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1 Suppleetary Iforatio -Breakdow of cotiuu fracture echaics at the aoscale- Takahiro Shiada,,* Keji Ouchi, Yuu Chihara, ad Takayuki Kitaura Departet of echaical Egieerig ad Sciece, Kyoto Uiversity, Nishikyo-ku, Kyoto , Japa Istability ode Aalysis: A theory of echaical istabilities i arbitrary atoic systes, Let us cosider a arbitrary atoic syste cosistig of N atos. The potetial eergy of the atoic syste, U, is a fuctioal of atoic coordiate, ad is give by U U (, ( where deotes the cofiguratio vector cosistig of atoic positios, ( ( ( ( ( ( ( α ( N ( N ( N ( r, r, r, r, r, r,..., r,..., r, r, r. ( x y z x y z i x y z Here, (α r i deotes the coordiate of ato α i the i directio. This is the geeral for to express the potetial eergy of atoic syste, i ay type of potetial fuctios ad force field icludig ab iitio ethod. The irreducible uber of degrees of freedo (DOFs i the atoic syste without ay costrait is 3N 6 because the DOFs of rigid-body traslatio (3 ad rotatio (3 is subtracted fro the total DOFs of atos (3N. Uder a displaceet costrait where soe of atos are fixed, the uber of DOFs is 3N 3 c 6, where c is the uber of costraied atos. Here, let us cosider the atoic syste uder static exteral load, i.e., the atos are located at their ow optial sites ad are i balace with the exteral load ad/or costrait. The total eergy Π of the equilibrated atoic cofiguratio ( uder exteral load ow cosists of the * E-ail: shiada@e.kyoto-u.ac.jp

2 potetial eergy U ad the work doe by exteral load W, ad is give by U W Π. (3 Assuig a ifiitesial deforatio to the equilibrated syste, the total eergy of slightly defored syste ( Π ca be described by the Taylor s series expasio of total eergy ( Π with respect to, ad is give by Π Π W U ( (... W U, (4 where deotes a copoet of cofiguratio vector icluded i the DOFs of syste, i.e., (α r i. Sice the first derivative of total eergy (i.e., force actig o atos is zero due to the syste at equilibriu, the secod ter o the right-had side ca be eliiated. Π W U (,, (5 Cosiderig that the exteral load is costat due to the static loadig, the work is proportioal to the displaceet of atos o which the exteral load is applied. Thus, we get W (,,,. (6 Usig Eqs. (4-(6 ad igorig the higher-order ters, the total eergy chage Π with respect to the ifiitesial deforatio is give by H T ( ( ( Π Π Π U, (7 where H is the Hessia atrix of potetial eergy Π with respect to ad T eas traspositio. The copoet of Hessia atrix, H ij, is

3 U H (,,,. (8 By solvig the eigevalue proble of the Hessia atrix H, H p η p, (9 where η ( η... η... η is the eigevalue of the Hessia atrix H, ad p is the correspodig eigevector. Usig the eigevector p, the Hessia atrix is diagoalized as η O T P HP P HP, ( O η where P p... p. Sice the eigevectors are a orthogoal basis set of -diesioal vector ( space, ay ifiitesial deforatio ca be expressed as a liear cobiatio of the eigevectors as u p Pu, ( where u is the copoet of i the p directio ad u u,..., u. Therefore, the total ( eergy chage by i Eq. (7 becoes T T Π u ( ( Π u H( Πu u( Π HΠ u η. ( Whe the iiu eigevalue is positive, η > (i.e., all the eigevalues are positive, the atoic syste is cofired to be echaical stable because Π is always positive with respect to ay ifiitesial deforatio. I cotrast, whe the iiu eigevalue is egative, Π becoes egative with respect to p, idicatig that the atoic syste is echaical ustable alog the deforatio path of p. Therefore, the echaical stability/istability of give atoic syste ca be deteried by the sig of iiu eigevalue η of Hessia atrix, ad the deforatio ode at the oset of echaical istability ca be idetified by the 3

4 correspodig eigevector p. Sice the cleavage fracture focused i the preset study accopaies bod breakig at the crack-tip, this is clearly oe of itrisic echaical istabilities i a brittle atoic syste, ad the atoistic ad discrete ature of brittle fracture at the crack-tip ca be described by eigevector p fro our approach, as show i Fig. 3a ad Suppleetary Fig. S. 4

5 Suppleetary Figure S Critical stress itesity factors at fracture obtaied fro fracture tests based o the quatu-echaics (Q vs. classical olecular statics (S siulatios. The critical stress itesity factor at fracture f K I is evaluated fro the cotiuu stress distributio ear the Si( crack-tip at a critical tesile load or bedig oet obtaied by the Q-based coputatioal fracture experiets usig the first-priciples desity-fuctioal theory (DFT calculatios withi the geeralized gradiet approxiatio, ad the S-based fracture tests usig the bod-order potetial for brittle fracture of silico. For the detailed theory ad odels of Q siulatios, see the ethod sectio ad suppleetary Fig. S3. 5

6 Suppleetary Figure S Copariso of fracture ode displaceet obtaied by quatu-echaics (Q ad classical olecular statics (S siulatios. The fracture ode displaceet obtaied by our istability ode aalysis, the details of which are described i this Suppleetary Iforatio above. The quatu-echaics results are based o the first-priciples desity-fuctioal theory (DFT calculatios withi the geeralized gradiet approxiatio. For the detailed theory ad odels, see the ethod sectio ad Suppleetary Fig. S3. 6

7 Suppleetary Figure S3 Siulatio odel of the ( crack i silico for quatu-echaics (Q desity-fuctioal theory calculatios. Si atos at odel surfaces are passivated by H-teriatio. The odel cosists of 46 Si atos ad 86 H atos. The positio ad displaceet of Si atos at the two atoic layers fro the surface are cotrolled i order to feedback the elastic strai field outside the odel. The preset odels ad procedure is previously validated. 3 7

8 efereces. Kitaura, T., Ueo, Y. & Fushio,. Istability criterio of ihoogeeous atoic syste. ater. Sci. Eg. A 379, 9-33 (4.. Shiada, T., Okawa, S., iai, S. & Kitaura, T. Siplified evaluatio of echaical istability i large-scale atoic structures. ater. Sci. Eg. A 53-54, 66-7 (9. 3. Pérez,. & Gubsch, P. A ab iitio study of the cleavage aisotropy i silico. Acta ater. 48, (. 8