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
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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