On the critical behavior of the Percus-Yevick equation for nontruncated potentials

On he c i ical beha io o he Pe c us- Ye ick equa ion o non unca ed po en ials J. J. B ey and A. San os Depa amen o de F sica Te6 ica, Facul ad de F sica, Uni e sidad de Se illa, Apdo. Co eos 1065, Sec o Su , Se illa, Spain (Recei ed 3 Augus 1984; accep ed 17 Decembe 1984) We p esen a quali a i e analysis on he in luence o unca ing a long- anged po en ial on he c i ical beha io o a luid desc ibed by he Pe cus-Y e ick equa ion. I is shown ha a nonclassical equa ion o s a e o unca ed po en ials can be compa ible wi h a classical one in he long- ange limi . Ou main assump ion is ha - he dominan pa o he di e ence be ween bo h equa ions o s a e is a egula unc ion d i en by he asymp o ic beha io o he di ec co ela ion unc ion. The esul s a e applied o he case o a Leona d-Jones po en ial. Compa ison wi h a ailable nume ical esul s is qui e sa is ac o y. I. INTRODUCTION In he las ew yea s, a g ea a en ion has been de o ed o he s udy o he beha io p edic ed in he c i ical egion by he app oxima e in eg al equa ions o luids. One o he mos widely s udied is he Pe cus- Ye ick (PY) equa ion. Bo h analy ical1-3 and nume i- cal4-6 s udies show ha he PY app oxima ion p edic s classical alues o he c i ical exponen s. Howe e , he exac solu ion o he PY equa ion o he so-called s icky- ha d~sphe e model1 gi es ise o a comp essibili y equa ion o s a e leading o nonclassical scaling unc ions in he c i ical egion.2 As a consequence, some nonclassical ea u es occu , one o hem being a s ong asymme y o he c i ical iso he m wi h espec o he c i ical poin .1•2 A ecen analysis o he PY equa ion o a la ice gas model wi h nea es neighbo in e ac ion e eals he same nonclassical ea u es. 3 Fo mo e ealis ic in e ac ions, nume ical p ocedu es a e needed o sol e he PY equa ion. F om he nume ical solu ion o a unca ed Lenna d-Janes (TLJ) po en ial by Hende son and Mu phy,4 Fishman and Fishe poin ed ou he possible exis ence o a ce ain asymme y on he c i ical iso he m. A s onge asymme y seems o a ise when a po en ial wi h an a ac i e Yukawa ail is consid- e ed. 5 On he o he hand, a nume ical s udy o he ion unca ed Lenna d-Janes (LJ) po en ial6 indica ed a pu ely classical beha io o he PY app oxima ion. These esul s suppo he idea2 ha he PY app ox- ima ion exhibi s.a nonuni e sal c i ical beha io in which, al hough he c i ical exponen s a e always classical, he ampli ude a ios, namely he one measu ing he "deg ee" o asymme y o he c i ical iso he m, ake alues depen- den on he de ails (in pa icula , on he ange) o he in e ac ion po en ial. Ou conjec u e is ha , in he limi o long- anged po en ials, he scaling unc ions o he equa ion o s a e become classical and he c i ical iso he m is symme ical. He e, an in e ac ion po en ial is said o be sho anged i i asymp o ically decays as e han any nega i e powe o he dis ance. So, a po en ial wi h an a l:lc i e Yukawa ail o any unca