Solvation in binary mixtures of dipolar hard sphere solvents: Theory and simulations

J. Chem. Phys. 113, 2360 (2000); h ps://doi.o g/10.1063/1.482051 113, 2360 © 2000 Ame ican Ins i u e o Physics. Sol a ion in bina y mix u es o dipola ha d sphe e sol en s: Theo y and simula ions Ci e as: J. Chem. Phys. 113, 2360 (2000); h ps://doi.o g/10.1063/1.482051 Submi ed: 16 Feb ua y 2000 . Accep ed: 10 May 2000 . Published Online: 28 July 2000 M. Mo illo, Claus Denk, F ancisco Sánchez-Bu gos, and An onio Sánchez ARTICLES YOU MAY BE INTERESTED IN In a ian Expansion o Two-Body Co ela ions: The modynamic Func ions, Sca e ing, and he O ns ein—Ze nike Equa ion The Jou nal o Chemical Physics 56, 303 (1972); h ps://doi.o g/10.1063/1.1676864 In a ian Expansion. II. The O ns ein-Ze nike Equa ion o Nonsphe ical Molecules and an Ex ended Solu ion o he Mean Sphe ical Model The Jou nal o Chemical Physics 57, 1862 (1972); h ps://doi.o g/10.1063/1.1678503 In a ian expansion III: The gene al solu ion o he mean sphe ical model o neu al sphe es wi h elec os a ic in e ac ions The Jou nal o Chemical Physics 58, 3295 (1973); h ps://doi.o g/10.1063/1.1679655 Sol a ion in bina y mix u es o dipola ha d sphe e sol en s: Theo y and simula ions M. Mo illo and Claus Denk Uni e sidad de Se illa, Fı ´sica Teo ´ ica, Apa ado 1065, E-41080 Se illa, Spain F ancisco Sa ´nchez-Bu gos and An onio Sa ´nchez Uni e sidad de Se illa, Quı ´mica Fı ´sica, Facul ad de Quı ´mica, C/ P o esso Ga cı ´a Gonza ´lez s/n, E-41012 Se illa, Spain 共Recei ed 16 Feb ua y 2000; accep ed 10 May 2000兲 The s uc u e o mix u es o dipola ha d sphe e luids wi h componen s o equal size bu di e en dipole momen s a ound a single ion is s udied. The sol a ion ene gy and he pola iza ion a ound he ion is ob ained in he amewo k o he mean sphe ical app oxima ion 共MSA兲. Ou heo e ical esul s and he esul s o o he wo ke s a e compa ed wi h simula ion da a ob ained om Mon e Ca lo simula ions. An in e p e a ion o he meaning o p e e en ial sol a ion is gi en in e ms o he con as ing beha io s o pa ial pola iza ion in he bulk and nea he ion. © 2000 Ame ican Ins i u e o Physics. 关S0021-9606共00兲50830-7兴 I. INTRODUCTION The sol a ion o ions by pola sol en s has been he subjec o nume ous heo e ical and expe imen al s udies. One o he undamen al quan i ies is he ee ene gy o sol- a ion. I s simples desc ip ion elies on he conside a ion o he sol en as a mac oscopic dielec ic con inuum ha be- comes pola ized by he p esence o he ion. The in e ac ion ene gy be ween he sol en pola iza ion and he ield c ea ed by he ion is hen cha ac e ized by he dielec ic cons an o he sol en . Al hough he mac oscopic app oxima ion cap- u es he basic ing edien s o he ee ene gy o sol a ion, hei quan i a i e p edic ions a e o en a a iance wi h he expe imen al indings, in pa icula o solu e ions whose sizes a e no e y la ge compa ed wi h he size o he sol en molecules. The limi a ions o he dielec ic con inuum ea men o he Bo n ee ene gy o sol a ion has led o he in oduc ion o o he al e na i es ha con empla e he molecula desc ip- ion o he solu e–sol en sys em. S a ing om he in eg al equa ions o liquid heo y, and unde sui able app oxima- ions, analy ical exp essions o he sol a ion ee ene gy ha e been ound. In pa icula , o a model o a ha d sphe e solu e ion in a ba h o sol en molecules o med by ha d sphe es wi h poin dipoles, Chan e al. de i ed a o mula o he Bo n ee ene gy o sol a ion wi hin he mean sphe ical app oxima ion 共MSA兲.1Thei exp ession amoun s o eplac- ing he sol en ha d sphe e diame e in he classical Bo n exp ession by an e ec i e one, which depends on he dipole numbe densi y, he dipole momen , and he dielec ic con- s an o he pu e sol en . Thei heo e ical ea men gi es suppo o empi ical exp essions widely used o i expe i- men al da a. A molecula heo y o sol a ion based on he densi y unc ional heo y has been pu o wa d by Chand a and Bagchi,2leading o an exp ession o he ee ene gy o sol a ion which has o be e alua ed nume ically. Pa ey and co-wo ke s3ha e used he linea ized hype ne ed chain clo- su e 共LHNC兲 o sol e he O ns ein–Ze nike equa ion o he same model o ha d sphe e dipoles embedding a ha d sphe e poin cha ge. The esul ing exp essions a e mo e compli- ca ed han he co esponding ones ound in he MSA, hei solu ions equi ing a non i ial nume ical ea men . I is ound ha he sol a ion ene gy ob ained wi hin he LHNC ag ees well wi h ha ob ained om he MSA, excep o solu e ions whose diame e s a e much la ge han ha o a sol en molecule. O he heo e ical app oaches a e based on he use o pe u ba ion heo y. In ecen wo k, Pade ´app ox- iman echniques ha e been exploi ed o unca e he pe u - ba ion expansion o he sol a ion chemical po en ial o a dipole in dipola liquids.4Ex ensions o he dielec ic con- inuum heo y ha include dielec ic sa u a ion and elec os- ic ion e ec s ha e been de eloped in Re . 