Minimization of the Free Energy under a Given Pressure by Natural Iteration Method

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1 Mateals ansactons, Vol. 5, No. 3 (011) pp. 48 to 43 #011 he Japan Insttute of Metals Mnmzaton of the Fee Enegy unde a Gven Pessue by Natual Iteaton Method Naoya Kyokane 1; * and etsuo Moh ;3 1 Dvson of Mateals Scence and Engneeng, Gaduate School of Engneeng, Hokkado Unvesty, Sappoo , Japan Dvson of Mateals Scence and Engneeng, Faculty of Engneeng, Hokkado Unvesty, Sappoo , Japan 3 Reseach Cente fo Integatve Mathematcs, Hokkado Unvesty, Sappoo , Japan Natual teaton method wthn the Cluste Vaaton Method s coupled wth zeo potental to mnmze the fee enegy unde the constant of a constant pessue. he calculated esults epoduce the ones obtaned by the conventonal method n whch a gadent of the fee enegy cuve s estmated at each atomc dstance. It s noted that the computaton tme s much educed n the pesent scheme and, theefoe, t has a potental applcablty n the futue calculatons wth a fee enegy contanng a lage numbe of vaables. [do:10.30/matetans.mbw01011] (Receved Octobe 9, 010; Accepted Novembe 6, 010; Publshed Febuay 5, 011) Keywods: cluste vaaton method, natual teaton method, zeo potental, smple cubc lattce 1. Intoducton Among vaous appoaches to calculate confguatonal entopy, Bagg-Wllams appoxmaton 1) has been often employed to calculate phase equlba and phase dagams fo bnay and multcomponent systems. 5) he advantages of Bagg-Wllams appoxmaton ae summazed as ts physcal tanspaency, mathematcal smplcty, and expansblty to multcomponent systems. Howeve, the ode of phase tanston has been often msled by Bagg-Wllams appoxmaton, whch s ascbed to the ovesmplfed fee enegy expesson. In ode to mpove ths appoxmaton, t s necessay to consde the atomc coelaton whch plays an essental ole n the confguatonal entopy. Cluste Vaaton Method (heeafte abbevated as CVM), whch was devsed by Kkuch, 6) ncludes wde ange of atomc coelatons and has been accepted as one of the most elable theoetcal tools to appoxmate the confguatonal entopy. A fee enegy of the CVM s expessed as a functon of a set of cluste pobabltes, whch s fomally wtten as F ¼ f ðfx g; fy g; fz k g; Þ ð1þ whee, and k specfy the atomc speces, x, y, and z k ae cluste pobabltes of a pont, pa and thee-body clustes, espectvely, fo a confguaton specfed by subscpt(s). When a fee enegy of CVM s only descbed as a set of pont cluste pobablty, fx g, whch coesponds to the Bagg- Wllams appoxmaton, the mnmzaton of the fee enegy s a smple poblem. Howeve, a moe geneal expesson gven by eq. (1) s cumbesome snce t nvolves a numbe of cluste pobabltes and they ae not mutually ndependent n the themodynamc confguaton space. As an effcent method to mnmze the fee enegy of CVM, Kkuch developed Natual Iteaton Method 7) (heeafte abbevated as NIM). Newton-Raphson method has been wdely appled to mnmze a fee enegy. Howeve, n ode to mnmze the *Gaduate Student, Hokkado Unvesty fee enegy of the CVM by usng ths method, t s necessay to convet the cluste pobabltes to coelaton functons 8 10) whch fom a set of ndependent vaables, followed by the calculatons of the fst and second ode devatves of the fee enegy wth espect to coelaton functons. Snce CVM fee enegy s a multvalued functon of coelaton functons the second ode devatve s epesented by a matx, whle matx nveson equed by the Newton- Raphson method s not tval. An altenatve way, NIM, s a specal teaton pocedue unque to the CVM. NIM eques nethe the coelaton functons no the nveson of the devatve matx. It s futhe noted that the fee enegy always deceases as the teaton poceeds and the convegence of the teaton s guaanteed. 