ed po en ial has a sho ange, while he LJ po en ial is long anged. In his pape we p esen a simple phenomenological analysis showing he plausibili y o he abo e conjec u e. In Sec. II we show ha he classical PY equa ion o s a e o a long- anged po en ial can be compa ible wi h he . · nonclassical equa ion o s a e o a unca ed po en ial, p o ided ha he main di e ence be ween bo h equa ions o s a e comes om he asymp o ic beha io o he di ec co ela ion unc ion. Mo eo e , i is possible o ela e he c i ical coo dina es and ampli udes o he unca ed . po en ial o he ones co esponding o he long- anged po en ial. E en mo e, we a e able o de i e equa ions o he change o he c i ical pa ame e s as he unca ion dis ance inc eases. In Sec. III we apply ou analysis o he LJ po en ial. S a ing om he alues o he c i ical coo dina es and ampli udes co esponding o he TU conside ed by Hen- de son and Mu phy,4 we p edic he co esponding alues o he LJ po en ial, as well as o o he TLJ po en ials. Compa ison wi h p e ious esul s, when possible, is qui e sa is ac o y, despi e he poo accu acy o he alues o Re . 4. II. THE ANALYSIS The comp essibili y equa ion o s a e eads x- 1 =- -= 1 - 41 p d 2C( ), 1 (ap) Loo ksT op T 0 (l) .whe e Pis he p essu e, pis he numbe densi y, Tis he empe a u e, k8 is he Bol zmann cons an , and C( ) is he di ec co ela ion unc ion. The PY app oxima ion consis s o closing he O ns ein-Ze nike ela ion by means o he equa ion C( ) = g( )[ 1 - eu( )/ksT], (2) whe e u( ) is he in e ac ion po en ial and g( ) is he adial dis ibu ion unc ion, which ends o uni y when is la ge. An impo an p ope y o he PY app oxima ion is ha he asymp o ic beha io o he di ec co ela ion unc ion C( ) is gi en by he in e ac ion po en ial u( ). Mo e conc e ely, i one assumes ha he physical condi ion 4312 J. Che n. Phys. 82 (9), 1 May 1985 0021-9606/85/094312-05$02.10 © 1985 Ame ican Ins i u e o Physics J. J. B ey and A. San os: C i ical beha io o he Pe cus-Ye ick equa ion 4313 g( ) --> 1 as --> oo holds in he PY app oxima ion, Eq. (2) implies u( ) C( ) ~- keT (3) o la ge enough, and his will be assumed ue e en a he c i ical poin . Now, le u( ) be a long- anged po en ial, say he U po en ial. We in oduce he po en ial uR( ) ob ained by unca ing u( ) a = R, i.e., uR( ) = u( ), :s; R, = 0, >R. The comp essibili y equa ion o s a e o uR( ) is XR 1 = 1 - 41 p LR d 2 CR( ), (4) (5) CR( ) being he PY di ec co ela ion unC ion o his po en ial, and whe e we ha e aken in o accoun ha CR( ) = 0, > R. A gi en densi y and empe a u e, we ha e x- 1 - XR 1 = -41 p Leo d 2 LlR( ), whe e (6) (7) LlR( ) == C( ) - CR( ). (8) The asymp o ic beha io o his unc ion is gi en by ) u( ) - uR( ) LlR( ~ - keT . (9) To be conc e e, le us assume ha Eq. (9) holds o > 0 and conside R > 0• Then, LlR{ ) o 0 < :s; R is gi en by e ms ha a e negligible as compa ed wi h u( )/ kBT. We w i e o < :s; R, >R. (10) We expec , o such alues o R, C( ), and CR( ) o be e y close o < R. Mo e p ecisely, we assume ha ILR d nLlR( )l ~ I Leo d nLlR( )l ~leo d nl u( ) I R kBT (11) o n ;;;;.. 