5. Nume ical simula ions ha e also been p o usely used in he s udy o sol a ion p oblems.6–12 Gene al conside a ions abou he calcula ion o ee ene gies o sol a ion can be ound in Re . 13. Al hough in ecen yea s mos o he in e - es has ocused on he analysis o he dynamics o sol a ion,14–18 some wo k has been de o ed o he s udy o s uc u al and he modynamical p ope ies. Mon e Ca lo simula ions we e used in Re . 6 o analyze he dependence o he sol a ion ene gy in a pola liquid wi h he solu e cha ge and he in luence o dielec ic sa u a ion e ec s. In Re . 9, he dependence on he ionic cha ge o hyd a ion ee ene - gies o ions is s udied. A de ailed analysis o he he mody- namics o ion sol a ion in dipola luids using Mon e Ca lo simula ions and he mean eac ion ield me hod is p esen ed in Re . 12. Recen ly, simula ions o a dipole in a ba h o pola ha d sphe es ha e been p esen ed in Re . 4. In his pape we add ess he p oblem o calcula ing he sol a ion ee ene gy o an ion in a mix u e o pola luids. Mix u es o pola sol en s a e e y con enien om an ex- pe imen al poin o iew as he pola i y o he sol en can be easily con olled by a ying he composi ion o he mix u e. None heless, he analysis o sol a ion o ions in mix u es has ecei ed less a en ion han in pu e sol en s.2,19 In a p e ious JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 6 8 AUGUST 2000 23600021-9606/2000/113(6)/2360/9/$17.00 © 2000 Ame ican Ins i u e o Physics s udy,20 we analyzed he dependence o he eo ganiza ion ene gy o elec on ans e eac ions in pola mix u es wi h he mix u e composi ion, using Mon e Ca lo echniques. The simula ions clea ly indica e ha he e exis s an excess eo - ganiza ion ene gy: as a small amoun o he sol en wi h highe pola i y is added o he mix u e, he eo ganiza ion ene gy inc eases d as ically. The e, he inabili y o heo e i- cal exp essions o explain his phenomena was no iced. The calcula ion o he solu e–sol en adial dis ibu ion unc ion seemed o indica e ha he mic oscopic o igin o his phe- nomenon is he p e e en ial sol a ion o he ions by he mo e pola species. The heo e ical s udy o he in luence o p e - e en ial sol a ion on he eo ganiza ion ene gy is complex, as one has o deal wi h wo cha ge cen e s. He e, we conside he somewha simple p oblem o a single ion in a solu ion a in ini e dilu ion and ca y ou a heo e ical and simula ion analysis o a model sys em. We will see ha a key ing edien o unde s and he be- ha io o he ee sol a ion ene gy o an ion in a mix u e o pola sol en s is he p e e en ial sol a ion o he ion by he componen wi h he highe pola i y. The exis ence o p e e - en ial sol a ion has been ela ed in he li e a u e o de ia ions o he ee sol a ion ene gy om an ideal beha io , cha ac- e ized by a linea dependence o he ee ene gy wi h he mix u e composi ion.2We belie e ha de ia ion om ha assumed linea i y is no a signa u e o p e e en ial sol a ion. Ra he , we ocus on he beha io o he o al and pa ial pola iza ion densi ies o he wo componen s a ound he ion induced by he ion ields. They con ain he key solu e– sol en co ela ions and, based on hei knowledge, one can gi e a mic oscopic in e p e a ion o wha is no mally e med p e e en ial sol a ion. Namely, he composi ion o he mix- u e nea he ion is subs an ially di e en om he nominal composi ion o he liquid in he bulk, away om he ion. This ea u e can be cha ac e ized by he excess local mola ac ion pa ame e 共Ma cus兲.21,22 The componen wi h he highe pola i y con ibu es o he o al pola iza ion densi y o he i s wo sol a ion shells a ound he ion wi h a much g ea e alue han i s bulk mola ac ion would indica e. This local s uc u a ion o he mix u e in he i s ew sol a- ion shells gi es ise o p e e en ial sol a ion, and as he au ho s o Re . 