Real space models for microstructural evolution

Transcription

1 Real space models for mcrostructural evoluton P.-W. Ma, D.R. Mason, I. Rovell, T.D. Swnburne, S.L. Dudarev UK Atomc Energy Authorty, Oxfordshre, UK. Ths work has been carred out wthn the framework of the EUROfuson Consortum and has receved fundng from the Euratom research and tranng programme under grant agreement No and from the RCUK Energy Programme [grant number EP/P012450/1].

2 Integrated model for DEMO materals Plasma as a 3D neutron source Neutron spectra at varous locatons n a DEMO power plant Transmutatons Defect producton M.R. Glbert et al., Nucl. Fuson 52 (2012) ; J. Nucl. Mater. 442 (2013) S755; J. Nucl. Mater. 467 (2015) 121

3 Power densty deposted n blanket materals Power densty, deposted by neutrons, falls off quckly wth dstance nto the fuson blanket away from the plasma. Most of the energy s deposted n the frst 30cm of blanket structure. S. Sato, K. Mak, Fus. Eng. Desgn 65 (2003) 501 Energy from neutrons s mostly deposted n the frst 10cm of the blanket structure.

4 Annealng of radaton defects: 1 hour F. Ferron, X.Y, K. Arakawa, et al., Hgh temperature annealng of on rradated (1.5 dpa) tungsten, Acta Mater. 90 (2015) 380.

5 Annealng of radaton defects: 1 hour F. Ferron, X.Y, K. Arakawa, et al., Hgh temperature annealng of on rradated (1.5 dpa) tungsten, Acta Mater. 90 (2015) 380.

6 F. Ferron, X. Y, K. Arakawa et al., Hgh temperature annealng of on rradated (1.5 dpa) tungsten, Acta Mater. 90 (2015) 380. The move shows the dynamcs of recovery of radaton defects at 1100ºC.

7 Real space models for mcrostructure ), ( ), ( ), ( ) ( ), ( ), ( ), ( t T k t D c t J t I t J t t c v B v x x x x x x Here Γ defnes a dslocaton lne, so that. ) ( ) ( ] [ 3 ds f x d f x x In the dlute gas approxmaton for the chemcal potental of vacances, we arrve at the boundary value problem for a movng dslocaton lne. A dslocaton lne moves due to elastc forces actng on t, whch stmulate absorpton or emsson of vacances: );, ( ln ), ( t c T k t B v x x c c v b c D cl e v ) ( ) ( 2 x moble vacances moble vacances

8 Real space models for mcrostructure The central step s the converson of the boundary value problem nto an ntegral equaton c 1 b ( v d l') x) 4 c D x x' ( v Assume that a dslocaton lne forms a closed loop. At large dstances from the loop, where x-x >> loop sze, ths equaton acqures a smple form: Here 1 c( x) ( '). 4 b v d l c D x v drel b ( v d l'), and rel (t) s the volume of the dslocaton loop. dt.

9 Conservaton of loop volume durng glde rel ( t) ( b A( t)) Volume of a loop remans constant durng ts moton even f the drecton and magntude of the loop vector area changes. Left: smulaton of Brownan moton of a dslocaton loop. Below: expermental observaton of Brownan moton of a dslocaton loop by K. Arakawa. Molecular dynamcs smulaton of thermal Brownan moton of a ½(111) dslocaton loop n ron at 500K, courtesy of M.R. Glbert. d rel d ( b A ( t)) dt dt 0. Invarant of loop moton.

10 Real space models for mcrostructure The feld of vacances changes adabatcally, followng the evoluton of dslocaton loops (centres of loops are at ) rel 1 1 c( x) c 4 D x x v d dt Volume of a dslocaton loop wth Burgers vector and area vector s gven by the scalar product ( b A). Volume s postve for an ntersttal loop and negatve for a vacancy loop. A b x rel b. A A b Intersttal loop, rel 0 vacancy loop, 0 rel The above equatons for c(x) can also be formulated as a set of ODEs for the veloctes of nodal ponts on dslocaton lnes.

11 Dffuson to/from surfaces The computatonal effcency of the treatment developed by Y. Gu, Y. Xang et al., JMPS (2015) s fundamentally due to the use of free-space Green s functon G0( x, x') 4 D v 1. x x' Ths computatonal advantage s lost f, n order to take nto account the boundary condtons, we attempt to modfy these Green s functons. An alternatve approach s the Krchhoff ntegral approxmaton, whch retans the use of free Green s functons V dv S ds 2 2 b( x) a ( x) x) ( x) 2 b x x b ( x) a ( x) ( x) b( x) x x a ( 2 a Ths formula s known as Green s theorem (G. Green, 1828). It provdes the means for treatng surfaces and retans the advantages offered by the free Green s functon formalsm.

