Order parameters of crystals in LAMMPS

Transcription

1 Order parameters of crystals n LAMMPS Aula Tegar Wcaksono Department of Materals Engneerng, The Unversty of Brtsh Columba tegar@alumn.ubc.ca Wrtten on: July 19, 015 Abstract To dentfy atoms n a bcrystal cell based on the gran to whch they belong, LAMMPS s equpped wth a feature called order parameters,.e. the fx orent/fcc command [1]. The command has been adapted for b.c.c crystals n my thess. Ths document ams to descrbe the order parameter calculaton n detal, ncludng ts mplementaton n LAMMPS for b.c.c and f.c.c crystals 1 Formalsm 1.1 Nearest neghbours An atom n the bulk of b.c.c (or f.c.c) crystal A s expected to have 8 (or 1 for f.c.c) nearest neghbours j, each at a dstance a T 3/ (or at / for f.c.c) from where at s the lattce constant at temperature T. Let Rj A be the reference lst for gran A,.e. the lst of vector poston of neghbours j relatve to the poston of. The lst Rj A s called the. In the prncpal Cartesan frame,.e. a frame whose orthogonal axes (X, Y, Z) are ([100], [010], [001]), the neghbour lst Rj A contans the followng vectors [ ] u j v j w j, [ at a T a T ] [ ] [ ] [ ] at a T a T at a T a T at a T a T b.c.c :Rj A= [ ] [ ] [ ] [ ] at a T a T at a T a T at a T a T at a T a T (1) 1

2 [ at a T 0 ] [ ] [ ] [ at a T 0 at a T 0 at a T 0] f.c.c :Rj A= [ at ] [ at ] [ at ] [ at ] () [ 0 a T a T ] [ a T ] [ a T ] [ a T ] where j [1, 8] for b.c.c or j [1, 1] for f.c.c. In a non-standard frame, the orthogonal axes (X, Y, Z) are ([h x k x l x ], [h y k y l y ], [h z k z l z ]). The poston of nearest neghbours [ u j v j w j ] n R A j, see Eqs. (1) and (), can be transformed nto [u j v j w j ] accordng to the non-standard frame va u j = [ uj v j w j ] [hx k x l x ] h x + k x + l x [ ] [ ] [ ], v j = uj v j w j hy k y l y, w h y + k y + ly j = uj v j w j [hz k z l z ] h z + k z + lz (3) for all j (8 for b.c.c and 1 for f.c.c). The transformaton descrbed by Eq. (3) s partcularly useful when workng wth bcrystal cells. In a bcrystal cell, the orthogonal axes of gran A and B are (X A, Y A, Z A ) and (X B, Y B, Z B ), respectvely. As wll be clear later, the mplementaton of order parameter calculaton requres two supplementary fles whch contan vector postons of nearest neghbours of atoms n gran A and B. Eq. (3) s thus relevant for such purpose. 1. Order parameter An unscaled order parameter ξ A can be calculated for all atoms based on the dfference between the actual poston of neghbours j relatve to r j and the reference postons documented n the lst Rj A. It s defned as ξ A = 1 N actual ( ) N actual mn r j Rj A j=1 (4) where N actual N 1nn = 8. The mn functon means that the unscaled order parameter ξ A s calculated by assgnng to each actual j one of the vectors from the reference lst Rj A such that ther nteratomc dstance s mnmum. Ths means that f an atom s n the bulk of crystal A, all r j = R A j and ts unscaled order parameter wll be ξ A = 0. If there s another crystal B, ts reference beng Rj B, and atoms n crystal A are measured

3 relatve to crystal B, the unscaled order parameter s ξ AB = 1 N 1nn N 1nn j=1 ( ) mn Rj B RA j (5) Note( that ξ AB ) s not necessarly the maxmum possble value for ξ A because the term mn Rj B RA j s not equal for each j, makng ξ AB the average of ξ A. If an atom s only surrounded by one frst-nearest neghbour j and that j happens to be the furthest from any vectors n the reference lst R B j, the resultng ξ A wll defntely be hgher than ξ AB. In order to standardze the range of order parameter across dfferent gran boundares, the normalzed order parameter η A s defned,.e. 0 f ξ A < ξ lo = K lo ξ AB η A (ξ = A ξ lo ) f ξ A (K h K lo )ξ AB [ξ lo, ξ h ] (6) 1 f ξ A > ξ h = K h ξ AB where K lo, K h are cutoff values chosen arbtrarly between 0 to 1. A senstvty analyss of cutoff values was dscussed n [1]; typcally, they are set as 0.5 and 0.75 respectvely. 1.3 Artcal drvng force technque The order parameter η A can further be used to nduce gran boundary mgraton va the artfcal drvng force (ADF) technque [1, ]. In ths technque, the potental energy of atoms belongng to one gran of a bcrystal s rased. The added energy drves the boundary mgraton, shrnkng the volume of hgh-energy gran. The added energy of an atom, U ADF, vares wth the atomc local envronment, as represented by the order parameter η, va [1] U ADF = 1 U max (1 cos(πη )) (7) where U max s the maxmum energy added nto any ndvdual atom. The choce of trgonometrc functon s purely arbtrary and the resultng mobltes have been verfed to be nsenstve of the choce [3]. Only the dynamcs of few atoms are affected by such an energy ncrease because the extra force actng on each atom s calculated based on the energy gradent. A non-zero extra force s only experenced by boundary atoms and ther neghbours. Mathematcally, atoms wth η (0, 1) or those surrounded by at least 1 other atom j wth η j (0, 1) wll experence an extra force of F ADF, gven by: 3