2 poin ou , i s exis ence is no necessa ily ela ed o speci ic de ails o he molecula in e ac ions such as hyd ogen bonds. Fu he mo e, we no ice ha con inuum dielec ic heo ies o he ee sol a ion ene gy con aining he dielec ic cons an as inpu a e no alid, in gene al. The eason is ha he dielec ic cons an inco po a es jus he sol en –sol en co ela ions in he absence o solu e, and hese co ela ions a e no he ele an ones in ela ion wi h he sol a ion ee ene gy. The s uc u e o he pape is as ollows. In Sec. II, we p esen analy ical esul s o a model o a ha d sphe e solu e ion embedded in a mix u e o ha d sphe e dipola sol en s o equal adii and di e en pola i ies. The MSA heo y o he case o single componen sol en has been ho oughly de el- oped by Chan e al.1We ex end hei ideas o he case o a bina y sol en mix u e, ob aining analy ical exp essions o he pa ial and o al sol en pola iza ions and he ee ene gy o sol a ion. We also b ie ly e iew he heo y o Chand a and Bagchi2based on a linea iza ion o he ee-ene gy unc- ional o an inhomogeneous mix u e in he p esence o an ion ield. In Sec. III we desc ibe he me hodology used in ou Mon e Ca lo simula ions. Sec ion IV con ains he esul s o he simula ions and a compa ison wi h he analy ical p edic- ions. II. ANALYTICAL THEORIES A. MSA o ions in a mix u e o ha d sphe e sol en s The desc ip ion o he dielec ic p ope ies o sol en s wi hin he mean sphe ical app oxima ion has been ca ied ou by se e al au ho s s a ing om he wo k o We heim.23,24 Adelman and Deu ch25 ha e sol ed he p ob- lem o a mix u e o dipola ha d sphe es in he amewo k o he MSA. In he case o equally sized componen s, hey show ha he mix u e beha es like a monocomponen luid wi h he same ha d sphe e adius and empe a u e pa ame e as he o iginal mix u e, bu wi h e ec i e dipole momen ␮ ˜ 2⫽1 m兺 k ␮ k 2,共1兲 and densi y ␳ ˜ ⫽1 ␮ ˜ 2兺 k ␮ k 2 ␳ k,共2兲 whe e mis he numbe o componen s in he mix u e, and ␮ k and ␳ ka e he dipole momen s and numbe densi ies o each componen , espec i ely. The co ela ion unc ions a e com- ple ely de e mined by he co ela ion unc ions o We - heim’s solu ion o he e ec i e pu e pola luid. Chan e al.1ha e sol ed he p oblem o a mix u e o ions modeled by ha d sphe es wi h poin cha ges in a dipola sol en o ha d sphe es wi h poin dipoles, ob aining an ex- p ession o he ee ene gy o sol a ion 共Bo n ee ene gy o cha ging兲o an ion o cha ge ze a in ini e dilu ion, FB⫽⫺ 共ze兲2 2共1 2R1⫹Rs兲 冉 1⫺1 ⑀ 冊 .共3兲 He e, ⑀ is he dielec ic cons an o he pu e dipola luid, R1 is he ion diame e , and Rsis a co ec ion o Bo n’s exp es- sion ha is gi en in e ms o he dipole adius and he di- elec ic p ope ies o he pu e dipola luid. Ou aim is o de e mine he sol a ion ene gy o an ion in a bina y mix u e o dipola luids wi h equal adii bu di e - en dipola momen s. The esul o Adelman and Deu ch o he sol en mix u e in he absence o ions sugges s ha Eq. 共3兲could be applied, using he dielec ic p ope ies o he e ec i e luid, as gi en by Eqs. 共1兲and 共2兲. In wha ollows, we will ou line he necessa y s eps o gene alize he heo y o Chan e al. o a sol en consis ing o wo componen s wi h equal adii and di e en pola i y. We will ollow he no a ion o Chan e al. and will e e o hei pape whe e app op ia e. The ionic solu es a e ea ed as cha ged ha d sphe es wi h diame e R1; di e en species o ions ca y cha ges z ␣ and he elec oneu ali y condi ion 2361J. Chem. Phys., Vol. 113, No. 6, 8 Augus 2000 Sol a ion in dipola bina y mix u es 兺 ␣ ␳ ␣ z ␣ ⫽0共4兲 is assumed o hold, whe e ␳ ␣ deno es he numbe densi y o species ␣ . The ions a e sol a ed by a mix u e o ha d sphe es, each species ha ing diame e R2, numbe densi y ␳ i, and ca ying a dipole momen ␮ i. The in e ac ion po en- ials ha e a ha d co e epulsi e e m and u ␣␤ 共 兲⫽z ␣ z ␤ e2/ , ⬎R1,共5兲 u ␣ i共 , ␻ 2兲⫽⫺z ␣ e ␮ iE2/ 2, ⬎R12⫽1 2共R1⫹R2兲,共6兲 uij共 ␻ 1, , ␻ 2兲⫽⫺ ␮ i ␮ jD12 / 3, ⬎R2,共7兲 whe e we ha e adop ed G eek indices o ions and La in indices o dipoles. The unc ions Ei⫽ ␮ ˆ( ␻ i)• ˆ and Dij ⫽ ␮ ˆ( ␻ i)(3 ˆ ˆ⫺I) ␮ ˆ( ␻ j) a e he angula dependen pa s o he in e ac ion po en ials. In hese exp essions, eis he el- emen a y cha ge, Iis he 3⫻3 uni enso , and he uni ec- o ˆ poin s om he molecule deno ed by he i s index owa ds he one deno ed by he second index. The o ien a- ion o a dipole momen is de ined by he solid angle ␻ . The O ns ein–Ze nike 共OZ兲equa ions o an ion–dipole mix u e a e gi en by h ␣␤ 共 兲⫽c ␣␤ 共 兲⫹兺 ␥ ␳ ␥ 冕 dsc ␣␥ 共 兩 ⫺s 兩 兲h ␥␤ 共s兲 ⫹兺 k ␳ k 冕 ds 具 c ␣ k共 ⫺s兲hk ␤ 共 ␻ 3,s兲 典 ␻ 3,共8兲 h ␣ j共 , ␻ 2兲⫽c ␣ j共 , ␻ 2兲⫹兺 ␥ ␳ ␥ 冕 dsc ␣␥ ⫻共 兩 ⫺s 兩 兲h ␥ j共s, ␻ 2兲⫹兺 k ␳ k 冕 ds ⫻ 具 c ␣ k共 ⫺s, ␻ 3兲hkj共 ␻ 3,s, ␻ 2兲 典 ␻ 3,共9兲 hi ␤ 共 ␻ 1, 兲⫽ci ␤ 共 ␻ 1, 兲⫹兺 ␥ ␳ ␥ 冕 dsci ␥ ⫻共 ␻ 1, ⫺s兲h ␥␤ 共s兲⫹兺 k ␳ k 冕 ds ⫻ 具 cik共 ␻ 1, ⫺s, ␻ 3兲hk ␤ 共 ␻ 3,s兲 典 ␻ 3,共10兲 hij共 ␻ 1, , ␻ 2兲⫽cij共 ␻ 1, , ␻ 2兲 ⫹兺 ␥ ␳ ␥ 冕 dsci ␥ 共 ␻ 1, ⫺s兲h ␥ j共s, ␻ 2兲 ⫹兺 k ␳ k 冕 ds 具 cik共 ␻ i, ⫺s, ␻ 3兲hkj ⫻共 ␻ 3,s, ␻ 2兲 典 ␻ 3,共11兲 whe e his he o al and c he di ec co ela ion unc ion. We ha e used he no a ion 具典 ␻ ⫽1/4 ␲ 兰 d ␻ o he angula con- olu ion. In he MSA, he closu e condi ions a e ob ained om Eqs. 