caillard, Acta Mater. 58, 3493; 3504 (2010).

Elasticity: reminder Reference state

dx x dl Displacement:

Strained state Deformation

dx ' x ' dl ' u = x′ − x

Strain (small deformation):

Stress:

1  ∂ui ∂u j  ε ij =  +  2  ∂x j ∂xi 

1  ∂F  σ ij =   V  ∂ε ij T

Elasticity constitutive law:

Equilibrium:

fi +

∂σ ij ∂x j

=0

1 F (T , ε) = F0 (T ) + VCijkl ε ij ε kl 2 ∂ 2uk σ ij = Cijkl ε kl + fi = 0 and Cijkl ∂x j ∂xl

Elasticity: reminder Green’s function: solution to a unit point force

∂ 2Gkn Cijkl + δ inδ ( x ) = 0 ∂x j ∂xl

∫ f ( x′)dx′ u ( x ) = ∫ G ( x − x′) f ( x′)dx′ σ ( x ) = C ∫ G ( x − x′) f ( x′)dx′

Force distribution k

ij

kn

ijkl

n

kn ,l

Isotropic elasticity:

1 Gkn ( x ) = 16πµ (1 −ν ) x

n

 xk xn  (3 − 4ν )δ ij + 2  x  

1 Gkn (r ) = g (θ , φ ) r 1 Gkn ,i (r ) = 2 h(θ , φ ) r 1 Gkn ,ij (r ) = 3 f (θ , φ ) r

2. Elasticity theory of dislocations Linear elasticity theory

∂ 2 uk Cijkl + fi = 0 ∂x j ∂xl Dislocation definition:

σ ij = Cijkl ε kl

and

1  ∂ui ∂u j  ε ij =  +  2  ∂x j ∂xi 

b = ∫ du

ζ

n

SD

Elastic field created by a dislocation loop in an infinite body

ui ( x ) = C jklmbm ∫ D nl Gij ,k ( x − x′)dS ′

b (Volterra’s formula)

S

σ ij ( x ) = Cijkl ∫ εlnhC pqmnbmζ h ( x′)Gkp ,q ( x − x′)dl ′

(Mura’s formula)

D

L

with ζh unit vector along the line

displacement: surface integral depends on history stress: line integral state variable computer implementation of stress calculations: DD simulations divergence close to the dislocation line: ln(r) for displacement

1/r

for stress

2. Elasticity theory of dislocations

Elastic field created by a dislocation loop in an infinite body

ui ( x ) = C jklmbm ∫ D nl Gij ,k ( x − x′)dS ′

ζ

n

S

= lim C jklm ∫ I ε Gij ,k ( x − x′)dV ′  Ω h→0  * ml

Dislocation loop equivalent to an Eshelby inclusion

Ω I = hS 1 * ( bm nl + bl nm ) • eigenstrain ε ml = − 2h • volume

SD h

b ΩI

2. Elasticity theory of dislocations

Elastic field created by a dislocation loop in an infinite body far from the loop

ui ( x ) = C jklmbm ∫ D nl Gij ,k ( x − x′)dS ′ S

= C jklmbm Sl Gij ,k ( x )

σ pq ( x ) = C pqinC jklmbm Sl Gij ,kn ( x )

1 ui ( x ) ∝ 2 x 1 σ ij ( x ) ∝ 3 x

Infinitesimal dislocation loop can model a point defect D elastic dipole:

Pij = −Cijkl bk Sl

(first order of multipole expansion)

ζ

n

SD b

2. Elasticity theory of dislocations Interaction with an applied stress using Gauss theorem

ζ

n

SD b

E inter

D ∂ u = ∫ σ ijAε iDj dV = ∫ σ ijA i dV V V ∂x j

∂σ ijA

∂ uiD dV =∫ σ ijA uiD ) dV − ∫ ( V ∂x V ∂x j j =∫

S

D+

∪S

D-

σ ijA uiD dS j

=0 (equilibrium)

