NpT EMD Smulaons of Daomc Lquds Bull. Korean Chem. Soc. 7, ol. 8, No. 697 Molecular Dynamcs Smulaon Sudy forgtranspor Properes of Daomc Lquds Song H Lee Deparmen of Chemsry, Kyungsung Unversy, Busan 68-736, Korea. E-mal: shlee@ks.ac.kr Receved Sepember 7, 7 We presen resuls for ranspor properes of daomc fluds by sohermal-sobarc (NpT) equlbrum molecular dynamcs (EMD) smulaons usng Green-Kubo and Ensen formulas. As he molecular elongaon of daomc molecules ncreases from he sphercal monaomc molecule, he dffuson coeffcen ncreases, ndcang ha longsh shape molecules dffuse more han sphercal molecules, and he roaonal dffuson coeffcens are almos he same n he sascal error snce random roaon decreases. The calculaed ranslaonal vscosy decreases wh he molecular elongaon of daomc molecule whn sascal error bar, whle he roaonal vscosy ncreases. The oal hermal conducvy decreases as he molecular elongaon ncreases. Ths resul of hermal conducvy for daomc molecules by EMD smulaons s agan nconssen wh he earler resuls of hose by non-equlbrum molecular dynamcs (NEMD) smulaons even hough he mssng erms relaed o roaonal degree of freedom no he Green-Kubo and Ensen formulas wh regard o he calculaon of hermal conducvy for molecular fluds are ncluded. Key Words : Dffuson, Shear vscosy, Thermal conducvy, Daomc lquds, Molecular dynamcs smulaon Inroducon The earles molecular dynamcs (MD) calculaons on polyaomc fluds have been carred ou by Harp and Berne usng a Sockmayer-ype poenal o smulae CO and N and by Rahman and Sllnger o smulae H O. For dumbbell daomc molecules, a very smple exenson of he hard-sphere model s o consder a daomc composed of wo hard spheres fused ogeher, 3 bu more realsc models nvolve connuous poenals. Thus, N, F, Cl ec. have been depced as wo 'Lennard-Jones aoms' separaed by a fxed bond lengh. 4-6 Lee and Cummngs 7 repored resuls of non-equlbrum molecular dynamcs (NEMD) smulaons for shear vscoses of pure quadrupolar fluds, a pure dpolar quadrupolar flud, non-quadrupolar/quadrupolar mxures, and quadrupolar/quadrupolar mxures. They found ha he addon of quadrupolar neracons o he pure Ar and o he pure dpolar Ar leads o a hgher vscoses as was observed n he addon of dpolar neracon o he pure Ar. 8 They connued o repor resuls of NEMD smulaons for shear vscoses of pure daomc fluds, monaomc/daomc mxures, and daomc/daomc mxures. 9 I was found ha he neracon beween daomc molecules s less aracve han ha beween sphercal molecules whch leads o lower vscoses as was observed n he expermenal fac ha he vscosy of normal alkanes s less han ha of branched alkanes. Recenly, Tokumasu e al. have suded he effec of molecular elongaon on he hermal conducvy of daomc lquds usng a NEMD mehod. I was found ha he reduced hermal conducvy ncreases as molecular elongaon ncreases. Dealed analyss of he molecular conrbuon o he hermal conducvy revealed ha he conrbuon of he flux caused by energy ranspor and by ranslaonal energy ransfer o he hermal conducvy s ndependen of he molecular elongaon, and he conrbuon of he hea flux caused by roaonal energy ransfer o he hermal conducvy ncreases wh he ncrease n he molecular elongaon. More recenly, rabec e al. calculaed shear vscosy and hermal conducvy of en fluds, modeled by he wo-cener Lennard-Jones plus pon quadrupole (CLJQ) par poenal, usng equlbrum molecular dynamcs smulaon wh he Green-Kubo formalsm. They found ha a low emperaure and hgh densy saes, he Green-Kubo negral for shear vscosy shows slow convergence. Ths problem was overcome by a new