2, when u( ) is a long- anged po en ial. As a ma e o ac , ou de ini ion o long- anged po en ials implies ha he e is a alue no such ha he igh -hand side o Eq. (11) di e ges o n > no. Howe e , al hough he alidi y o he inequali y ( 11) o a gi en n implies ha i holds o n + 1, he e e se is no ue. The assump ion (11) o n = 2 allows us o w i e -1 -1 p X - XR ~ - kBT WR, (12) whe e wR == -411" Leo d 2u( ). (13) Equa ion (12) is ou main physical ansa z. I s plau- sibili y lies on he long- ange cha ac e o he in e ac ion po en ial u( ), and on he beha io (10), which is a consequence o he law (3). Fo sho - ange po en ials, he igh -hand side o Eq. (11) exis s o all n and he e is no eason o expec inequali y ( 11) o hold. O cou se, deepe heo e ical and nume ical analysis is needed in o de o check he alidi y o Eq. (12). No ice ha we canno use Eq. (12) o w i e -1 -1 p ( ) XR' - XR ~ -kBT wx - wR , (14) unless R' --> oo o a gi en R. The eason is ha e ms ha ha e been neglec ed upon w i ing Eq. (12) can be ele an as compa ed wi h he igh -hand side o Eq. (14). Now, suppose ha he PY comp essibili y equa ion o s a e o uR( ) akes in he egion a ound he c i ical poin (Pc,R, Tc,R) he o m ob ained o s icky ha d sphe es1•2 and also o he la ice gas model wi h nea es neighbo in e ac ion,3 i.e., kBTXR1 = {[BR(T- Tc,R) + A~(A.R + 1)2(p - Pc,R) 2] 112 -AR(A.R- 1)(p- Pc,R)} 2, (15) whe eAR, BR, and XR a e c i ical ampli udes. Along he c i ical isocho e p = Pc,R, one has kBTXR1 = BR(T- Tc,R), p = Pc,R> T- Tc,R--> o+, (16) which co esponds o he classical c i ical exponen 'Y = 1. The c i ical exponen o also akes i s classical alue (o = 3) since kBTXR1 ;____ 4A~(p- Pc,R) 2, T = Tc.R> p- Pc.R--> o+, T = Tc.R, p - Pc.R--> o-. (17) Ne e heless, he c i ical iso he m is asymme ical a ound he c i ical poin , unless A.R = 1. The coo dina es (Pc.R, Tc,R) o he c i ical poin depend on he ange R o he po en ial. Fo TU po en ials, Wa s7 showed ha bo h he c i ical densi y and empe - a u e inc ease as R does. Le us de ine he shi s o he c i ical coo dina es as (18) (19) whe e (Pc, Tc) is he c i ical poin co esponding o he non unca ed po en ial. In he spi i o ou ansa z ( 12), we admi ha XR ami R a e small enough, so ha he c i ical poin (pc, Tc) and i s immedia e icini y lie in he c i ical egion a ound (Pc.R> Tc,R), whe e Eq. (15) holds. In ac , he esul s epo ed by Hende son and Mu phy o a TU po en ial4 show ha laws ( 16) and ( 17) ex end J. Che n. Phys., Vol. 82, No.9, 1 May 1985 4314 J. J. B ey and A. San os: C i ical beha io o he Pe cus-Ye ick equa ion un il, a leas , T ~ 1.3Tc,R and p ~ 1.2Pc,R• and he nume ical s udy o Re . 