共5兲–共7兲, hIJ共 ␻ 1, , ␻ 2兲⫽⫺1, ⬍RIJ c, cIJ共 ␻ 1, , ␻ 2兲⫽⫺ ␤ uIJ共 ␻ 1, , ␻ 2兲 ⬎RIJ c;共12兲 I⫽ ␣ ,i;J⫽ ␤ ,j. He e we ha e in oduced uppe case La in indices which un o e bo h G eek and La in indices. In his no a ion he a gu- men s o he co ela ion unc ions ha e o be adjus ed de- pending on he cu en index: i wo G eek indices a e p esen , he a gumen is . A G eek and a La in index ca y and an angula dependence. In his case, i he La in index is he second one, his angula dependence is ␻ 2, o he wise i is ␻ 1. Two La in indices ca y he ull a gumen as in Eq. 共12兲. The con ac dis ance RIJ cis R1 o wo G eek indices, R2 o wo La in indices, and R12 o one G eek and one La in index. The angula unc ions 1, Ei,Dij, and ⌬ij⫽ ␮ ˆ( ␻ i) • ␮ ˆ( ␻ j) o m a closed se unde he angula con olu ion 具 A( ␻ 1, ␻ 3)B( ␻ 3, ␻ 2) 典 ␻ 3. Mul iplica ion ables o hese quan i ies can be ound in Re . 1. Using his angula decom- posi ion, we use an ansa z o he o m ␣␤ 共 兲⫽ 11共 兲⫹z ␣ z ␤ C共 兲, ␣ j共 , ␻ 2兲⫽ 12共 兲⫹z ␣ ␮ ˜ j E共 兲E2, 共13兲 i ␤ 共 ␻ i, 兲⫽ 21共 兲⫺z ␤ ␮ ˜ i E共 兲E1, ij共 ␻ i, , ␻ 2兲⫽ 22共 兲⫹ ␮ ˜ i ␮ ˜ j ⌬共 兲⌬12⫹ ␮ ˜ i ␮ ˜ j D共 兲D12 o he co ela ion unc ions, whe e ⫽c,h, and ␮ ˜ i ⫽ ␮ i/ ␮ ˜ . This con enien ly in oduces he dipole momen o he e ec i e luid as de ined in Eq. 共1兲. No e ha ou ansa z is essen ially he same as he one in Re . 1; he angula pa s o he co ela ion unc ions ha e been scaled acco ding o he di e en in e ac ion po en ials. The closu e condi ions 共12兲 in e ms o hese unc ions a e he ha d co e condi ions h11 ⫽h12⫽h21⫽h22⫽⫺1 o dis ances sho e han con ac , and cC共 兲⫽⫺ ␤ e2 , ⬎R1,共14兲 cE共 兲⫽ ␤␮ ˜ e2 2, ⬎R12 ,共15兲 cD共 兲⫽ ␤␮ ˜ 21 3, ⬎R2.共16兲 These condi ions a e he same as in Re . 1, when ␮ in Re . 1 is eplaced by he dipole momen o he e ec i e pu e luid ␮ ˜ . The h ee-dimensional Fou ie ans o m ˜ 共 ␻ 1,k, ␻ 2兲⫽ 冕 d eik• 共 ␻ 1, , ␻ 2兲共17兲 o he OZ equa ions in ou no a ion may be w i en as h ˜ IJ共 ␻ 1,k, ␻ 2兲 ⫽c ˜ IJ共 ␻ 1,k, ␻ 2兲 ⫹兺 K ␳ K 具 c ˜ IK共 ␻ 1,k, ␻ 3兲h ˜ KJ共 ␻ 3,k, ␻ 2兲 典 ␻ 3.共18兲 The angula decomposi ion o ˜ IJ( ␻ 1,k, ␻ 2) is he same as in Eq. 共13兲, when is eplaced by kand by ˜ . The 2362 J. Chem. Phys., Vol. 113, No. 6, 8 Augus 2000 Mo illo e al. k-dependen angula unc ions will be deno ed by 1, E ˜ i,⌬ ˜ ij, and D ˜ ij. No e ha he coe icien s o hese angula unc ions may be ob ained by one-dimensional Hankel ans o ms om hei -dependen coun e pa s ˜ 共k兲⫽4 ␲ 共⫺i兲n 冕 0 ⬁d jn共k 兲 共 兲,共19兲 whe e jn(x) is he sphe ical Bessel unc ion o o de nand he coe icien s a e ans o med acco ding o ˜ 共k兲⫽ ˜ D共k兲;n⫽2, 共20兲 ˜ 共k兲⫽ ˜ E共k兲;n⫽1, 共21兲 ˜ 共k兲⫽ ˜ C共k兲, ˜ ⌬共k兲, ˜ ␣␤ 共 ␣ , ␤ ⫽1,2兲;n⫽0. 共22兲 In o de o ob ain decoupled equa ions o hese quan i- ies, i is con enien o in oduce wo new o hogonal angula unc ions J ˜ ⫹⫽1/3共⌬ ˜ ⫹D ˜ 兲,共23兲 J ˜ ⫺⫽1/3共2⌬ ˜ ⫺D ˜ 兲,共24兲 wi h coe icien s ˜ ⫹⫽ ˜ ⌬⫹2 ˜ D,共25兲 ˜ ⫺⫽ ˜ ⌬⫺ ˜ D.共26兲 Ca ying ou he angula con olu ion in Eq. 共18兲, using he elec oneu ali y condi ion 共4兲, and se ing ␳ c⫽兺 ␥ ␳ ␥ z ␥ 2,共27兲 ␳ I⫽兺 ␥ ␳ ␥ ,共28兲 ␳ d⫽兺 k ␳ k,共29兲 we ob ain he ollowing coupled se o equa ions o he co ela ion unc ions: h ˜ 11共k兲⫽c ˜ 11共k兲⫹ ␳ Ic ˜ 11共k兲h ˜ 11共k兲⫹ ␳ dc ˜ 12共k兲h ˜ 21共k兲,共30兲 h ˜ 12共k兲⫽c ˜ 12共k兲⫹ ␳ Ic ˜ 11共k兲h ˜ 12共k兲⫹ ␳ dc ˜ 12共k兲h ˜ 22共k兲,共31兲 h ˜ 21共k兲⫽c ˜ 21共k兲⫹ ␳ Ic ˜ 21共k兲h ˜ 11共k兲⫹ ␳ dc ˜ 22共k兲h ˜ 21共k兲,共32兲 h ˜ 22共k兲⫽c ˜ 22共k兲⫹ ␳ Ic ˜ 21共k兲h ˜ 12共k兲⫹ ␳ dc ˜ 22共k兲h ˜ 22共k兲,共33兲 h ˜ C共k兲⫽c ˜ C共k兲⫹ ␳ cc ˜ C共k兲h ˜ C共k兲⫺1 3 ␳ ˜ c ˜ E共k兲h ˜ E共k兲,共34兲 h ˜ E共k兲⫽c ˜ E共k兲⫹ ␳ cc ˜ C共k兲h ˜ E共k兲⫹1 3 ␳ ˜ c ˜ E共k兲h ˜ ⫹共k兲,共35兲 h ˜ ⫹共k兲⫽c ˜ ⫹共k兲⫺ ␳ cc ˜ E共k兲h ˜ E共k兲⫹1 3 ␳ ˜ c ˜ ⫹共k兲h ˜ ⫹共k兲,共36兲 h ˜ ⫺共k兲⫽c ˜ ⫺共k兲⫹1 3 ␳ ˜ c ˜ ⫺共k兲h ˜ ⫺共k兲.共37兲 This se o equa ions can be di ided in o h ee uncoupled g oups: g oup A, Eqs. 共30兲–共33兲, oge he wi h he co e- sponding closu e condi ions 共12兲de ines a mix u e o ha d sphe es o diame e R1a densi y ␳ Iand diame e R2a densi y ␳ dwi h he Pe cus–Ye ick closu e. G oup B, Eqs. 共35兲–共36兲, desc ibes he angula co ela ion o he luid. Fi- nally, Eq. 共37兲is decoupled om he o he equa ions. When compa ing ou equa ions wi h he esul s o Chan e al.,we no e ha he angula co ela ions 共g oup B兲a e desc ibed by he same OZ equa ions, when ␳ is eplaced by he densi y o he e ec i e luid ␳ ˜ . As has been no ed abo e, he closu e condi ions a e also iden ical, eplacing ␮ by ␮ ˜ . Thus, in wha ollows, we will use he esul s o Chan e al. o h ˜ ⫹ and h ˜ E. The a e age in e ac ion ene gy o a single ion wi h he pola mix u e is gi en by E⫽1 4 ␲ 兺 k ␳ k 冕冕 d d ␻ u ␣ k共 , ␻ 兲g ␣ k共 , ␻ 兲 ⫽⫺ 4 ␲ z ␣ 2e 3 ␳ ˜ ␮ ˜ 冕 R12 ⬁hE共 兲,共38兲 whe e we ha e used Eqs. 共6兲and 共13兲. The Bo n ee ene gy o sol a ion o an ion o cha ge ze is hen ob ained as1 FB⫽⫺ 1 2 共ze兲2 1 2R1⫹Rs 冉 1⫺1 ⑀ 冊 , Rs⫽R2 冉 1 2⫺3 ␰ ˜ 共1⫹4 ␰ ˜ 兲 冊 ,共39兲 whe e ␰ ˜ is he solu ion o Q共2 ␰ ˜ 兲⫺Q共⫺ ␰ ˜ 兲⫽34 ␲ 9 ␤ • ␳ ˜ ␮ ˜ 2.