= ∫ D σ ijA bi dS j S

using analogy with Eshelby inclusion

E inter = − ∫ I Cijkl ε ij* ( x )ε klA ( x )dV Ω

with

Ω = hS I

D

E inter = ∫ D σ ijA bi dS j S

and

ε

* ml

bm nl =− h

2. Elasticity theory of dislocations Interaction with an applied stress

ζ

n

SD b

E inter = ∫ D σ ijA bi dS j S

Energy variation δE when the dislocation sweeps

an infinitesimal surface area δSD Peach Koehler force

F PK = ( σ A b ) × ζ

2. Elasticity theory of dislocations Dislocation self energy

ζ

n

SD b

1 E = ∫ σ ijDε iDj dV 2 V D

needs to introduce a cutoff distance, core radius rc, to avoid divergence in the core inside the core region: core energy outside: elastic energy can be obtained by a double line integral1 + tractions on the core cylinder2

other solution = spreading of the Dirac peak defining the density of Burgers vector: Peierls Nabarro, standard core3, non singular theory4 the self energy induces a self stress 1. J. Lothe, Philos. Mag. A 46, 177 (1982). 2. E. Clouet, Philos. Mag. 89, 1565 (2009). 3. J. Lothe, Dislocations in Continuous Elastic Media, 187 (1992). 4. W. Cai, A. Arsenlis, C. R. Weinberger and V. V. Bulatov, J. Mech. Phys. Solids 54, 561 (2006).

2. Elasticity theory of dislocations Dislocation self energy: line tension approximation

1 E = ∫ σ ijDε iDj dV 2 V

ζ

n

D

SD E D = ∫ D E ∞ (θ )dl

b

b

θ

L

with

R 1 2 E (θ ) = K (θ )b log   2  rc  ∞

energy of an infinite straight dislocation of same character θ

R characteristic size of the loop

E D ∝ R [ log( R) + cste ] Self stress

Γ τ =− ρb

line tension

∂ 2 E (θ ) Γ = E (θ ) + ~ µb 2 ∂θ

dislocation curvature radius ρ

ζ

2. Elasticity theory of dislocations Infinite straight dislocation: screw in isotropic elasticity cylindrical symmetry

F S screw

b ζ

θ u ( ρ ) = b ez ∝ log( ρ ) 2π 1 σ = µ b ∝  θ z 2πρ ρ  σ ρρ = σ θθ = σ zz = σ ρ z = σ ρθ = 0 

R µb2 log   Strain energy contained in a cylinder of radius R: E = 4π  rc  Edge dislocation

u ( ρ ) ∝ log( ρ )

σ∝

1

ρ

µb2

R E= log   4π (1 −ν )  rc 

higher energy

2. Elasticity theory of dislocations Infinite straight dislocation: anisotropic elasticity1-3 same radial variation as in isotropic elasticity but angular dependence changes (non null term)

u ( ρ ) ∝ log( ρ )

σ∝

1

ρ

R 1 Strain energy contained in a cylinder of radius R: E = bi K ij b j log r  2  c defined by Stroh matrix K ij

0  1 0 µ 0 1 K ij = 0   2π (1 −ν )   0 0 1 − ν  ( x , y , z )

isotropic elasticity:

for z along the dislocation line (screw orientation)

1. J. D. Eshelby, W. T. Read and W. Shockley, Acta Metall. 1, 251 (1953). 2. A. N. Stroh, Philos. Mag. 3, 625 (1958). 3. A. N. Stroh, J. Math. Phys. (Cambridge, Mass.) 41, 77 (1962).

2. Elasticity theory of dislocations 1 111 {110} 2 edge dislocation in Fe

E (eV .Å-1) 1.5

Atoms fixed by linear anisotropic elasticity

1 111 {110} 2

1

24331 + 1952 atoms (×3)

0.5

Data Fit

r 0

0

40

80

120

r (Å)

∆E (eV.Å-1)

Atoms relaxed using empirical potential

1 ∆E ( r ) = E ( r ) − bi K ijb j ln  r  2  rc 

0.3

Excess energy stored in a cylinder (radius r)

E (r ) = E fit

core

R 1 + bi K ij b j ln   2  rc  known

(linear anisotropic elasticity)

