What Drives Microstructural Evolution:
Energies of Defects or Elastic Strains
and Stresses?
S.L. Dudarev, P.-W. Ma, D.R. Mason, and F. Hofmann
UK Atomic Energy Authority, Culham Centre for Fusion Energy, Oxfordshire, UK
Department of Engineering, University of Oxford, Parks Road, Oxford OX1 3PJ, UK||

JET tokamak

ITER

Plasma-facing tiles (beryllium)

Tritium breeding blanket
modules (FM steel)

Divertor (tungsten +CuCrZr)

JET

Fusion experiments: scaling up
3|

Hierarchical multiscale modelling – a conventional way to macroscale simulations.

4|The structure of elementary defects

A self-interstitial atom defect in body-centred cubic (bcc) iron.

5|

The structure of elementary defects

A self-interstitial atom defect in body-centred cubic (bcc) iron.

6|

The structure of elementary defects

A self-interstitial atom defect in body-centred cubic (bcc) iron.

7|

Formation and migration energies

8|

Annual Review of Materials Research 43 (2013) 35

Structure of self-interstitial defects
Fe

110 dumbbell
does not occur

9|tetrahedral

Tungsten, vanadium, etc.

111 crowdion
does not occur

octahedraldoes not occur100 crowdion

Dynamics of self-interstitial defects

Thermal migration of a 110
dumbbell. Occurs in Fe andferritic steels.

110 dumbbell

111 crowdion

10 |Thermal migration of alinear 111 type defect.
Occurs in non-magnetictransition metals(tungsten, vanadium,molybdenum).

11 |Lattice distortions are of primary significance to engineering, where they areknown as “strains and stresses”.

Stress field produced by a cascade

Stress field in the vicinity of defects formed in a collision cascade.

12 |D. R. Mason et al., Journ. Appl. Phys. (2019) in press

Elastic fields of defects

Green’s function of elasticity: the field of atomic displacements generated bya point source. Different from the Coulomb law because in a solid there are
two different velocities of sound, longitudinal and transverse.

13 |Elastic fields of defects

elastic dipole tensor – a 3x3 symmetric real matrix containing six independent
parameters (three eigenvalues defining the shape of the defect, and three
angles defining its orientation).

14 |Elastic fields of defects

Amazingly, the elements of this tensor are almost always computed, by DFT or
molecular statics, in any simulation involving atomic relaxations.

15 |E. Clouet et al., Acta Materialia 56 (2008) 3450;
P.-W. Ma and S.L. Dudarev, Phys. Rev. Mat. 3 (2019) 013605

Elastic fields of defects

relaxation volume tensor: the sum of its diagonal elements gives therelaxation volume of the defect

16 |S.L. Dudarev et al., Nucl. Fusion 58 (2018) 126002

Dipole and relaxation volume tensors

17 |P.-W. Ma and S.L. Dudarev, Phys. Rev. Mat. 3 (2019) 013605

Elastic fields of defects

Averaging over orientations produces a diagonal tensor

18 |Strain field of defects
In applications, we are interested in the elastic field produced by many
defects => self-averaging

Anisotropic crystallographic effects are also not significant for large
structural components where the orientations of grains are randomstraindimensionless

19 |ω(r) is the density of relaxation volumes of defects.

Stress field of defects
The next (key) step: computing stresses.

Strain tensor does not enter the equilibrium conditions and not convenientin the context of FEM. Computing stresses requires convoluting the strain
tensor with the tensor of elastic constants:

non-local

local

20 |

S.L. Dudarev et al., Nucl. Fusion 58 (2018) 126002

Stress field of defects

Using this, it is possible to compute the derivative of stress – which is known
as the “body force”

B is the bulk modulus of the material.

21 |S.L. Dudarev et al., Nucl. Fusion 58 (2018) 126002

Condition of global equilibrium
Condition of equilibrium includes gravity, thermal expansion, andswelling due to defects

defects
gravity

thermal expansion

Defects, as opposed totemperature, generatecompressive or tensile strains:
this agrees with DFTW. Hertz et al., Phys. Letters 43A (1973) 289

22 |A finite element model implementation

Traction free boundary
conditions at surfaces

New equations have been implemented in the ABACUS finite element code. The
FEM implementation also includes the various conventional body and surface
forces, for example applied external stresses.