7) Hence, a seach of fee enegy mnmum s ease compaed to Newton-Raphson method and NIM has been wdely used fo the mnmzaton of fee enegy of the CVM. In ode to fomulate the ntenal enegy, phenomenologcal atomc potentals ae often employed. Fo example, Lennad-Jones type pa potental n whch two potental paametes ae detemned fom expemental cohesve enegy and lattce constant s appled to calculate phase boundaes and t s confmed that the calculated phase dagam well epoduces the expemental one. 11) Hence, n the pesent study, we employed a Lennad-Jones type pa potental (heeafte abbevated as L-J potental) of whch detals wll be descbed shotly. Unlke the conventonal pocedue, to mnmze the fee enegy solely wth espect to cluste pobabltes, the employment of L-J potental demands addtonal mnmzaton wth espect to volume, snce the ntenal enegy s explctly gven as a functon of volume (equvalently atomc dstance). Hence, the constancy of the pessue s mposed as an addtonal constant. he conventonal mnmzaton pocedue fo such a fee enegy poceeds wth two steps; one s the mnmzaton of the fee enegy fo each atomc dstance, and the second one s the calculaton of a pessue fom the gadent of mnmzed fee enegy at each atomc dstance. he pocedue s epeated untl the mposed

2 Mnmzaton of the Fee Enegy unde a Gven Pessue by Natual Iteaton Method 49 α sublattce β sublattce able 1 Lennad-Jones paametes employed n the pesent study. e A A A B B B Fg. 1 Odeed phase n the smple cubc stuctue. pessue s attaned. Such two-folds mnmzaton pocess demands heavy computatons. In ode to educe the amount of calculatons, we popose an altenatve pocedue of mnmzaton of the fee enegy unde the constant of a constant pessue n the pesent epot. he coe of the pocedue s to employ NIM coupled wth zeo potental. he oganzaton of the pesent epot s as follows. In the next secton, the confguatonal fee enegy wthn the pa appoxmaton of the CVM s ntoduced fo the sake of completeness. he pesent mnmzaton pocedue and the conventonal pocedue ae also summazed n the next secton. he esults ae demonstated and dscussed n the thd secton followed by the conclusons n the last secton.. heoetcal Calculaton.1 Confguatonal fee enegy wthn cluste vaaton method Fo the sake of smplcty, the odeed phase on a smple cubc lattce s consdeed n the pesent epot. A and B atoms ae aanged altenatvely n the neaest neghbo lattce ponts and the stochometc composton of the pesent odeed phase s ffty atomc pecent. he ente lattce s dvded nto two sublattces and as shown n Fg. 1 whee ðþ-sublattce s defned as the pefeental lattce pont fo AðBÞ atom. hen, the long ange ode (heeafte abbevated as LRO) paamete,, s defned as ¼ x A x A ; ðþ whee x A s the cluste pobablty of fndng A atom n the sublattce desgnated by. he accuacy of the confguatonal entopy s mpoved by ntoducng wde ange of atomc coelatons. Howeve, the pa appoxmaton s suffcent to develop a new mnmzaton pocedue unde a gven pessue, and the confguatonal entopy of pa appoxmaton s gven as ( 5 X S ¼ k B Lðx Þþ5 X Lðx Þ 3 X ) Lðy Þþ ð3þ ; whee k B s the Boltzmann constant, y s pa cluste pobabltes of fndng atomc aangements on the sublattces specfed by subscpts and supescpts, espectvely, and L opeato s defned as LðxÞ x ln x x: ð4þ he devaton of eq. (3) has been amply demonstated n the pevous publcatons 1) and s not epeated hee. Note that the confguatonal entopy, the ntenal enegy and the fee enegy ae defned pe lattce pont thoughout the pesent publcaton. he ntenal enegy n the pesent study s assgned by the sum of neaest neghbo atomc pa nteacton eneges and s descbed as EðÞ ¼3 X ; e ðþy whee 3 s one half of the coodnaton numbe fo a smple cubc lattce, e the neaest neghbo atomc pa nteacton enegy between speces and, and s the neaest neghbo atomc dstance. It s noted that the neaest neghbo atomc dstance s equvalent to the lattce constant fo the smple cubc lattce. Hence, togethe wth the confguatonal entopy gven by eq. (3), the Helmholtz enegy s expessed as F ¼ 3 X ; k B e ðþy ð5þ ( 5 X Lðx Þþ5 X Lðx Þ 3 X ) Lðy Þþ ; whee s the tempeatue. o peseve the symmety, the subscpt of the pont cluste pobablty on sublattce was tansfomed fom n eq. (3) to n eq. (6). Fo the neaest neghbo pa potental, e ðþ, the L-J type potental s employed, whch s descbed as e ðþ ¼e m m n whee e and expess the depth of the potental and equlbum atomc dstance, espectvely, and the exponents m and n ae elated wth stffness of the lattce aganst the compesson and expanson, espectvely. In the pesent study, m ¼ 8 and n ¼ 4 ae assgned and the values employed fo e and ae summazed n able 1. houghout the pesent manuscpt, the enegy and the atomc dstance ae nomalzed wth espect to e AB and AB, espectvely. Because e AB s lage than e AA and e BB, odeng eacton can be guaanteed.. Mnmzaton pocedue wo mnmzaton pocedues unde the constant of a constant pessue ae dscussed n ths secton. One s the conventonal pocedue n whch a gadent of the fee enegy cuve s calculated at each lattce constant. he othe s the pesent pocedue whch employs zeo potental wthn NIM. All of numecal calculatons n the pesent epot ae pefomed by a wokstaton, Intel Xeon Conventonal mnmzaton pocedue In the conventonal pocedue, gand potental,, s fomulated by the Legende tansfomaton on the Helmholtz enegy, F, wth espect to the numbe of patcles, and s gven as n ð6þ ð7þ

