E ec o dipola in e ac ions on he phase beha io o he Gay–Be ne
liquid c ys al model
Mohammed Houssa and Luis F. Rull
Depa amen o de Fı
´sica A o
´mica, Molecula y Nuclea , A ea de Fı
´sica Teo
´ ica, Uni e sidad de Se illa,
Ap do 1065, Se illa 41080, Spain
Simon C. McG o he
School o Chemical Enginee ing, No h Ca olina S a e Uni e si y, Raleigh, No h Ca olina 27695-7905
~Recei ed 29 May 1998; accep ed 21 Augus 1998!
A compu e simula ion s udy o he phase beha io o he dipola Gay–Be ne liquid c ys al model
is p esen ed. The phase ansi ions a e de e mined wi h iso he mal–isoba ic ~NPT!Mon e Ca lo
simula ions, u ilizing he eac ion ield me hod. The elec os a ic o ces a e ound o ha e a
conside able e ec on he na u e o he obse ed phases, bu he densi y a which he iso opic luid
becomes uns able wi h espec o pa ially o de ed phases is seen o be ema kably insensi i e o he
s eng h o he dipole. We pay pa icula a en ion o he s uc u e o he mesophases, combining
in o ma ion om se e al single and pai dis ibu ion unc ions o build up an accu a e pic u e o he
molecula a angemen o he sys ems. © 1998 Ame ican Ins i u e o Physics.
@S0021-9606~98!51544-9#
I. INTRODUCTION
I is becoming inc easingly e iden ha he liquid c ys-
alline phase beha io o complex molecules is dic a ed,
p incipally, by he nonsphe ici y o he in lexible egions o
he cons i uen molecules. Since he 1940’s, i has been ec-
ognized ha he simples liquid c ys al ~ he nema ic!could
esul om excluded olume e ec s only. In mo e ecen
yea s, o he , mo e complex, liquid c ys alline phases ha e
been epo ed o model molecules in e ac ing only ia an-
iso opic epulsi e po en ials. Bo h smec ic1and columna 2
phases ha e been obse ed o hese simple models. E en
he spon aneous pola i y o he il ed chi al smec ic-C*phase
has been a ionalized pu ely in e ms o packing e ec s o
he epulsi e co es.3
O cou se, i is only in hese idealized heo e ical models
ha he a ac i e and epulsi e elemen s o he in e molecu-
la in e ac ion can be conside ed in isola ion. In any eal
sys em, bo h o ces will be p esen . The e a e, howe e , col-
loidal sys ems ~e.g., obacco mosaic i us!, in which he non-
sphe ical molecules ha e a negligible a ac ion o each
o he and he phase ansi ions a e hough o be almos en-
i ely d i en by excluded olume e ec s. T ue o heo e ical
p edic ion, such sys ems do indeed exhibi a numbe o me-
sophases a app op ia e concen a ions and empe a u es. Bu
hese sys ems a e excep ional; in all he mo opic mesogens,
a ac i e o ces a e no only p esen , bu equen ly com-
p ise complica ed componen s such as mul ipola o ces and
p
-elec on in e ac ions, in addi ion o he usual London dis-
pe sion e ec s. I is impo an , he e o e, ha he ole o he
a ac i e o ces in s abilizing he mesophases is in es i-
ga ed.
F enkel and co-wo ke s ha e shown ha ha d p ola e
and obla e ellipsoids o e olu ion4and ha d sphe o-
cylinde s5ha e a ich phase diag am wi h nema ic, and in
some cases smec ic and columna phases, depending on he
molecula size and shape. These indings ein o ce he ideas
o Onsage ,6,7 ha he aniso opic shape o he molecules is
he p incipal d i ing o ce behind mesophase o ma ion. The
a ac i e o ces ~ an de Waals, mul ipola , dispe sion, e c.!
a e o seconda y impo ance. Despi e hei lesse ole, an
in es iga ion o he in luence o such a ac i e o ces is nec-
essa y i a ull unde s anding o liquid c ys alline phase be-
ha io is o be achie ed. P obably he simples idealized liq-
uid c ys al ~LC!model which inco po a es bo h epulsi e
and a ac i e e ms, and consequen ly he mos widely s ud-
ied, is he Gay–Be ne ~GB!po en ial. Depending upon he
choice o pa ame e s his model may be ei he obla e ~disc-
like!o p ola e ~ odlike!. The majo i y o eal mesogens a e
od shaped, hus we ocus in his s udy on he p ola e Gay–
Be ne model. Bo h he shape- and ene ge ic aniso opy a e
adjus able. A minimum elonga ion is equi ed be o e o ien-
a ionally o de ed luid phases become he modynamically
s able. Simila ly, he e is a minimum ene ge ic-aniso opy,
below which no spa ially inhomogeneous luid phases a e
obse ed. These wo minimum alues a e pa ially coupled.