共40兲 He e, Q( ␩ ) is he Pe cus–Ye ick ha d sphe e in e se com- p essibili y and ⑀ is he dielec ic cons an o he pu e e ec- i e luid25 共see he Appendix兲. In he ollowing sec ions, we will compa e he p edic ions o Eq. 共39兲wi h he simula ion esul s. The sol a ion ene gy basically depends on he sol en longi udinal pola iza ion densi y a ound he ion. The sol en pola iza ion cha ac e izes he esponse o he sol en o he ield o he sol a ed ion. By de ini ion, he pola iza ion den- si y o he kspecies in he mix u e induced by he ion is gi en by Pk共 兲⫽ ␳ k 4 ␲ 冕 d ␻ g ␣ k共 , ␻ 兲 ␮ k共 ␻ 兲• ˆ⫽ ␳ k ␮ k 2z 3 ␮ ˜ hE共 兲, 共41兲 and he o al pola iza ion densi y is P共 兲⫽兺 kPk⫽1 3 ␳ ˜ ␮ ˜ zhE共 兲.共42兲 Fa om he ion, he pola iza ion densi y akes i s asymp o ic alue gi en by he mac oscopic exp ession Pmac⫽ze 2 1 4 ␲ 冉 1⫺1 ⑀ 冊 .共43兲 I will be illus a i e o inspec he a io Pk( )/Pmac( ). This quan i y ep esen s he de ia ion o he pola iza ion densi y wi h espec o he mac oscopic model, and i empha- 2363J. Chem. Phys., Vol. 113, No. 6, 8 Augus 2000 Sol a ion in dipola bina y mix u es sizes he ela i e con ibu ion o each species. The e alua ion o he unc ion hE( ) has been ca ied ou in Re . 1 and i leads o Pk共 兲 Pmac共 兲⫽ ␳ k ␮ k 2 ␳ ˜ ␮ ˜ 2 冉 1⫺ dF共 兲 d ⫹F共 兲 冊 ,共44兲 whe e he quan i y F( ) is he solu ion o an in eg al equa- ion which can be gene a ed nume ically by a apezoidal ule. The o al pola iza ion densi y can hen be w i en as P共 兲 Pmac共 兲⫽1⫺ dF共 兲 d ⫹F共 兲.共45兲 The ela i e con ibu ion o species kis scaled by he ac o ␳ k ␮ k 2. Fo he special case o a bina y mix u e wi h ␳ 1 ␮ 1 2/ ␳ 2 ␮ 2 2⫽1, bo h species con ibu e equally o he o al pola iza ion densi y P( ). These esul s will be used la e on o discuss he meaning o p e e en ial sol a ion o an ion in a mix u e o luids o med by ha d sphe e dipoles. B. The heo y o Chand a and Bagchi Chand a and Bagchi2de i ed an exp ession o he en- e gy o sol a ion o an ion in a bina y mix u e o pola liq- uids. Thei wo k is based on an expansion o he ee-ene gy unc ional o he inhomogeneous dipola mix u e in he p es- ence o an ex e nal ield 共 he ion ield兲. Fo he longi udinal pola iza ion o he wo species, hey ob ain Pi共k兲⫽3 4 ␲ ␥ i共k兲E0共k兲,i,j⫽1,2, 共46兲 whe e E0(k)⫽E0(k)k ˆis he ba e elec ic ield o he ion, E0(k)⫽i4 ␲ zej0(kR)/k, and ␥ i共k兲⫽YiiXjj共k兲⫺Xij共k兲Yjj Xii共k兲Xjj共k兲⫺Xij共k兲Xji ,i⫽j,共47兲 Xij共k兲⫽ ␦ ij⫺ ␮ i ␮ j ␳ i 4 ␲ cij m共110,k兲,共48兲 Yij⫽4 ␲ ␤ ␳ i ␮ i ␮ j/9. 共49兲 He e, he cij m(110,k) a e coe icien s in he expansion o he wa e ec o -dependen sol en –sol en di ec co ela ion unc ion in e ms o sphe ical ha monics in he molecula ame 共 his choice o ame is indica ed by he supe sc ip m兲.26 Namely, 共k, ␻ 1 ⬘, ␻ 2 ⬘兲⫽兺 l1l2m m共l1l2m,k兲Yl1m共 ␻ 1 ⬘兲Yl2m ¯ 共 ␻ 2 ⬘兲, m ˜ ⫽⫺m.共50兲 The di ec co ela ion unc ion co esponds o he homoge- neous dipola mix u e in he absence o solu e ions and i may be ob ained in analy ical o m om he MSA solu ion o he p oblem o he e ec i e liquid.25 Thus, he coe i- cien s cij m(110,k) can be eadily e alua ed wi hin he MSA in e ms o he Pe cus–Ye ick solu ion o a sys em o nonpo- la ha d sphe es wi h an e ec i e densi y which depends on he pola i y o he ac ual sol en . In wha ollows, we will es ic ou sel es o a mix u e o equally sized dipoles wi h diame e R⫽R2. Fo his case, he esul o Chand a and Bagchi o he sol a ion ene gy is FB⫽⫺ 1 2 1 共2 ␲ 兲3 冕 dkP共k兲•E0共k兲 ⫽3共ze兲2 ␲ 冕 dk共 ␥ 1共k兲⫹ ␥ 2共k兲兲j0 2共kR兲,共51兲 and he pola iza ion a ound he ion is gi en by P共 兲⫽P1共 兲⫹P2共 兲⫽6ze 共2 ␲ 兲2 冕 dkkj1共k 兲j0共kR兲 ⫻共 ␥ 1共k兲⫹ ␥ 2共k兲兲.共52兲 Using he de ini ion o ␥ i(k) and Eq. 共A1兲, i is easy o show ha P1共k兲 P2共k兲⫽ ␥ 1共k兲 ␥ 2共k兲⫽ ␳ 1 ␮ 1 2 ␳ 2 ␮ 2 2,共53兲 i.e., as in he p e ious app oach, he a io o he pa ial po- la iza ion densi ies is ixed by he sys em pa ame e s. In o - de o nume ically e alua e Eqs. 共51兲and 共52兲, he coe i- cien s cij m(110,k) a e equi ed. In he Appendix, hese unc ions a e gi en in e ms o he Pe cus–Ye ick di ec co ela ion unc ion. The in eg ands in Eqs. 共51兲and 共52兲a e di icul o ea nume ically. I is con enien o s udy ␥ 12(k)⫽ ␥ 1(k)⫹ ␥ 2(k) o he limi ing cases k→0 and k →⬁共see he Appendix兲and o ca y ou he in eg a ion in h ee pa s, i.e., FB⫽3共ze兲2 ␲ 冋 ␥ 12共0兲 冕 0 k1dk j0 2共kR兲 ⫹ 冕 k1 k2dk ␥ 12共k兲j0 2共kR兲⫹ ␥ 12共⬁兲 冕 k2 ⬁dk j0 2共kR兲 册 , 共54兲 whe e he pa ame e s k1and k2a e chosen such ha he esul does no depend on hese pa ame e s. The in eg als o e j0 2(kR2) can be exp essed in e ms o sine in eg als and he nume ical in eg a ion can now be es ic ed o a ini e ange o alues kwhe e he in eg ands a e nume ically well beha ed. III. MONTE CARLO SIMULATIONS We now u n o he calcula ion o he sol a ion ee en- e gy om Mon e Ca lo 共MC兲simula ions o he model sys- em conside ed in he p e ious sec ion. Pe iodic bounda y condi ions wi h he minimum image con en ion27,28 we e ap- plied o a cubic simula ion box o side leng h L. Con e - gence o he algo i hm is accele a ed by swapping sol en pa icles o di e en species. De ails o ou simula ion me hod a e desc ibed in Re . 