12 Dffuson to/from surfaces We choose one of the functons n Green s theorem as the vacancy concentraton feld. The other s free Green s functon. Ths formula below shows that vacancy feld can be evaluated everywhere, f c(x) and ts normal dervatves at surfaces are known. 2 2 G 0( x, x') c( x') G0 ( x, x') c( x') dv ' c( x') G0 ( x, x') ds' c( x') n G0 ( x, x') n 2 2 x' x' x' x' V S In the rght-hand sde of ths equaton, vacancy concentraton at a pont x, stuated at a surface, can be evaluated usng the same approach as the one developed for the dslocaton loops. Evaporaton of vacances from dslocaton loops s drven by elastc self-stress. In the case of surfaces, t s drven by surface tenson. Arrve at a system of coupled ODEs for the veloctes of nodes on dslocaton lnes and at surfaces. Ths fully defnes the dynamcs of dffuson-medated evoluton of loops and cavtes/surfaces.

13 Dffuson to/from surfaces Equatons, descrbng the vacancy dffuson-medated evoluton of dslocaton loops, cavtes and the external surface, have the form: G0 ( x, x') c( x') ( x) c( x) Dv ds' c( x') n G0 ( x, x') n x' x' S Here ( x) 1 n the bulk, ( x) 1/ 2 at surfaces (ths comes from the ntegraton of a delta-functon at the surface) and ( x) 0 n the vacuum. Surfaces also nclude the torodal surfaces wrapped around dslocaton lnes. Ths anmaton shows evoluton of vods, ntersttal and vacancy dslocaton loops, evolvng through the evaporaton and exchange of vacances n tungsten at 1750K.

14 Applcatons: evoluton tmescales Vacancy concentraton profles plotted as a functon of dstance from a typcal object. Vacancy loops (red) and vods (blue) generate local zones wth hgh concentraton of dffusng vacances, exceedng the background concentraton by over two orders of magntude. Intersttal loops produce local vacancy depleted zones. Evoluton of 20 cavtes, 20 ntersttal loops (yellow) and 20 vacancy loops (red) randomly dstrbuted wth the average number densty 5x10-6 nm -3, n a sphercal sample of radus R=142 nm. Objects rad are ntally normally dstrbuted: wth the mean of 3.2 nm and standard devaton of 1 nm (loops); wth the mean of 1 nm and standard devaton of 0.1 nm (18 smaller cavtes); wth the mean of 2.2 nm and standard devaton of 0.22 nm (for the two larger cavtes).

15 Elastcty-drven self-clmb of dslocatons Dslocaton clmb may also occur due to dffuson of atoms around the permeter of dslocaton loops, ndependent of the vacancy atmosphere. At relatvely low temperatures ths vacancy-free clmb s many orders of magntude faster than vacancy-dffuson-medated clmb. T.D. Swnburne et al., Scentfc Reports 6 (2016) 30596

16 Langevn dynamcs of nteractng defects 3D Langevn dynamcs for nteractng defects: dr dt D k T B E r ( t), ( t) ( t') 2D ( t t') j T j Dffuson coeffcents are parameters of the model, related to defect mobltes va the fluctuaton-dsspaton theorem. Interacton between defects s derved from elastcty (sotropc or ansotropc) ths s not a parameter of the model. Fluctuatng thermal force s an essental ngredent of the model, t s requred by the fluctuaton-dsspaton theorem. S.L. Dudarev et al., Phys. Rev. B 81, (2010).

17 Real-tme dynamcs of nteractng defects Langevn dynamcs of nteractng defects dr dt D k T B E r ( t), ( t) ( t') 2D ( t t') j T j Ths set of lnked stochastc equatons s exactly equvalent to a many-body dffuson equaton of the form P t D r P r P k T B E r ; where P P( r1, r2,..., rn, t) The procedure goes beyond the statement of equvalence. The many-body dffuson equaton s not solvable. The mappng s smlar to mappng a quantum many-body problem of nteractng partcles onto a set of the Kohn- Sham equatons for ndependent quas-partcles. The Langevn equatons can be ntegrated usng algorthms developed for molecular dynamcs. S.L. Dudarev et al., Phys. Rev. B 81, (2010).

18 Strans and stresses produced by defects u ( r) Pjk Gj ( r); r ( x1, x2, x3) x k A formula, central to the treatment of elastc felds, strans and stresses produced by defects. G j (r) s Green s functon of elastcty equatons, and P j s the elastc dpole tensor of a defect. From the elastc dpole tensor of a defect, t s possble to evaluate ts relaxaton volume, and compute swellng resultng from the accumulaton of defects n a materal. There s a beatful mathematcal formula for the relaxaton volume of a dslocaton loop. It s also possble to evaluate the energy of nteracton beween any two defects E( r) P (1) k P (2) jl 2 x x k l G j ( r)

19 n Interacton between crowdon defects r e The energy of nteracton between the two defects depends on the dstance between them, and on ther orentaton. Smlarly to the case of a two small dslocaton loops, t s possble to represent ths nteracton n a closed analytcal form.