23 |Nucl. Fusion 58 (2018) 126002

Case studies: a R=3m steel shell
Case study 1

Analytical solutions of the equilibriumelasticity equations and numerical finiteelement solutions agree exactly. Highstresses develop even if swelling is low.

24 |Nucl. Fusion 58 (2018) 126002

Statistics of generation of defects
A 200 keV cascade in tungsten.

X. Yi et al., EPL 110 (2015) 36001

A.E. Sand et al., EPL 103 (2013) 46003

simulations|Analysis of hundreds of collisioncascade simulations, and tens of𝐴
𝐹 𝑁 = 𝑆 ; 𝑆 ≈ 1.8
𝑁large defect thousand of events recorded inelectron microscope images showsclusters
that the statistics of sizes of defectpoor or no visibility
experimentclusters follow a power law – likeearthquakes or avalanches.

Relaxation volumes of complex defects

Interstitialloops

VoidsVacancyloops

26 |This spans the poor or no visibility size range.

D. R. Mason et al., Journ. Appl. Phys. (2019) in press

Relaxation volume of a dislocation loop

independent of the distanceto the surface

I. Rovelli et al., Phys. Rev. E 98 (2018) 043002;
S.L. Dudarev and P.-W. Ma, Phys. Rev. Materials
2 (2018) 033602

27 |The relaxation volume of a dislocation loop
equals the volume of the same number ofatoms as the number of defects forming
the loop. Can be positive or negative (SIAor vacancy). Invariant if the loop glides.

Relaxation volume of a helium bubble

p is the pressure of gas inside the bubble and γ is the average surface
energy density. If p=0, the relaxation volume of a void is:

It is negative - hence material containing only voids and no dislocation
loops, contracts.
D. R. Mason et al., Journ. Appl. Phys. (2019) in press

28 |Microstructure driven by elastic forces

Dislocation climb may occur due to the diffusion of atoms around the perimeter of adislocation loop, independent of the vacancy atmosphere. At low temperatures thisself-climb is orders of magnitude faster than vacancy-diffusion-mediated climb.

29 | T. D. Swinburne et al., Sci. Repts. 6 (2016) 30596

Stochastic dislocation dynamics

30 |Y. Li, M. Boleininger, C. Robertson et al. (2019)

Recovery of microstructure
Including effects of surfaces requires using Kinchoff’s formula relating derivativesof Green’s functions in the volume of the material and at surfaces, to fully define
the field of vacancies everywhere in the sample 2G0 (x, x' )
 2 c ( x' )  G0 (x, x' )  c(x' ) dV'c(x')−G(x,x')=dS'c(x')
n−G(x,x')

n 

00
V 
x'2
x'2  S 

x'
x'



In the right-hand side of this equation, vacancy
concentration at a point x’ at a surface can beevaluated just like at dislocation lines.
Evaporation of vacancies from dislocations isdriven by elastic self-stress. At surfaces,evaporation is driven by surface tension.
A system of coupled ODEs for the velocities ofnodes on dislocation lines and at surfaces. The
dynamics of diffusion-mediated evolution ofdislocations and cavities/surfaces is fully defined.

31 | I. Rovelli et al., Journ. Mech. Phys. Solids 103 (2017) 121

Recovery of microstructure

Estimated timescales for the evaporation ofVacancy loop in tungsten, evaporating
vacancy dislocation loops, vacancy clusters
due to its own self-stress at T~1600 C.
(voids) and self-interstitial loops in Be, Fe andInitial size of the loop is 100 A, theW at various temperatures. Note the hightotal evaporation time is ~10 s.
temperature sensitivity of the estimated values.
I. Rovelli et al., Physical Review E98 (2018) 043002

32 |A. Breidi et al. (2019)

Summary

A fundamental condition of elastic equilibrium, containing no free parameters,
and including effects of gravity, thermal expansion and swelling due to defects.
Macroscopically, accumulation of defect produce body forces similar tothermal expansion, but the effect can be positive or negative. Note that theaccumulation of defects itself depends on temperature.
Relaxation volumes of defects (the third term) includes invisible defects andcan be computed numerically (DFT or MD). There are also analytical formulae
for the relaxation volume of a dislocation loop, a void or a gas bubble.

33 |