6 o he LJ po en ial leads o Tc ~ 1.01 Tc,R and Pc ~ 1.04Pc,R. In summa y, Eq. (12) implies ha , i Eq. (15) desc ibes he asymp o ic PY equa ion o s a e in he c i ical egion o a unca ed po en ial, we ha e koTx-' = -pwR + {[BR(T- Tc + R) + A1U'R + 1) 2(p - Pc (20) o he non unca ed po en ial. The p esence o he e m -pwR on he igh -hand side o Eq. (20) makes xR and R o be nonze o, and, he e o e, x- 1 becomes a egula unc ion o p and Ta he c i ical poin (Pc, Tc). So, nea he c i ical poin , Eq. (20) educes o koTx- 1 = 4A2 (p - Pc) 2 + B(T- Tc), .. ,... (21) whe e (22) (23) and he c i ical poin is gi en by x- 1 = l.- = o. I a _,1 c ap c (24) Equa ions (24) allow us o ob ain xR and R in e ms o he pa ame e s desc ibing he c i ical egion o he un- ca ed po en ial: WR 1 + 2AR(>'R- l)(pc wR) 112 XR = AR 8ARAR- (AR- 1)(wRIPc) 112 ' w2(A + 1)2 _ R R R- BR 16A1AR(Pc wR)- 4AR(AR- 1)(pc wR) 112 -1 x--~~~~----~~--~~~----- [8ARAR- (AR- 1)(wR Pc) 112 (25) (26) S ic ly speaking, Eq. (25) is an implici equa ion o xR, as Pc = Pc,R + xR. In he same way, one ge s BR 8ARAR- (AR- 1)(wRIPc) 112 B = - --:.C......::-=-----'-c:..:_--7'--.:..::....:....::.:..__ 2AR (AR + 1)2 (27) 2 wRIPc{ B 2 4A = -·- 4-1 + BR [16ARAR(pc wR) - 4AR(AR - 1)(pc wR) 112 - l]} . (28) The asymp o ic equa ion o s a e (21) is ully classical. In pa icula , he c i ical iso he m is symme ical. In o he wo ds, AR _, 1 when R _, oo, i.e., when wR _, 0. In his limi , we also ha e xR _, 0, R _, 0, AR _, A, and BR _,B. Equa ions (25)-(28) allow us o p edic he alues o Pc, To A, and B om he knowledge o Pc,R• Tc,R, AR, BR, and AR o a gi en R. This p ocess canno be e e sed, as one o he pa ame e s, say AR, would be le unde e mined. The physical eason is ha we ha e been able, wi h he help o assump ion (12), o de i e Eq. (21) om Eq. (15), bu i is impossible o ge Eq. (15) om Eq. (21). Ne e heless, one can s udy he way in which he c i ical coo dina es and ampli udes beha e as R ends o in ini y. The s uc u e o Eqs. (25)-(28) sugges s w i ing (29) AR = A + A(l>w} Z + A< 2 >wR + O(wJ Z), (30) BR = B + B(l>w}/ 2 + B< 2 >wR + O(wJ{Z), (31) xR = wR[x<o> + x(l>w}/ 2 + O(wR)], (32) R = wR[ <O> + (l>w}/ 2 + < 2 >wR + O(wJ{Z)], (33) whe e he coe icien s a e independen o R. Subs i u ion o Eqs. (29)-(33) in o Eqs. (25)-(28) allows o exp ess all he coe icien s in e ms o wo o hem, say A (I) and A (2). The esul is A< 1> = - :i_ A(l) 2 ' (34) A(2) = :i_ (AM - A(2) + 3A(l) ) 2 8AVPc ' (35) B<'> = 0, (36) B(2) = !!_ ( A<l) 2 + ~) 4 2AVPc ' (37) x<o> = Pc (A (I)+ _1_) 4A 2AVPc ' (38) x<l) = Pc (A< 2 >- A0 )2) 4A ' (39) (O) = Pc B' (40) (l) = 0, (41) <2> = - Pc (A<!)+ _1_) . 8AB 2AVPc (42) In he nex sec ion, we will use hese exp essions o ob ain nume ical alues o he LJ po en ial. Ill. APPLICATION TO THE LENNARD-JONES (6, 12) POTENTIAL Le us conside he LJ (6, 12) po en ial u( ) = 4( - 12 - - 6 ), (43) whe e usual uni s o leng h and ene gy ha e been chosen. The co esponding pa ame e wR, de ined in Eq. (13), is hen . 