3 430 N. Kyokane and. Moh ð; V; f gþ ¼ F X x ð8þ whee s the chemcal potental of speces and V s the volume whch s equvalent to 3 fo a smple cubc lattce. Based on the gand potental, the pessue, p, s deved as ¼ ;f g ¼ ;f g ;f g ð9þ he mnmzaton pocedue s as follows. Fstly, n addton to the tempeatue and a set of chemcal potental, the pessue, p n and the atomc dstance, n, ae specfed. houghout the couse of ths study n (out) n the supescpts ndcates nput (output) value. Secondly, the neaest neghbo atomc nteacton eneges between speces and ae calculated by substtutng n nto eq. (7). hdly, the gand potental s mnmzed by applyng NIM, and the teaton s epeated untl the followng cteon of convegence, c X x n x out þ X x n x out ; ð10þ becomes less than 10 1 fo all the pont clustes. When c does not satsfy the convegence cteon, the nput of atomc dstance s slghtly changed and the second step and the thd step ae pefomed, usng the output cluste pobabltes as the nput cluste pobabltes. Fom these pocedues, the gadent of the gand potental, = ¼ n, s obtaned fo a gven tempeatue and the chemcal potentals. By substtutng the gadent of the gand potental nto eq. (9), the pessue at the specfed atomc dstance, p out, s obtaned. hen, the teaton poceeds untl a convegence cteon fo the pessue, P p n p out ; ð11þ becomes less than Pesent mnmzaton pocedue Zeo potental 13) s employed n ths calculaton. By Legende tansfomaton of the gand potental wth espect to the volume, zeo potental s obtaned as Zð; p; f gþ ¼ þ pv ¼ F X x þ pv: ð1þ By substtutng eq. (6) nto eq. (1), zeo potental of the pesent study s fomulated. It s noted that the zeo potental of the pesent calculaton s also the functon of the atomc dstance snce the ntenal enegy s descbed as L-J type potentals. he mnmzaton pocedue s as follows. Fstly, the tempeatue, chemcal potentals and pessue ae specfed. Secondly, one gves ntal guess fo cluste pobabltes, and the atomc dstance, out, s calculated by substtutng gven cluste pobabltes nto ð@z=@þ ;p;f g ¼ 0. hen, the mnmzaton of the zeo potental s pefomed n the same manne as the second and thd steps of the conventonal pocedue descbed n the secton..1, usng out as n. hese steps ae teated untl the followng cteon of the convegence s satsfed,, p, µ α out x α n x, x β out c X, x þ X ; β n, y x n, y αβ out αβ n y n x out þ X x n y out < 10 1 : x out he summay of these pocedues s shown n Fg.. 3. Results and Dscusson f c > ε Z αβ y Z, p, µ, p, µ = 0 = 0 out out e Fg. Mnmzaton pocedue. In the pesent calculaton, as the convegence condton 10 1 s assgned to ". LRO paamete, η empeatue, Fg. 3 empeatue dependences of the LRO paamete, n eq. (), at a fxed zeo pessue and chemcal potentals. ð13þ Shown n Fg. 3 s the tempeatue dependences of the LRO paamete, n eq. (), at a fxed zeo pessue and chemcal potentals. he mnmzaton of the fee enegy s pefomed by applyng the pesent pocedue descbed n secton... Snce LRO paamete behaves vey senstvely to the change of the tempeatue aound a tanston tempeatue, we pefom caeful numecal calculatons wth hgh accuacy, and confm that the LRO paamete successvely deceases as the tempeatue nceases. hs ndcates that the ode of the tanston s of the second ode, whch agees wth the pevous esults. 14,15) Hence, the tanston tempeatue, k B C ¼ 0:449, s detemned as the tempeatue at whch ðþ falls zeo, whee the tempeatue s nomalzed by the L-J paamete, e AB. By epeatng the same pocedue fo dffeent chemcal potentals and pessue, one can deve a phase dagam. Shown n Fgs. 4(a) and (b) ae calculated phase boundaes at the nomalzed pessue p ¼ 0:0 and p ¼ 0:05, espectvely. Squaes n Fg. 4 ae detemned by the pesent pocedue, whle coss maks () ae obtaned by the conventonal pocedue descbed n secton..1. One can confm that these two esults concde satsfactoly. Hence, the valdty of the mnmzaton pocedue of the pesent method s confmed. he computaton tme s much educed n the pesent scheme. he pesent pocedue eques about fve hous to k B