Ob iously he e is a as pa ame e space o explo e; i is
necessa y, he e o e, o limi ou in es iga ion o a single
ene ge ic aniso opy
k
855~
k
8is he a io o he po en ial
well dep hs o side-by-side and end- o-end con igu a ions!,a
alue which yields a ich phase diag am o elonga ions o
in e es .
S udies o he Gay–Be ne po en ial wi h
k
53~
k
is he
molecula elonga ion, and is de ined as he a io o he wo
p incipal axes o he ellipsoidal co e!and
k
855 ha e p o en
ha he iso opic ~I!, nema ic ~N!, smec ic-B ~SmB!, and
c ys alline phases can be o med along di e en iso he ms.8,9
Upon inc easing he elonga ion o he GB pa icle ~i.e.,
k
.3!, he smec ic-A ~SmA!phase appea s be ween he nem-
a ic and smec ic-B phases, o ce ain empe a u es. Fo ha d
ellipsoids wi h an elonga ion a/b53~wi h aand b he
JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER 21 1 DECEMBER 1998
95290021-9606/98/109(21)/9529/14/$15.00 © 1998 Ame ican Ins i u e o Physics
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leng hs o he wo p incipal molecula axes!, he iso opic
and c ys alline phases a e sepa a ed only by a nema ic e-
gion. Thus we can con iden ly s a e ha he exis ence o he
smec ic phases is di ec ly ela ed o he a ac i e o ces in
he GB model ~ he so ness o he epulsion is unlikely o
cause a quali a i e di e ence in phase beha io !.
S udies o dipola liquid c ys al models ha e also been
epo ed in ecen yea s. Weis, Le esque, and Za agoicho-
echea, in a se ies o pape s,10–13 s udied how he p esence o
poin dipoles in luenced he s uc u e o he LC phases o a
a ie y o ha d-co e models. Selec ed s a e poin s, ep esen-
a i e o a ious mesophases, we e simula ed wi h Mon e
Ca lo, employing he compu a ionally expensi e Ewald sum-
ma ion me hod, p incipally in he canonical ensemble. Two
majo conclusions we e d awn: cen al dipoles p omo e lay-
e ed s uc u es and e minal dipoles ha e li le e ec on he
s uc u e o he phase. Li le could be concluded as o he
s abili y o he a ious phases, since only a e y limi ed
numbe o s a e poin s we e in es iga ed. McG o he e al.
cons uc ed ull iso he ms o se e al sphe ocylinde plus di-
pole models. Fo cen al dipoles,14 sho - anged o de in he
iso opic phase made he I–LC phase ansi ion di icul o
pinpoin ; indeed, o su icien ly s ong dipoles, comp ession
led only o me as able glassy phases and no ue LC phases
we e obse ed. A much g ea e deg ee o hys e esis is no ed
a he I–N ansi ion han in he ha d sphe ocylinde sys em.
The p esence o he cen al dipole ~bo h longi udinal14 and
ans e se15!unambiguously s abilizes he smec ic-A phase.
This laye ed phase is seen o be s able a p essu es and den-
si ies co esponding o less o de ed phases in he nonpola
sys em. When he dipole esides in he e minal posi ion,16
di e en beha io is no ed. Again hys e esis is seen a he
I–N ansi ion, bu now he smec ic-A phase is des abilized,
and he N–SmA ansi ion is pos poned o highe densi y and
p essu e as a esul o he elec os a ic in e ac ion. In all
cases no e oelec ic phases we e no ed.
Fo he Gay–Be ne model, some nume ical esul s o
dipola sys ems ha e been o e ed. P incipally Sa oh
e al.17,18 ha e used he eac ion ield me hod o c ea e se -
e al isocho es o such sys ems. Fo GB molecules ~
k
53
and
k
855!wi h cen al longi udinal dipoles,17 he I–N an-
si ion is again seen o be insensi i e o he dipole s eng h.