Data Fit

0.25

Ecore = 286 ± 1 meV.Å-1 0.2

rc = b

0

40

80

120

r (Å)

3. Atomistic simulations vallées de Peierls

σ=0

∆E 0

σP contrainte de Peierls ∆E WP barrière de Peierls

λP

0

σ = σP

σ>0

- b σ xDislo xDislo

xDislo

Dislocations: • champ élastique à longue distance • perturbation importante du réseau cristallin au voisinage de la ligne: cœur Mobilité des dislocations: propriété liée au cœur - glissement: plan de glissement, énergie et contrainte de Peierls - montée: absorption ou élimination de défauts ponctuels - glissement dévié: changement de plan de glissement

3. Atomistic simulations • Description de la structure du cœur des dislocations simulations atomiques • Champ élastique à longue distance • Séparation énergie de cœur / énergie élastique couplage avec théorie élastique (anisotrope)

Informations transposables aux échelles supérieures (DD, plasticité cristalline…) - contraintes de Peierls - lois de mobilité …

• Potentiel empirique (EAM) • Ab initio (DFT)

500

Natomes

Précision

- plan de glissement - énergie d’activation

need to go to a small scale to have a precise description of the atomic bonding (transition metals)

3. Atomistic simulations The displacement discontinuity opened by the dislocation has to be closed a simulation box with periodic boundary conditions cannot contain a single dislocation

Solutions: - simulation box with surfaces - dislocation dipole with PBC

3. Atomistic simulations 1. Cluster approach with fixed boundary Isolated dislocation in a cylinder

Inner atoms relaxed according to atomic forces Outer atoms fixed to elastic solution (Volterra)

Main drawbacks: - the Volterra solution is only the leading term of the elastic field created by the dislocation. The core field cannot be accommodated.

1 u ( ρ ) = uV log( ρ ) + O   ρ - back-stress exerted by the boundary when the dislocation moves. ok for empirical potentials, but not for ab initio

3. Atomistic simulations 2. Cluster approach with relaxed boundary Isolated dislocation in a cylinder Inner atoms relaxed according to atomic forces Intermedidate atoms fixed and used to calculate atomic forces due to the dislocation Outer atoms fixed, only here to screen the surface Lattice Green’s functions (inverse of force constants matrix) are used to relax atomic forces that build in the intermediate zone correct harmonic, thus elastic, relaxation of dislocation core Main drawback: the energy contribution of the dislocation cannot be isolated from the surface one in ab initio calculations (no cutoff distance for atomic interactions) ok for atomic structure, for Peierls stress, but not for energy J. E. Sinclair, P. C. Gehlen, R. G. Hoagland and J. P. Hirth, J. Appl. Phys. 49, 3890 (1978). C. Woodward and S. I. RaoPhys. Rev. Lett. 88, 216402 (2002). J. A. Yasi and D. R. Trinkle, Phys. Rev. E 85, 066706 (2012).

3. Atomistic simulations 3. Dipole approach Periodic array of dislocations (periodic boundary conditions) All atoms relaxed according to atomic forces Main drawback: interaction between dislocations and their periodic images can be minimized with quadripolar arrangement (but still a live)

b −b

If the interaction is only caused by Volterra elastic field, it can be calculated within linear elasticity theory with proper account of PBC Interaction energy between 2 dislocations at a distance D

E

inter

≃ −b K b (1) i

(2) ij j

D log    rc 

W. Cai, V. V. Bulatov, J. Chang, J. Li and S. Yip, Philos. Mag. 83, 539 (2003).

3. Atomistic simulations 3. Dipole approach

Screw dislo Fe

3n - 1

ab initio (L. Ventelon)

E (eV/Å) < 101 >

Eexcess < 12 1 >

< 111 >

z = 2b/3

z = b/3

z=0

0.4

Eelastic

3n - 1

0.2

Ecore 0

0

100

200

300

Natoms

core energy does not depend on simulation setup E. Clouet, L. Ventelon and F. Willaime, Phys. Rev. Lett. 102, 055502 (2009).