approach whch s based on he adjusmen of a suable funcon descrbng he long me behavor of he auo-correlaon funcon and yelds relable resuls whou he need of excessvely long smulaons runs. In he presen paper, we repor equlbrum MD smulaons for he sysems of sphercal monaomc and several dumbbell daomc molecules. The prmary sudy goal s o analyze he dependence of ranspor properes of daomc molecules on molecular elongaon. Ths paper s organzed as follows : We presen he molecular models and he echncal deals of MD smulaon n he followng secon, some heorecal aspecs n Secon III, our resuls n Secon I, and concludng remarks n Secon. Molecular Models and MD Smulaon Mehods The daomc molecule s modeled as he wo-cener Lennard-Jones poenal. 6 The oal neracon s a sum of parwse conrbuons from dsnc aoms a n molecule, a poson r a, and b n molecule j, a poson r jb : u j ( r j ) u ab r ab a b, ()
698 Bull. Korean Chem. Soc. 7, ol. 8, No. Song H Lee Theorem As dynamc properes, we consder dffuson consan (D), vscosy (η), and hermal conducvy (λ) of daomc lqud sysems. Dffuson consan. Translaonal dffuson consan (D ) can be obaned hrough wo roues: he Green-Kubo formula from velocy auo-correlaon funcons (AC) : Fgure. Monaomc and daomc molecules. where r ab s he ner-se separaon r ab r jb r jb and u ab s he par poenal acng beween ses a and b: u ab σ d 4ε d ------ r ab σ ------ d r ab r ab. () Here ε d and σ d are he Lennard-Jones (LJ) parameers for each se of he daomc molecule. The neraomc separaon n a daomc molecule, l, s chosen such as he volume of he daomc molecule s he same as ha of a sphere of dameer σ s. Snce he volume of a daomc molecule of wo spheres of dameer s d s gven by 3 d --πσ, (3) 6 d --πσ + 4 d l + -----πl 3 for a gven neraomc separaon l Lσ d, σ d s deermned by equang d s, and can be expressed n he form of σ d cσ s. In hs sudy, we have chosen as L, /, /6, /4, and /3, and he correspondng c s deermned as.,.966,.989,.9, and.877, respecvely, as shown n Fgure. The oher LJ parameer ε d s chosen as ε s /4. The LJ parameers, σ s and ε s, for he sphercal Ar are chosen as.345 nm and.996 kj/mol, respecvely. All EMD smulaons were carred on 78 molecules n sohermal-sobarc (NpT fxed) ensemble and fully equlbraed for a leas 5, me seps of 5 second ( femo second). The equlbrum properes were hen averaged over 5 blocks of, me seps for a oal of,, me seps ( nano second), and he confguraons of molecules were sored every me seps for laer analyss of srucural and dynamc properes. The nermolecular poenals were subjec o a sphercal cuoff as follows: he cuoff dsance was.5σ s for pure daomc fluds. The equaons of ranslaonal moon n NpT ensemble were devsed by a consran mehod and solved usng a ffh-order, predcor-correcor, Gear negraon, 3 and he equaons of roaonal moon abou he cener of mass for molecular fluds n NpT ensemble were derved usng quaernons. 4-6 6 -- < (4) 3 v ( ) v ( ) > d and he Ensen formula from mean square dsplacemens (MSD) : d < r D -- r ( ) > lm --------------------------------------------------. (5) 6 d The conrbuon o dffuson by roaonal moon of daomc molecule s represened by roaonal dffuson consan: τ and l D --- d < e ( ) e ( ) > r lm --------------------------------------------------. (7) 4 d Here w () and e () are he angular velocy and he un orenaon vecor of daomc molecule, respecvely. The denomnaors of and 4 n Eqs. (6) and (7) are due o degrees of freedom of roaonal moon. Shear vscosy. Shear vscosy by ranslaonal moon s calculaed by a modfed Green-Kubo formula for beer