20. The simula ions p esen ed in his pape ha e been ca ied ou using N⫽863 sol en pa icles. We used a packing ac ion ␩ ⫽0.42, co espond- ing o a dense liquid. A cu o o 4 molecula diame e s has been applied o he long- anged dipola o ces; con ibu- ions om ou side he cu o sphe e ha e been aken in o 2364 J. Chem. Phys., Vol. 113, No. 6, 8 Augus 2000 Mo illo e al. accoun using a gene alized eac ion ield me hod.29 Fo each sys em, 108MC con igu a ions we e gene a ed; mean alues we e calcula ed a e 107equilib a ion con igu a ions. The dielec ic cons an o he eac ion ield has been ob ained om simula ions o he pu e sol en . The ee ene gy o cha ging o he solu e ion in he so- lu ion is gi en o mally by he he modynamic in eg a ion exp ession30,31 ⌬F共0→1兲⫽ 冕 0 1d␭ 具 V 典 ␭,共55兲 which desc ibes he ee-ene gy di e ence be ween he un- cha ged s a e o he solu e (␭⫽0) and i s inal cha ged s a e wi h o al cha ge ze(␭⫽1). V ep esen s he o al solu e– sol en in e ac ion ene gy and he angula b acke s indica e an a e age aken wi h an equilib ium ensemble desc ibing he o e all sys em when he solu e cha ge akes he ic i ious alue ␭ze. The nume ical e alua ion o he abo e exp ession equi es, in p inciple, he knowledge o he in eg and o a la ge numbe o poin s, hus equi ing a subs an ial numbe o in e media e simula ions. As poin ed ou by Humme and Szabo,31 ano he al e na i e o e alua e he ee-ene gy di - e ence is o make use o he in o ma ion abou he s a is ical dis ibu ion o he alues o Va ailable om compu e simu- la ions o jus he ini ial and inal s a es. Following hei analysis, we e alua e he ee-ene gy di e ence using he exp ession ⌬F共0→1兲⬇1 2共 具 V 典 0⫹ 具 V 典 1兲⫺ ␤ 12共 具 共V⫺ 具 V 典 0兲2 典 0 ⫺ 具 共V⫺ 具 V 典 1兲2 典 1), 共56兲 which depends on he knowledge o he i s momen s o he s a is ical dis ibu ion o Vin he ini ial and inal s a es. As shown in Re . 31, his exp ession is exac o ou h o de in he ee-ene gy pe u ba ion expansion. In o de o check he accu acy o Eq. 共56兲, we ha e ca ied ou simula ions o in e media e poin s; in all cases he co ec ions o he ee ene gy a e ound o be wi hin he s a is ical e o limi s. IV. RESULTS AND DISCUSSION In Fig. 1 we plo he Bo n ee ene gy o sol a ion o an ion wi h alence z⫽1 as a unc ion o he mola composi ion o a mix u e o wo ha d sphe e dipole liquids wi h equal adii and dipole momen s ␮ L⫽0.17eÅ and ␮ H⫽0.34eÅ. The adius o he solu e ion is aken o be he same as he sol en adius, R1/2⫽R2/2⫽1.44Å. E en o he cases o a single componen sol en 共 H ⫽0 and H⫽1兲, he sol a ion ene gy ob ained om he MSA and he Chand a and Bagchi heo y di e s om he alue ob ained in he simula ions. The de ia ion o he he- o e ical esul s om he simula ion da a is la ge o he pu e sol en wi h highe pola i y ( H⫽1). The simula ion esul s show wo main ea u es: i s , a s eep inc ease o FBcan be no ed when a small ac ion o high pola i y sol en is added o he pu e low pola i y sol en ( H⬍0.1). Second, he alue o he ee ene gy o sol a ion becomes p ac ically indepen- den o he mola ac ion o H⬎0.5. This second ea u e o he sol a ion ene gy is be e desc ibed by he MSA esul s and seems o be absen in he Chand a and Bagchi app oach. Bo h heo ies ep oduce he de ia ion o FB( H) om an ‘‘ideal’’ law which implies a linea beha io wi h he mola ac ion H. This de ia ion, as measu ed by he excess ee ene gy ⌬FB共 H兲⫽FB共 H兲⫺FB共0兲⫺ H共FB共1兲⫺FB共0兲兲,共57兲 has been used in Re . 2 as an indica ion o p e e en ial sol- a ion. In Fig. 2 we plo ⌬FB( H) o he same pa ame e s as in Fig. 1. In his ep esen a ion he de ia ion om he ideal beha io can clea ly be seen o bo h he heo e ical and he simula ion esul s. Howe e , he d as ic inc ease o FB o small mola ac ions H共which gi es ise o a p onounced asymme y in ⌬FB兲in he simula ion esul s is no ep o- duced by he heo ies. The MSA app oach yields a sligh ly highe alue o he excess ee ene gy when compa ed wi h he esul s o Chand a and Bagchi, al hough bo h heo ies unde es ima e ⌬FB. To u he in es iga e he di e ences be ween he simu- la ion esul s and he heo e ical p edic ions, we now u n ou a en ion o he induced pola iza ion densi y P( ) a ound he FIG. 1. The Bo n ee ene gy o sol a ion o he pa ame e alues gi en in he main ex . The ci cles a e he simula ion esul s; he solid line is an in e pola ion spline o hese poin s. The dashed line is he heo e ical p e- dic ion o Eq. 