3. Atomistic simulations Screw Dislocation in Zr: Core Structure Differential displacement maps1 (Vitek): arrow proportional to difference of atomic displacement projected in the direction of the Burgers vector measure of strain Dislocation density2 (Nye tensor): Nye tensor α

bi = ∫ α ij n j dS A

αij: density of dislocation with line along

direction i and Burgers vector along j

α = −∇ × F with distortion F extracted from atomic positions 1. V. Vitek, R. C. Perrin and D. K. Bowen, Philos. Mag. 21, 1049 (1970). 2. C. S. Hartley and Y. Mishin, Acta Mater. 53, 1313 (2005).

3. Atomistic simulations Screw Dislocation in Zr: Core Structure Disregistry (in prismatic plane) 1) Atomic simulations: disregristy displacement difference between above and below glide plane

∆u ( x ) = u up ( x) − u down ( x )

b

uup udown

2) Fit Peierls-Nabarro model dislocation position x0 dissociation length d partial spreading δ/π

box S 7x7 (196 atoms)

b   x − x0 − d / 2   x − x0 + d / 2   ∆u ( x) = π − atan  − atan     δ δ 2π      R. Peierls, Proc. Phys. Soc. 52, 34 (1940). F. R. N. Nabarro, Proc. Phys. Soc. 59, 256 (1947). D. Rodney and J. Bonneville, Dislocations in Pysical Metallurgy (2014).

3. Atomistic simulations Screw Dislocation in Zr: Core Structure

box S 7x7 (196 atoms)

dissociation in prismatic plane in 2 partial dislocations b/2

3. Atomistic simulations Dislocation dissociation: Elastic energy

b

(1)

+b

(2) 2

E D ∝ b2

> b

(1) 2

+b

(2) 2

if

b (1) .b (2) > 0 dissociation reduces elastic energy … but creation of a stacking fault

∆u

∆u

b

b

0

0

x

d

Energy variation due to dissociation: minimal for

d eq =

γ

∆Ediss (d ) = −b K b

bi(1) K ij b (2) j γ

(1) i

(2) ij j

d  ln   + γd  rc 

Stacking fault needs to be stable

Generalization (γ-surface and dislo distrib.): Peierls-Nabarro model

3. Atomistic simulations Generalized Stacking Fault: • the crystal is sheared in a plane with different fault vectors • atoms are relaxed perpendicularly to the fault plane look for a minimum on the surface energy Hcp Zr

b

basal plane

Slight minimum in 1 3[1100] γb = 13.3 meV/Å2

prism plane

b

Minimun in 1 6[1210] γp= 13.2 meV/Å2

3. Atomistic simulations Screw Dislocation in Zr: Core Structure

Different stable core structures of the same dislocation can exist need to know their energy to conclude on their stability

3. Atomistic simulations Zr: stability mobility

Simulation cell: 192 atoms (6x8)

Mobility • prismatic planes easy glide • pyramidal planes (Peierls) nucleation of kink pairs b

c

d

e

3. Atomistic simulations Zr

Ti

3. Atomistic simulations Kink Formation and Migration Dislocation glide by nucleating and propagating kinks

MD simulations in Fe (Mendelev potential) 77 K

C. Domain and G. Monnet, Phys. Rev. Lett. 95, 215506 (2005).

3. Atomistic simulations Mobility of screw dislocation in bcc iron Activation energy for kink nucleation determined from the Peierls potential dependent on the applied stress (glide and non-glide components)

Thermal activation law of the plastic strain rate

γɺ = ∑ γɺ 0α exp  −Hα (σ) / kT  α

L. Proville, L. Ventelon and D. Rodney, Phys. Rev. B 87, 144106 (2013). L. Dezerald, L. Proville, L. Ventelon, F. Willaime and D. Rodney, Phys. Rev. B 91, 094105 (2015).

Conclusions I. Elasticity Theory II. Inclusions, Inhomogeneities and Point Defects III. Dislocations 1. Dislocation: lattice defect 2. Elasticity theory of dislocations 3. Atomistic simulations 4. Dislocation dynamics 5. Crystal plasticity IV. Plane interfaces