sascal accuracy 7 : η ------ d < P, (8) kt αβ ( ) P αβ > where P αβ s he αβ componen of he molecular sress ensor, P, of parcle by ranslaonal moon : P αβ -- mv α v β ( ) + r α f β There s anoher formula for P αβ : P αβ (6). (9). () where αβ xy, xz, yx, yz, zx, or zy. The equaly of Eqs. (9) and () wh Σ s dscussed n Ref.. 8 I s recommended o use Eq. () n a smulaon ha employs perodc boundary condon. Shear spn vscosy by roaonal moon s calculaed by a smlar way o ranslaonal moon 9 : η r where wo formulas for P αβ r : < > d l D r --- w w [ ] -- mv α v β ------ d kt < P r αβ + r jα ( )f jβ j r ( ) P αβ >, ()
NpT EMD Smulaons of Daomc Lquds Bull. Korean Chem. Soc. 7, ol. 8, No. 699 and r P αβ r P αβ -- [ mv α Iw β + r α N β ] -- mv α Iw β + r jα N jβ (),(3) where N denoes he orque on he molecules I. Thermal conducvy. Thermal conducvy s calculaed by a modfed Green-Kubo formula for beer sascal accuracy 7 : λ -------- d < q, (4) kt α ( )q α > where α x, y, and z, and he oal hea flux by molecule s q α -- ε v α -- r (. (5) jα )v f j w p N p + [ + j ] j Here, he superscrp p ndcaes he prncple axs frame and he oal energy of molecule s gven by ε --m. (6) v + --Iw + -- Φ [ r j ] where Φ(r j ) denoes he poenal energy beween molecules and j. The hea flux by each molecule, Eq. (5), wh he energy of molecule, Eq. (6), consss of fve conrbuons : r q α q α + q α + q α + q α, (7) and here are wo dfferen molecular mechansm responsble for hea flux n lquds, namely, energy ranspor due o molecular moon and energy ransfer due o molecular neracon. These mechansms correspond o he frs and second erms on he rgh-hand sde of Eq. (5), respecvely, whch are called he ranspor erm and he neracon erm. The hea flux caused by molecular moon consss of hree conrbuons of ranslaonal, roaonal and poenal energy ranspor, whch correspond o he frs, second and hrd erms n Eq. (6), respecvely. The conrbuons o he hea flux due o ranslaonal, roaonal and poenal energy ranspor are defned, respecvely, by p he neracon erm n Eq. (5). The hea flux caused by ranslaonal and roaonal energy ransfer s defned by q α --, () --r jα ( v f j ) and q r p p α --, () --r jα ( w N j ) respecvely. Hence, he hermal conducvy, Eq. (4), consss of fve conrbuons : λ o λ + λ r + λ p + λ + λ r Resuls and Dscusson. (3) NpT EMD smulaons for monaomc and daomc molecular sysems carred ou wh a careful consderaon of he long-range correcon due o he sphercal cu-off he poenal, whch s a al correcon esmang he conrbuon from pars of parcles whose dsance apar s greaer han he cu-off dsance., Thermodynamc properes for monaomc and daomc molecules a 94.4 K n NpT ensemble obaned from our EMD smulaons are lsed n Table. The Lennard-Jones(LJ) energy and oal energy decrease negavely wh ncreasng neraomc separaon n a daomc. Ths means ha he neracon beween daomc molecules becomes less aracve wh he Table. Lennard-Jones energy (E LJ n kj/mol), oal energy (E o n kj/mol), pressure (p n am), volume ( n nm 3 ) and of daomc molecules a 94.4 K n NpT ensembles. Unceranes (sandard devaons) n he las repored dg(s) are gven n parenhess. Properes L L / L /6 L /4 L /3 -E LJ 5.339(5) 5.96(8) 4.455(5) 3.555(9).44(6) -E o 4.6(5) 3.34(8).493(5).59(9).5(6) p.47(3).969().3().5(3).9(4) 84.(49) 78.8(44) 79.93(5) 88.(4) 3.() q α --, (8) --m v v α and r q α -- --Iw v α, (9) p q α --. () -- Φ ( r j ) v α The hea flux caused by molecular neracon consss of wo conrbuons of ranslaonal and roaonal energy ransfer, whch corresponds o he frs and second erms n Fgure. Energy dsrbuons.