1611' _3( R- 6) WR = -3- R 1 - -3- ' (44) so ha w}/ 2 - R- 312 • J. Che n. Phys., Vol. 82, No.9, 1 May 1985 J. J. B ey and A. San os: C i ical beha io o he Pe cus..:.Ye ick equa ion 4315 F om a nume ical solu ion o he PY equa ion o he LJ po en ial, in which beha io (3) was assumed o hold o > 0 = 5, a classical c i ical beha io o he o m gi en by Eq. (21) was ound.6 The c i ical coo dina es and ampli udes we e Pc ~ 0.288, Tc ~ 1.320, A~ 2.013, B ~ 2.474. (45). (46) (47) (48) I mus be said ha hese alues migh be a ec ed by e o s because o he nume ical algo i hm, 8 namely he choice o 0• On he o he hand, Hende son and Mu phy4 ob- ained, o a TU po en ial wi h R = 6, Pc,R ~ 0.278, Tc,R ~ 1.311. ,-- •(49) (50) These au ho s do no quo e alues o he c i ical ampli- udes. Ne e heless, using hei Figs. 3 and 4, we ha e es ima ed AR ~ 1.847, AR ~ 1.245, BR ~ 2.459. (51) (52) (53) Ob iously, no all he igu es a e signi ican . As a ma e o ac , Fishman and Fishe es ima ed A.R = 1.28 ± 0.03. As ou calcula ions in his sec ion ha e a mainly quali a i e and illus a i e ·cha ac e , we a e no in e es ed in he s udy o he p opaga ion o e o s coming om he unce ain ies o he alues (49)-(53). Al hough he alue R = 6 is p obably no la ge enough io apply in de ail h,e analysis o Sec. II, we can inse alues (49)-(53) in o Eqs. (25)-(28) in o de o es ima e he alues p edic ed o he U po en ial. The esul is Pc ~ 0.284, Tc ~ 1.320, A~ 2.027, B ~ 2.413. (54) (55) (56) (57) The ag eemen wi h he alues (45)-(48) is ai ly sa is ac o y. Now, we a e going o es ima e he coe icien s in he expansions (29)-(33). By aking ad an age om he ac ha Bo> = 0 [Eq. (36)], 'one could compu e B< 2> om BR and B, p o ided ha e ms o o de highe han wR in Eq. (31) can be neglec ed. Al hough o R = 6 i is wl/ 2 ~ 0.28, we ha e used Hende son and Mu phy's esul s as hey a e he only ones we a e awa e o . In his way, one ge s B< 2> ~ 0.595 5. (58) As A.R ~ 1, A_(Il mus be non-nega i e. The posi i e oo o Eq. (37) is A_(l) ~ 0.779 7. (59) Now, subs i u ion o alues (52) and (59) in o Eq. (29) yields A_( 2) ~ 0.358 9. (60) Finally, Eqs. (34), (35), (38)-(40), and (42) gi e A0l ~ -0.790 3, A< 2> ~ 0.526 6, x<o> ~ 0.081 68, x(ll ~ 0.003 615, (O) ~ 0.117 8, (2) ~ -0.016 93. (61) (62) (63) (64) (65) (66) As a es o consis ency, le us no ice ha om Eqs. ( 61 )- (66) one eob ains Eqs. (49)-(51). We can make use o he alues (56)-(62) o es ima e he c i ical ampli udes when R is la ge enough (o , equi alen ly, wR is small enough). Table I shows he esul s o se e al alues o R, as well as he co esponding c i ical coo dina es ob ained om Eqs. (25) and (26). Le us emphasize ha he alues lis ed in Table I mus be seen as me ely indica i e, because o he unce ain ies o he s a ing alues (49)-(53) and he ac ha he alues o R he e conside ed a e smalle han hose o which ou analysis is expec ed o apply. Howe e , compa ison wi h he esul s ob ained om nume ical solu ions o he PY equa ion is wo hwhile.The ag eemen in he c i ical TABLE I. C i ical coo dina es imd ampli udes o Lenna d-Jones po en ials wi h se e al unca ion dis ances R ob ained, by heme hod desc ibed in he ex , om he esul s gi en in Re . 