4 F ADF = UADF tot r = πu max ηn actual = r [( Nactual j U ADF δ j δ j ) sn(πη ) + N actual j ( δ )] j δ j sn(πη j) (8) where the followng defnton apples: δ j = R A j r j δ j = R A j r j and r j s the actual locaton of atom j relatve from atom, also r j = r j. Note that the net force due to the ADF technque s non-translatng,.e. F ADF = 0 where the summaton ndex s over all solvent atoms. LAMMPS mplementaton.1 Overvew There are 3 man commands n LAMMPS relevant for the order parameter and the ADF formulaton: 1. The man fx orent command.. The assocated fx_modfy, relevant for energy calculaton. 3. The thermodynamc output f_id and the per-atom output f_id[1] andf_id[]. The descrpton of each command s explaned n detal n the followng sub-sectons.. The fx orent command Syntax: fx [ID] [grp] orent/bcc <nstats> <dr> <latp> <Umax> <Klo> <Kh> <fle0.txt> <fle1.txt> 4

5 Anatomy: [ID] ID of the fx command [grp] Group of atoms to whch the fx s appled. Use all only f there s 1 type of atoms. If there are more than 1 type of atoms, e.g. a bnary alloy, one must frst create an ndependent grp that encompasses atoms that consttute the crystal (solvent, not solute). The exstng fx command n LAMMPS wll not see ths dfference. The adapted fx orent/bcc has also been modfed to now consder the possblty of havng multple type of atoms. orent/bcc The b.c.c verson of orent/fcc nstats Number of tme-steps at whch the statstcs of the fx command s prnted. Use 0 as a default. dr The drecton of the gran boundary mgraton. Use dr of 1 f you want to use fle0.txt as the reference orentaton. Use dr of 0 f you want to use fle1.txt as the reference orentaton. latp The lattce parameter used to compute the cutoff dstance. LAMMPS by default defnes r_cutoff_fcc = latp sqrt(). The use of 1.57 s arbtrary snce what LAMMPS does s, after collectng all neghbours wthn r_cutoff_fcc, sortng them out and then storng only the closest 1. If ths defnton makes you uncomfortable, you can always rescale the cutoff dstance by modfyng the latp latp /1.57. The same apples for r_cutoff_bcc. Here, the r_cutoff_bcc = 1.57 * 0.5 * <latp> * sqrt(3) and only the closest 8 wll be stored. Umax The maxmum energy added to an atom, see Eq. (7). The unt s ev for unts metal. Klo The low-threshold for the order parameter, see Eq. (6), where Klo = 0.5 n [1]. Kh The hgh-threshold for the order parameter,see Eq. (6), where Kh = 0.75 n [1]. fle0.txt fle1.txt Each fle contans the vector postons (x, y, z) of neghbour j of atom for gran A (fle0.txt) and B (fle1.txt). The coordnates (x, y, z) must be n the actual dstance unt, e.g. Å. There are 4 lnes n both fles for b.c.c (6 for f.c.c) bcrystals, representng half of the total number of nearest neghbours n a perfect crystal. Ths s because the rest of nearest neghbours are symmetrcal to the orgn. In other 5

6 words, f (x j, y j, z j ) denotes one neghbour, there s another neghbour located at ( x j, y j, z j ). To dentfy the proper lst of nearest neghbour postons, the transformaton descrbed by Eq. (3) can be used..3 The fx_modfy command Syntax: fx_modfy [ID] energy yes Anatomy: Ths command allows one to dsplay the net potental energy from the ADF technque contrbuton to the system, Utot ADF. When ths command s added to the nput fle, the total potental energy wll ncrease by an amount of Utot ADF. To prnt out only the extra potental energy, Utot ADF, see f_id below..4 The outputs f_id, f_id[1] and f_id[] Syntax: Thermodynamc output: thermo_style custom f_id Per-atom output: dump [ID] all custom nstats f_id[1] f_id[] Anatomy: Thermodynamc output: when enabled, f_id dsplays U ADF tot defned n Eq. (8). Per-atom output: when enabled, f_id[1] and f_id[] dsplay the unscaled order parameter ξ, see Eq. (4), and the normalzed order parameter η, see Eq. (6). Important note: LAMMPS does not nclude the per-atom ADF energy calculated from Eq. (7) nto ts pe/atom output. However, one can easly know ths per-atom quantty smply by substtutng f_id[] nto Eq. (7). 6

7 References [1] K.G.F. Janssens, D. Olmsted, E.A. Holm, S.M. Foles, S.J. Plmpton, P.M. Derlet, Computng the moblty of gran boundares. Nat. Mater. 5, 14 (006). do: /nmat1559 [] D.L. Olmsted, E.A. Holm, S.M. Foles, Survey of computed gran boundary propertes n face-centered cubc metals, II. Gran boundary moblty. Acta Mater. 57(13), 3704 (009). do: /j.actamat [3] M.J. Rahman, H.S. Zurob, J.J. Hoyt, A comprehensve molecular dynamcs study of lowangle gran boundary moblty n a pure alumnum system. Acta Mater 74, 39 (014). do: /j.actamat