7 Bull. Korean Chem. Soc. 7, ol. 8, No. Song H Lee Table. Translaonal (D ) and roaonal dffuson coeffcens (D r, 5 cm /sec) of monaomc and daomc molecules a 94.4 K obaned hrough he Green-Kubo and Ensen formulas. Unceranes (sandard devaons) n he las repored dg(s) are gven n parenhess D L L / L /6 L /4 L /3 D, Eq. (4).55(4).77(5) 3.65(3) 5.48(3) 9.93(83) D, Eq. (5).54(4).9(7) 3.8(3) 5.4() 8.53(68) D r, Eq. (6) 36(9) 9.(4) 9.8(4) Fgure 3. Angular speed dsrbuons. Fgure 4. Radal dsrbuon funcons. ncrease n he molecular elongaon. The calculaed pressures n NpT EMD smulaons are much close o am excep he monaomc molecular sysem. The modfcaon from he sphere of dameer s s o he daomc sysem of L / brngs a volume decrease from 84 nm 3 o 79 nm 3 and hen he volume ncreases wh he value of L. Also noe ha beyond he elongaon of L /3, for example, n he daomc molecular sysems of L 5/ and /, he sysem becomes nfne dluon, keepng he pressure of he sysem as am. Fgure shows he energy dsrbuons of he poenal energy (P.E.), ranslaonal knec energy (T.E.), and roaonal knec energy (R.E.) of monaomc and daomc molecules. The P.E. decreases negavely wh ncreasng neraomc separaon n a daomc as seen n Table. The dsrbuon of T.E. dsplays a ypcal Maxwell-Bolzman energy dsrbuon [f(ε) ~ ε e ε/kt ] whle ha of R.E. ndcaes a lo of daomc molecules do no roae a T 94.4 K. However, hs may lead a msundersandng of roaonal speed dsrbuon, whch s a ypcal Maxwell- Bolzman energy dsrbuon as shown n Fgure 3. Fgure 4 shows he cener-cener radal dsrbuon funcons, g(r), of monaomc and daomc molecules a 94.4 K n NpT ensemble. As he molecular elongaon ncreases, he frs and second peaks n he cener of mass g(r) dmnsh gradually and he mnma ncrease. I s also observed ha he neares dsance beween ceners of daomc molecules becomes shorer wh he ncrease n he molecular elongaon. Ths s because he ceners of mass of daomc molecules come closer as he molecular elongaon ncreases. Table lss he ranslaonal and roaonal dffuson coeffcens of monaomc and daomc molecules a 94.4 K obaned hrough he Green-Kubo and Ensen formulas. The velocy auo-correlaon funcon and mean square dsplacemen of monaomc and daomc molecules are well-behaved (daa no shown) as seen n he sandard devaon for he ranslaonal dffuson coeffcens. As he molecular elongaon of daomc molecule ncreases from he sphercal monaomc molecule, he ranslaonal dffuson coeffcen ncreases. Ths means ha a rod-lke molecule dffuses more han a sphercal molecule as seen n n-buane and -buane. Employng wo formals, Eqs. (6) and (7), o calculae he roaonal dffuson coeffcens, here exss wo dffcules. Frs, n he Green-Kubo formula, Eq. (6), he angular velocy auo-correlaon funcon for L / does no decay o zero n he long me as shown n Fgure 5 and he resulng roaonal dffuson coeffcen goes nfny. Ths s because he neraomc separaon s so shor ha he daomc molecule persss no o roae even hough he orque s large. Second, n he Ensen formula, Eq. (7), he mean square dsplacemens of un orenaon vecor e () are no a lnear funcon of me as seen n Fgure 6 because he value of e () s un. As a resul, we were no able o calculae he roaonal dffuson coeffcens from Eq. (7). The value of he mean square dsplacemens of un orenaon vecor approaches o as me goes nfny snce n he followng equaon : lm e e lm e + e e e [ ] lm e e (4)
NpT EMD Smulaons of Daomc Lquds Bull. Korean Chem. Soc. 7, ol. 8, No. 7 Table 3. Shear (η, mp) and shear spn vscoses (η r, 4 kg/ m sec) of monaomc and daomc molecules a 94.4 K obaned hrough he Green-Kubo and Ensen formulas. Unceranes (sandard devaons) n he las repored dg(s) are gven n parenhess. L L / L /6 L /4 L /3 η, Eq. (9) 3.9(3) 6.6(3) 6.8(5) 4.(3).5(35) η, Eq. () 3.8().8(5).5(3).8(5).6(6) ηr, Eq. ().35(8).86(9) 3.78() 6.3(4) ηr, Eq. ().34(8).43(6).