共39兲, he do ed-dashed line is he nume ical esul ob ained om he Chand a and Bagchi app oach, Eq. 共54兲. The long dashed line ep esen s he mac oscopic Bo n o mula. The a iable His he mola ac ion o he componen wi h highe pola i y. FIG. 2. The excess ee ene gy, Eq. 共57兲, o he same pa ame e s and line ypes as in Fig. 1. 2365J. Chem. Phys., Vol. 113, No. 6, 8 Augus 2000 Sol a ion in dipola bina y mix u es ion. In Fig. 3 we ep esen he pa ial pola iza ion densi ies PL( )/Pmac( ) and PH( )/Pmac( ) o he case o a mix u e wi h ␳ L ␮ L 2/ ␳ H ␮ H 2⫽1, i.e., H⫽0.2. Fo his choice o pa am- e e s, he pa ial pola iza ions o he low and high pola i y species 关PL( ) and PH( )兴ob ained wi hin he MSA 关see Eq. 共44兲兴 a e iden ical. The same holds ue o he Chand a and Bagchi app oach 关see Eq. 共53兲兴. In Fig. 3, his is mani- es ed by he coincidence o he wo pa ial pola iza ion den- si y plo s o each heo e ical desc ip ion. Bo h heo e ical esul s gi e a simila beha io , excep o alues o close o con ac . The i s maximum o Pk( )(k⫽L,H) lies a ⫽Rin he MSA me hod, while he Chand a and Bagchi esul shows a i s maximum o alues sligh ly la ge han R. Fo alues →⬁, bo h heo ies p edic he expec ed mac- oscopic bulk alue o he pola iza ion densi y PL( )/Pmac( )⫽PH( )/Pmac( )⫽1/2. The ailu e o he heo e ical pola iza ions o desc ibe he simula ion esul s a e clea ly seen in he igu e. The pa - ial pola iza ion densi ies Pk sim( ) show oscilla ions, e en o in e molecula dis ances a which he heo e ical esul s ha e al eady eached hei asymp o ic alue. Fu he mo e, he po- si ions o he maxima and minima a e displaced. Thus, he s uc u e o he i s ew sol a ion shells is no p edic ed well by he heo ies. Ano he disc epancy mani es s i sel when s udying he i s and second sol a ion shell. While he heo- ies p edic PL( )⫽PH( ) o all , he simula ion indica es ha he con ibu ion o PHp e ails o small in e molecula dis ances. This ea u e is mo e p onounced o he i s sol- a ion shell. An in e es ing conclusion ha can be d awn om he esul s discussed abo e conce ns he meaning o p e e en ial sol a ion. In he wo k o Chand a and Bagchi, p e e en ial sol a ion is de ined in e ms o he de ia ion o he sol a ion ene gy om i s ideal beha io wi h espec o he mola ac- ion. Le us suppose o a momen ha he ion–dipole pai dis ibu ion unc ion o species k⫽L,His de ined as g ␣ k共 , ␻ 2兲⫽g ␣ k共 兲⫹z ␮ k ␮ ˜ hk E共 兲E2 k,共58兲 i.e., bo h he adial and he angula componen o he co e- la ion unc ion depend on he index k. In his case, he angu- la a e age o ␮ k• ˆ is gi en by 具 ␮ kE2 k 典 共 兲⫽ 1 4 ␲ 兰 d ␻ 2 ␮ kE2 kg ␣ k共 , ␻ 2兲 1 4 ␲ 兰 d ␻ 2g ␣ k共 , ␻ 2兲 ⫽1 3z ␮ k 2 ␮ ˜ hk E共 兲g ␣ k共 兲,共59兲 and he pa ial pola iza ion due o species kis Pk共 兲⫽ ␳ k 4 ␲ 冕 d ␻ 2 ␮ kE2 kg ␣ k共 , ␻ 2兲 ⫽ ␳ k ␮ k 2 具 E2 k/ ␮ k 典 共 兲g ␣ k共 兲⫽ ␳ k ␮ k 2h ˆk共 , H兲,共60兲 whe e we ha e in oduced h ˆk( , H). The a gumen Hin his unc ion indica es ha h ˆk( , H) gene ally depends on he mola composi ion o he mix u e. The a io o he pa ial pola iza ions will hus be gi en by PH共 兲 PL共 兲 ⫽ ␳ H ␮ H 2h ˆH共 , H兲 ␳ L ␮ L 2h ˆL共 , H兲 .共61兲 By w i ing he pa ial pola iza ion densi ies in he o m o Eq. 共60兲, we ha e con enien ly sepa a ed he -dependen con ibu ions o Pk( ) ha desc ibe he angula co ela ions 具 E2 k/ ␮ k 典 ( ), on one hand, and he adial dis ibu ion unc- ions g ␣ k( ) on he o he . In he p e ious sec ions, we ha e shown ha o he wo heo ies discussed in his pape , he a io PH( )/PL( )isa cons an independen o :PH( )/PL( )⫽ ␳ h ␮ H 2/ ␳ L ␮ L 2. This a ises because in he amewo k o he wo heo e ical ea - men s men ioned in his pape , h ˆ( , H) has he same unc- ional o m o bo h componen s o he mix u e a a ixed mola ac ion, so ha h ˆH( , H)⫽h ˆL( , H)⫽h ˆ( , H). No e ha he unc ion h ˆ( , H) ha is implici in he wo k o Chand a and Bagchi di e s om h ˆ( , H) in he MSA. The o al pola iza ion in e ms o h ˆ( , H) is hen P共 兲⫽PL共 兲⫹PH共 兲⫽ ␳ d共 ␮ L 2⫹ H共 ␮ H 2⫺ ␮ L 2兲兲 共 , H兲. 共62兲 I h ˆ( , H) we e independen o H, Eq. 