20 Langevn dynamcs of nteractng defects Real-tme Langevn smulatons of thermally actvated mgraton of nteractng loops n pure ron. Left: n-stu electron mcroscope observaton of mgraton of loops (courtesy of Prof. K. Arakawa, Shmane Unversty, Japan), rght: a move derved from smulatons. The loop szes are the same n both cases, nteracton laws are derved from sotropc elastcty..

21 Real-tme dynamcs of nteractng defects Langevn dynamcs of nteractng defects Left: expermentally observed trajectores of three nteractng dslocaton loops (Arakawa). Rght: the smulated trajectores of the three loops. The loops szes and the dstances between the loops match expermental observatons. S.L. Dudarev et al., Phys. Rev. B 81, (2010)

22 Dsplacement (nm) Real-tme dynamcs of nteractng defects Langevn dynamcs of nteractng defects experment 1o.dat & 2o.dat smulaton Tme (s) Left: expermentally observed trajectores of two nteractng dslocaton loops (Arakawa). Rght: the smulated trajectores of the two loops. The loops szes and the dstance between the loops match expermental observatons. S.L. Dudarev et al., Phys. Rev. B 81, (2010)

23 Real-tme dynamcs of nteractng defects Langevn dynamcs of nteractng defects experment smulaton Left: expermentally observed trajectores of loops n on-rradated ron (Yao and Jenkns). Rght: trajectores of moton smulated usng Langevn dynamcs, takng nto account nteracton wth the nvsble vacancy clusters. Loops szes match those observed expermentally. Note that characterstc ~10 s tmescales charactersng the moton of defects observed n experment. S.L. Dudarev et al., Phys. Rev. B 81, (2010)

24 Relaxaton volume of a dslocaton loop where s the vector area of the loop, and b s the Burgers vector. The trace of a dpole tensor s proportonal to the relaxaton volume of the loop A smple calculatons shows that the elastc relaxaton volume of a dslocaton loop equals the product of ts Burgers vector and ts area

25 Is a crowdon an nfntesmally small loop? To answer the queston, we have computed P j of a crowdon usng DFT, and nvestgated whether the symmetry of the tensor s the same or dfferent from that of an nfntesmal loop. P j ev Snce tungsten s elastcally sotropc, had the crowdon been a small dslocaton loop, ts dpole tensors should have had the form (of that of an nfntesmal loop) P j 2bAnn BUT ths form does not ft the DFT data. j j 1 2 where n s a unt vector n the [111] drecton.

26 Is a crowdon an nfntesmally small loop? Further analyss shows that n order to parameterse the elastc dpole tensor of a crowdon, two ndependent parameters are requred. P j P j C jkl (1) n n j (2) j ev where the sum of these two parameters equals the relaxaton volume of the crowdon rel Ths shows that a crowdon s NOT a small dslocaton loop. (1) (2)

27 Parameterzaton of dpole tensors Accuracy of a 129 atom cell For a vacancy, rel / Surprsngly, we fnd that a 129 atom cell, often used for computng energes of defects, only delvers ~5% accuracy n dpole tensor evaluaton. F. Hofmann et al., Acta Mater. 89 (2015) 352

28 Full code s avalable to download from the journal webpage or the CCFE webste Tme evoluton of spns and atoms. In the absence of magnetsm, spnlattce dynamcs s equvalent to molecular dynamcs. Treats the effect of non-collnear magnetsm on nteratomc forces.

29 Applcatons Varaton of lattce T l, spn T s and electron T e temperatures n a collson cascade smulaton. Smulaton was performed usng a bcc Fe supercell contanng ~ atoms. The knetc energy of the prmaryknock-on atom (PKA) s 5 kev, ntally the lattce s at 300K. P.-W. Ma, S.L. Dudarev, C. H. Woo, Phys. Rev. B85 (2012) Laser pulse at t=0. Varaton of temperatures as functons of tme Dynamcs of demagnetzaton and recovery n Fe

30 BCC-FCC-BCC phase transtons n Fe The maxmum bcc-fcc free energy dfference over the bcc-fcc-bcc phase transton temperature nterval s 2meV per atom. The case looks hopeful: smlarly to how molecular dynamcs has evolved nto a predctve tool, spn-lattce dynamcs s evolvng nto an accurate smulaton method, capable of modellng dynamc processes at mev/atom level of accuracy. P.-W. Ma, S.L. Dudarev, J.S. Wrobel, Phys. Rev. B96 (2017)