(4).48() Fgure 5. Normalzed angular velocy auo-correlaon funcon. Fgure 7. Translaonal sress [Eq. ()] auo-correlaon funcon. Fgure 6. Mean square dsplacemens of un orenaon vecor. he second erm n he las equaon goes zero as me goes nfny. Rewrng Eq. (6) n erms of he normalzed angular velocy auo-correlaon funcon, D r l --- w w ( ) w ( ) d ------------------------------------, (5) w ( ) w ( ) snce he roaonal emperaure, T r Iw /, s a consan and he nera of momenum for daomc molecule s gven by I m(l/) ml /, l w /T r /m s a consan. The roaonal dffuson coeffcen depends on only he negral of he normalzed angular velocy auo-correlaon funcon n Fgure 5. As he molecular elongaon of daomc molecules ncreases, random roaon decreases and he roaonal dffuson coeffcens for L /4 and /3 are almos he same n he sascal error. Shear vscoses by ranslaonal moons and shear spn vscoses by roaonal moon calculaed from our equlbrum MD smulaons are ls n Table 3. Translaonal vscosy of he sphercal monaomc molecule a Fgure 8. Roaonal sress [Eq. (3)] auo-correlaon funcon. 94.4 K shows a very close resul wh he expermenal vscosy of lqud argon (.97 mp ). Two shear vscoses obaned from dfferen ranslaonal sresses, Eqs. (9) and Eq. (), and wo shear spn vscoses obaned from dfferen ranslaonal sresses, Eqs. () and Eq. (3) for daomc molecules show a dfference n order of magnude each oher. Referrng he shear vscosy of he sphercal monaomc molecule, he frs values of shear vscosy and
7 Bull. Korean Chem. Soc. 7, ol. 8, No. Song H Lee Table 4. Thermal conducves (λ n 7 cal/k cm sec) of monaomc and daomc molecules a 94.4 K obaned hrough he Green-Kubo. Unceranes (sandard devaons) n he las repored dg(s) are gven n parenhess. parenhess. L L / L /6 L /4 L /3 λ, Eq. (8) 45(7) 66() () 76(7) 368(9) λr, Eq. (9) 4(4) () 6() (3) λp, Eq. () 79(5) 87() 88(39) 75(4) 463(6) λ, Eq. () 773(6) 83(34) 74(3) 59(3) 346(53) λr, Eq. () 5.6() 53() 9(3) 9() λo, Eq. (3) 79 898 844 688 37 Fgure. Normalzed hea flux ( p q a ) auo-correlaon funcon. Fgure 9. Normalzed hea flux ( q a ) auo-correlaon funcon. Fgure. Normalzed hea flux ( q a ) auo-correlaon funcon. Fgure. Normalzed hea flux ( hose of shear spn vscosy for daomc molecules should be dscarded. Fgures 7 and 8 shows he ranslaonal and roaonal sresses, respecvely. As he molecular elongaon of daomc molecule ncreases from he sphercal monaomc molecule, he hegh of he ranslaonal sress auocorrelaon funcon decreases and accordngly he shear r q a ) auo-correlaon funcon. vscosy decreases, whle for he shear spn vscosy he oppose s observed. Energy ranspored va molecular moon governs hea conducon n gases, whle energy ransfer beween molecules due o molecular neracon s a domnan facor n hea conducon n lquds. Lqud molecules ranspor energy by molecular moon and ransfer her energy o oher molecules by molecular neracon. Each conrbuon o he oal hermal conducvy, Eq. (3), for he sphercal monaomc and daomc molecules are shown n Table 4. Thermal conducvy of he sphercal monaomc molecule a 94.4 K shows a very close resul wh he expermenal vscosy of lqud argon (74 7 cal/kcm sec ). Several normalzed hea flux auo-correlaon funcons are shown n Fgures 9-3. Each correlaon funcon corresponds o hea flux of Eqs. (8)-(). Snce he modfed Green-Kubo formula, Eq. (4), for hermal conducvy s rewren as : λ d -------- q kt α ( ) q α ( )q α () ------------------------------------, (6) q α ( )q α ( )
NpT EMD Smulaons of Daomc Lquds Bull. Korean Chem. Soc. 7, ol. 8, No. 73 Concluson Fgure 3. Normalzed hea flux ( s relaed o wo facors - he average of square of hea flux a me and he me negraon of he normalzed hea flux auo-correlaon funcon. The normalzed hea flux auo-correlaon funcons of hea fluxes due o ranslaonal, roaonal and poenal energy ranspor show a very smlar rend as seen n Fgures 9, and because each hea flux of each molecule conans he velocy erm, v, n Eqs. (8), (9) and (). Thermal conducvy due o ranslaonal energy ranspor monooncally ncrease as he molecular elongaon of daomc molecule ncreases from he sphercal monaomc molecule, whle for he hermal conducves due o poenal energy