4. The alues be ween b acke s co espond o nume ical solu ions o he Pe cus-Y e ick equa ion. R Pc.R Tc.R AR AR BR 3.5 0.225 1.276 1.739 1.628 2.645 (0.268") (1.275") 5.0 0.274 1.305 1.808 1.334 2.493 (0.276") (1.305•) 6.0 0.278b 1.311b 1.847c 1.245c 2.459d 6.4 0.279 1.313 1.861 1.220 2.451 (0.282<) (1.316") (2.540 ) 8.0 0.282 1.316 1.902 1.153 2.432 10.0 0.283 1.318 1.934 1.107 2.423 15.0 0.284 1.319 1.974 1.057 2.416 (X) 0.284 1..320 2.027 1.000 2.413 (0.2888) (1.3208) (2.0138) (l.OOOS) (2.4748) • Gi en in Re . 7. b Gi en in Re . 4. c Es ima ed om Fig. 4 o Re . 4. d Es ima ed om Fig. 3 o Re . 4. e Gi en in Re . 9. Es ima ed om Fig. 2 o Re . 9. 8 Gi en in Re . 6. J. Che n. Phys., Vol. 82, No.9, 1 May 1985 4316 J. J. B ey and A. San os: C i ical beha io o he Pe cus.,.Ye ick equa ion empe a u e is ai ly good, bu he e is a sys ema ic de ia ion in he c i ical densi y. Compa ison in he c i ical ampli udes is ha dly possible, as hey ha e been a ely s udied. Apa om he alues o he non unca ed po en ial, commen ed be o e, i is qui e good he ag ee- men in BR o R = 6.4.9 In summa y, we ha e shown ha in he PY app ox- ima ion a· nonclassical equa ion o s a e in he c i ical egion o a unca ed po en ial may be consis en wi h a ully classical one in he long ange limi . The main poin in ou analysis is ha he asymp o ic beha io (3) is inhe en o he PY app oxima ion a any he modynamical s a e. Thus, i he in e ac ion is long anged, i is jus i ied o expec he igh -hand side o Eq. (7) o be a egula unc ion o densi y and empe a u e, e en a he c i ical poin . This su ices o p o e he abo e consis ency. In o de o ca y ou explici calcula ions we ha e assumed he alidi y o Eq. (12). The e o e, he PY app oxima ion seems o p esen a nonuni e sal c i ical beha io , in such a way ha he c i ical iso he m ampli ude a io depe ids on he in e ac- ion ange, ending owa ds uni y when he ange becomes in ini y. We ha e explici ly s udied his o unca ed Leona d-Jones po en ials and ound a sa is ac o y ag ee- men wi h p e ious esul s. Ne e heless, accu a e nu- me ical solu ions o se e al unca ion dis ances l! e equi ed in o de o check he eliabili y o he analysis p esen ed he e. 1 R. J. Bax e , J. Che n. Phys . .49, 2770 (1968). 2 S. Fishman and M. E. Fishe , Physica A 108, 1 (1981). 3 A. Pa ola and L. Rea o, Physica A 125, 255 (1984). 4 D. Hende son and R. D. Mu phy, Phys. Re . A 6, 1224 (1972). 5 F. Galle ani, G. Lo Vecchio, and L. Rea o, Phys. Re . A ( o be published). 6 J. J. B ey, A. San os, and F. Rome o, J. Che n. Phys. 77, 5058 (1982). 7 R. 0. Wa s, J. Che n. Phys. 48, 50 (1968). 8 J. J. B ey and A. San os, J. Che n. Phys. 79, 4652 (1983). 9 M. I. Gue e o, G. Sa ille, and J. S. Rowlinson, Mol. Phys. 29, 1941 (1975). We mus poin ou ha , upon es ima ing BR"" 2.540, we ha e aken in o accoun a p in ing e o in he legend o Fig. 2. The e ical axis should ead (pkBT 1 (iJP/iJp)T. We hank J. S. Rowlinson o co espondence abou his poin . J. Chem. Phys .• Vol. 82, No.9, 1 May 1985