共62兲would yield a linea 共ideal兲beha io o he sol a ion ene gy wi h he mola ac ion. Thus, he nonideali y obse ed in Fig. 2 is a conse- quence o he dependence o h ˆ( , H) on he mola ac ion. One should no e ha , e en when using he Bo n o mula o he sol a ion ene gy, wi h a dielec ic cons an o he mix u e e alua ed ollowing he p esc ip ion o Adelman and Deu ch,25 a nonideal beha io is ob ained. This is o be ex- pec ed, as he dielec ic cons an is no a linea unc ion o H. This limi ed desc ip ion is no capable o desc ibing he in ui i e pic u e o p e e en ial sol a ion: when he ion is p e e en ially sol a ed by he mo e pola species, one usually has in mind ha he i s sol a ion shell is composed, o a FIG. 3. The pa ial pola iza ion densi ies Pk(R)/Pmac(R) a ound he ion o he same sys em pa ame e s as in Fig. 1. The educed adius Ris de ined as R⫽ /R1⫽ /R2. The composi ion o he mix u e is adjus ed by he ela ion ␳ L ␮ L 2/ ␳ H ␮ H 2⫽1, i.e., H⫽0.2. The simula ion esul s PH(R) and PL(R) a e ep esen ed by a solid line and a do ed line, espec i ely. Fo ␳ L ␮ L 2/ ␳ H ␮ H 2⫽1, bo h heo ies gi e PL(R)⫽PH(R). The MSA esul is plo ed as a dashed line, while a do ed-dashed line is used o he Chand a and Bagchi app oach. 2366 J. Chem. Phys., Vol. 113, No. 6, 8 Augus 2000 Mo illo e al. la ge ex en , o molecules o he mo e pola species. This has as a consequence ha g ␣ H( ) and g ␣ L( ) a e di e en , and he a io h ˆH( , H)/h ˆL( , H) is no longe uni y and will gen- e ally depend on . The simula ion esul s o he pa ial pola iza ions shown in Fig. 3 show jus his beha io . The a io h ˆH( , H)/h ˆL( , H) di e s om uni y in he i s wo sol a ion shells; o dis ances close o con ac we ob ain al- ues as high as h ˆH( ,0.2)/h ˆL( ,0.2)⬇25. I is ins uc i e o analyze he a ios 具 E2 H/ ␮ H 典 ⫻( )/ 具 E2 L/ ␮ L 典 ( ) and g ␣ H( )/g ␣ L( ) sepa a ely. In Fig. 4 we ha e plo ed hese a ios om he simula ion da a o a mola ac ion o H⫽0.2. I can clea ly be seen ha he main con ibu ion o he de ia ion o h ˆH( , H)/h ˆL( , H) om uni y s ems om he adial con ibu ion, i.e., om a spa ial eo ganiza ion o he i s wo sol a ion shells. The a io g ␣ H( )/g ␣ L( ) eaches alues o g ␣ H( )/g ␣ L( )⬇45 o dis ances close o con ac . This co esponds o he in ui- i e pic u e o p e e en ial sol a ion ha we ha e gi en abo e. The angula con ibu ion o he mo e pola species, 具 E2 H/ ␮ H 典 ( ), when de ined in his way, ac ually is smalle han i s coun e pa co esponding o he less pola species. Howe e , he angula co ela ion 具 E2 H 典 ( )⫽ 具 cos( ␮ ˆH• ˆ) 典 is la ge han 具 E2 L 典 ( ), as expec ed. The p e e en ial sol a ion pa ame e as de ined by Ma cus21,22 is ␦ H⫽ H loc⫺ H,共63兲 whe e H loc is he local mola ac ion o species Hin a sphe e o adius Rloc a ound he ion. No e ha his pa ame e is ze o bo h o he mac oscopic 共Bo n兲and he MSA desc ip ion as used in his wo k. In Fig. 5 we ha e ep esen ed his excess mola ac ion o Rloc⫽4.0Å, co esponding app oxima ely o he i s minimum o g ␣ H( ). The simula ion da a show a qui e la ge de ia ion om ze o o a wide ange o mola ac ions. The maximum alue o ␦ H⬇0.6 is eached o H⬇0.12, in ag eemen wi h he s eep inc ease o he sol a- ion ee ene gy o small mola ac ions, as has been no ed abo e. The simula ions indica e ha a be e heo e ical desc ip- ion o he sol a ion o ha d sphe e ions in bina y mix u es o ha d sphe e dipoles mus ake in o accoun he dependence o h ˆH( , H)/h ˆL( , H). Namely, an ion–dipole pai dis ibu- ion unc ion wi h he o m o Eq. 共58兲, which con ains sepa- a e adial and angula co ela ion unc ions o each species, should be used in o de o adequa ely desc ibe he phenom- enon o p e e en ial sol a ion. We will add ess his ques ion in u u e wo k. The app oach used he e 关see Eq. 共13兲兴,as well as he heo y o Chand a and Bagchi, do no ul ill his equi emen . Ne e heless, hey ep oduce he quali a i e be- ha io o he sol a ion ene gy wi h espec o he mola ac- ion and ep esen an imp o emen o e he mac oscopic de- sc ip ion. ACKNOWLEDGMENTS Suppo by he Di eccio ´n Gene al de Ensen ˜ anza Supe- io o Spain 共P ojec Nos. PB98-0423 and PB98-1120兲and he Conseje ia de Educa ion y Ciencia o he Jun a de An- dalucı ´a is g a e ully acknowledged. APPENDIX: MSA RESULTS FOR THE HOMOGENEOUS MIXTURE In he pape o Adelman and Deu ch,25 he di ec co e- la ion unc ion o a homogeneous dipole mix u e o equally sized molecules o diame e ␴ is gi en in e ms o he MSA solu ion o he e ec i e luid cij m共110,k兲⫽ ␮ ˜ ⫺2 ␮ i ␮ jcMSA m共110,k, ␳ ˜ , ␮ ˜ 兲,共A1兲 cMSA m共110,k兲⫽2共4 ␲ 兲2 ␬ 冕 0 ␴ d 2j0共k 兲cPY共 ,2 ␬ ␳ ˜ 兲, 共A2兲 cPY共 兲⫽c0⫹c1 冉 ␴ 冊 ⫹c3 冉 ␴ 冊 3 .共A3兲 He e, cPY is he Pe kus–Ye ick di ec co ela ion unc ion a densi y 2 ␬ ␳ ˜ , and ␬ is ob ained om he ela ion Q共2 ␬ ␩ ˜ 兲⫺Q共⫺ ␬ ␩ ˜ 兲⫽3y ˜ ⫽3共4 ␲ /9兲 ␤ ␳ ˜ ␮ ˜ 2,共A4兲 wi h Q共x兲⫽共1⫹2x兲2 共1⫺x兲4.共A5兲 FIG. 4. The a ios 具 E2 H/ ␮ H 典 (R)/ 具 E2 L/ ␮ L 典 (R) and g ␣ H(R)/g ␣ L(R) as ob- ained om he simula ion. The pa ame e s a e he same as in Fig. 3. FIG. 5. The p e e en ial sol a ion pa ame e as de ined in Eq. 共63兲. The local mola ac ion has been de e mined om he adial dis ibu ion unc- ions inside a sphe e wi h adius Rloc⫽4.0 Å a ound he solu e. The ci cles ep esen he simula ion esul s; he solid line is a spline in e pola ion o hese poin s. 2367J. Chem. Phys., Vol. 113, No. 6, 8 Augus 2000 Sol a ion in dipola bina y mix u es