4 Mnmzaton of the Fee Enegy unde a Gven Pessue by Natual Iteaton Method 431 (a) (b) empeatue, k B empeatue, k B Concentaton, x Concentaton, xa Fg. 4 Calculated phase dagams at the pessue p ¼ 0:0 (a) and p ¼ 0:05 (b). Squaes ae obtaned by the pesent pocedue, whle coss maks () ae by the conventonal pocedue. detemne one pont n the phase bounday. Whle, the conventonal pocedue takes moe than thty tmes of the pesent one. hs s because the conventonal pocedue eques mnmzaton of the fee enegy wth espect to cluste pobabltes fo each specfed atomc dstance. In contast, the pesent scheme s vey effectve fo the case n whch the atomc nteacton enegy depends on the atomc dstance. he tanston tempeatue at p ¼ 0:05 at the stochometc composton s slghtly hghe than the one at p ¼ 0:0. hs s because the effectve neaest neghbo pa nteacton enegy, v ¼ e AA þ e BB e AB,atp ¼ 0:05 s hghe than the one at P ¼ 0:0 as shown n Fg. 5. Fo the sake of smplcty, the confguatonal entopy n the pesent epot s gven as the pa appoxmaton wthn the CVM. In ode to mpove the accuacy, t s necessay to consde wde ange of atomc coelatons. Howeve, the numbe of vaables n a fee enegy dastcally nceases and the computatonal buden becomes heave wth the ntoducton of a wde ange of atomc coelatons though lage clustes. Fo nstance, when the confguatonal entopy s descbed as the cubc appoxmaton fo the smple cubc lattce, 14) the fee enegy s a functon of 80 knds of cluste pobabltes. It s expected that the computaton tme s much educed by applyng the pesent mnmzaton pocedue. One of the defcences of the CVM s the fact that the local lattce dstoton s not ntoduced. hs s because the confguatonal entopy wthn the CVM s calculated on the assumptons that an atom always occupes a Bavas lattce pont and that Bavas lattce unfomly expands o contacts despte the fact that an atom s dsplaced fom a Bavas A Effectve nteacton enegy, v lattce pont 16 18) dependng upon the dffeence n atomc sze of consttuent elements. In ode to ntoduce local atomc dsplacement, Kkuch and hs cowokes devsed Contnuous Dsplacement Cluste Vaaton Method 19 1) (heeafte abbevated as CDCVM). In ths method, addtonal ponts, whch ae temed quas lattce ponts, ae ntoduced aound a Bavas lattce pont and an atom can be dsplaced to those ponts. Dsplaced atoms n the CDCVM ae egaded as dffeent atomc speces and the entopy asng fom the feedom of dsplacements s calculated by mappng t onto confguatonal entopy of a multcomponent system fo whch atoms ae located only at Bavas lattce pont. Hence, the numbe of cluste pobabltes fa moe nceases and the mnmzaton of fee enegy becomes futhemoe cumbesome. Due to the computatonal buden, CDCVM has not been well employed n the phase equlba calculatons, howeve the pesent mnmzaton pocedue may apply to the CDCVM and educes the computaton tme. Applcaton of the pesent scheme to CDCVM s left fo the futue nvestgaton. 4. Concluson he mnmzaton pocedue of the fee enegy unde the constant of a constant pessue s nvestgated n the pesent wok. hs s pefomed by applyng Natual Iteaton Method coupled wth zeo potental. he calculated esults epoduce the ones obtaned by the conventonal method n whch a gadent of the fee enegy cuve s estmated at each lattce constant, and the computaton tme s much educed. he pesent scheme has a potental applcablty fo the mnmzaton of a fee enegy wth a lage amounts of cluste pobabltes typcally faced n Contnuous Dsplacement Cluste Vaaton Method wth less computatonal buden. REFERENCES P = 0.05 P = empeatue, k B Fg. 5 empeatue dependence of the neaest neghbo pa nteacton enegy, v,at A ¼ B ¼ 0. he uppe and lowe cuve ndcate the one at p ¼ 0:05 and p ¼ 0:0, espectvely. 1) W. L. Bagg and E. J. Wllams: Poc. R. Soc. London A 145 (1934) ) H. Ohtan and M. Hllet: CALPHAD 15 (1991) ) H. Ohtan, M. Yamano and M. Hasebe: CALPHAD 8 (004) ) H. Ohtan, Y. akeshta and M. Hasebe: Mate. ans. 45 (004)

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