ranspor he oppose s observed snce he poenal energy decreases wh he molecular elongaon of daomc molecule as shown n Fgure. The conrbuon of roaonal energy ranspor s relavely slgh. The normalzed hea flux auo-correlaon funcons of hea fluxes caused by ranslaonal and roaonal energy ransfer, Fgures and 3, are dfferen from hose auocorrelaon funcons of hea fluxes due o energy ranspor. Thermal conducvy due o ranslaonal energy ransfer monooncally decreases wh he molecular elongaon of daomc molecule snce he ranslaonal energy ransfer by molecular neracon s relaed o he poenal energy, whle for he hermal conducves due o roaonal energy ransfer he oppose s observed and he magnudes are relavely small compared wh oher conrbuons for hermal conducvy. The oal hermal conducvy obaned our NpT ensemble EMD smulaon ncreases as he molecular elongaon of daomc molecule decreases from he sphercal monaomc molecule (Table 4). Ths resul s nconssen wh he NEMD smulaon resul for wo-cener Lennard-Jones molecules. Includng wo erms for hermal conducvy whch are relaed o he roaonal degree of freedom of daomc molecule, he calculaed hermal conducvy gves almos he same resul o our prevous EMD smulaon sudy 3 n whch he erms relaed o he roaonal degree of freedom of daomc molecule were mssng. r q a ) auo-correlaon funcon. Isohermal-sobarc(NpT) molecular dynamcs smulaons for daomc molecule sysems are carred ou a 94.4 K. The daomc molecules are modeled by equang s volume o monaomc molecule lke argon and by ncreasng he dsance beween nucle. As he molecular elongaon of daomc molecules ncreases from he sphercal monaomc molecule, Lennard-Jones poenal energy and he oal energy decrease, he volume ncreases. The dsrbuon of ranslaonal energy shows a ypcal Maxwell-Bolzmann, ndcang he smulaon sysems are well-equlbraed. The dsrbuon of roaonal energy has he maxmum a energy, whle he dsrbuon of roaonal speed shows a ypcal Maxwell-Bolzmann. Translaon dffuson coeffcens obaned from velocy auo-correlaon funcons (AC) by Ensen relaon are n good agreemen wh hose obaned from mean square dsplacemens (MSD) by Green- Kubo relaon for he sphercal monaomc and daomc molecule sysems. As he molecular elongaon of daomc molecules ncreases from he sphercal monaomc molecule, he dffuson coeffcen ncreases, ndcang ha longsh shape molecules dffuse more han sphercal molecules. Roaonal dffuson coeffcens of daomc molecules are no obaned from mean square dsplacemens of un orenaon vecor for all cases due o non-lnear behavor, and for he L / case snce angular velocy auocorrelaon funcon does no decay o zero. As he molecular elongaon of daomc molecules ncreases, random roaon decreases and he roaonal dffuson coeffcens are almos he same n he sascal error. Translaonal vscosy of he sphercal monaomc molecule a 94.4 K shows a very close resul wh he expermenal vscosy of lqud argon. As he molecular elongaon of daomc molecules ncreases from he sphercal monaomc molecule, ranslaonal vscosy decreases. Roaonal vscosy s much less han ranslaonal vscosy, ndcang he pressure on he wall s conrbued by ranslaon moon no by roaonal moon. In general roaonal vscosy ncreases as he molecular elongaon of daomc molecules ncreases. Thermal conducvy of he sphercal monaomc molecule a 94.4 K shows a very close resul wh he expermenal vscosy of lqud argon. As he molecular elongaon of daomc molecules ncreases from he sphercal monaomc molecule, he oal hermal conducvy by energy ranspor decreases even hough he ranslaonal dffuson ncreases, he oal hermal conducvy by energy ransfer also decreases, and accordngly he oal hermal conducvy ncreases. However, hermal conducves by ranslaonal energy ranspor and by roaonal energy ransfer ncrease wh he molecular elongaon of daomc molecule. Acknowledgmen. Ths work was suppored by Korea Research Foundaon Gran (KRF-4-4-C58). References. Harp, G. D.; Berne, B. J. Phys. Rev. 97, A, 975.
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