On he o he hand, he N–SmA ansi ion empe a u e is seen
o inc ease wi h he dipole momen , i.e., he smec ic-A phase
is s abilized. The empe a u e a which c ys alliza ion occu s
does no a y much wi h he s eng h o he mul ipole. In
e ms o s uc u e, he laye ing in he smec ic-A phase is
obse ed o be much sha pe in he dipola sys em, a low
empe a u es; nea he N–SmA ansi ion, he e ec is less
p onounced. When he longi udinal dipole is placed in he
e minal posi ion,18 he I–N ansi ion empe a u e TIN in-
c eases wi h inc easing dipole momen
m
*5(
m
2/«0
s
0
3)1/2
whe e
s
0is he con ac dis ance, and «0is he ene gy ~ig-
no ing he dipole!o a pai o GB molecules in he side-by-
side a angemen . TNS exhibi s simila
m
dependence. The
s uc u e o he esul an smec ic phase is seen o be bilay-
e ed, wi h a no able deg ee o in e digi a ion.
In e es ing phases we e ound by Be a di e al.19 when
he longi udinal dipole is in ei he he cen al o e minal
posi ion. This s udy ocused p ima ily on he smec ic phases,
and as such la ge sys em sizes we e employed ~N51000 and
8000!. Th ee empe a u es we e selec ed a a ixed densi y
*5
s
0
350.3, co esponding, in he absence o dipola in-
e ac ions, o iso opic, nema ic, and smec ic-A phases, e-
spec i ely. The addi ion o poin dipoles wi h
m
*52 is seen
o ha e no in luence on he o e all na u e o he obse ed
phase, a leas a he h ee chosen empe a u es. In bo h LC
phases, he p esence o he dipole in ei he posi ion is seen o
inc ease he o ien a ional o de sligh ly. Fo cen al dipoles,
he smec ic has he usual monolaye s uc u e, bu a cha ac-
e is ic spli o he second peak in he adial pai dis ibu ion
unc ion leads Be a di e al. o desc ibe his phase as
smec ic-B. Shi ing he dipole owa d one end o he mol-
ecule elimina es his hexagonal o de wi hin he plane, bu
se e al new ea u es a e no ed. The p ojec ion o he pai
dis ibu ion unc ion shows a spli ing in he i s peak, which
is indica i e o a bilaye s uc u e. The i s peak is a a
dis ance which sugges s signi ican in e digi a ion. Pa icles
in each laye a e locally e oelec ic ~i.e., he dipoles all
poin in he same di ec ion!, and he di ec ion o pola iza ion
al e na es be ween successi e laye s. These indings sugges
ha he phase is a bilaye ed an i e oelec ic smec ic phase.
Howe e , simula ion o huge sys ems (N58000), and isu-
aliza ion o he phase, shows ha he si ua ion is somewha
mo e complex. In each o he smec ic laye s, he e is a no-
iceable disloca ion, a which he cen e o he laye shi s by
a ound a molecula leng h ~due o he o oidal bounda ies,
he e is second disloca ion in each laye , which cancels he
e ec o he i s !. E en s a ing in a pe ec smec ic-A
phase, he sys em was seen o elax in o he eloquen ly
named modula ed an i e oelec ic bilaye s ipe domains.
Gwo
´z
´dz
´e al.20 used molecula dynamics ~MD! o s udy
he in luence o ans e se cen al dipoles on he phase dia-
g am. These au ho s ake a sligh ly di e en app oach: a he
han cooling along an isocho e, hey comp ess along an iso-
he m, o only one alue o he dipole momen (
m
*
50.5). They ind ha he I–SmA ansi ion occu s a he
same densi y wi h o wi hou he dipola o ces p esen . The
only dis inc ions be ween he pola and nonpola case a e he
highe deg ee o p e ansi ional o de in he dipola sys em,
and be e de ined posi ional o de in he smec ic phase. The
au ho s epo smec ic-C o de in he dipola case, bu he il
angle is small and he il disappea s as he densi y is in-
c eased s ill u he .
Mos ecen ly Houssa e al.9s udied a GB model deco-
a ed wi h a cen al longi udinal dipole. These au ho s epo
he comple e supp ession o he nema ic phase o su i-
cien ly s ong dipoles. The phase sequence o he nonpola
sys em is I–N–SmB, bu wi h he inclusion o a dipole o
educed momen
m
*52.5, he iso opic phase spon aneously
o de s di ec ly o he smec ic-B phase upon comp ession.
Thus om ea lie s udies o dipola LC models we can
conclude ha he dipole causes only a mild pe u ba ion o
he ini ial I–LC ansi ion, which appea s o be almos en-
i ely dependen upon he sho – ange epulsi e o ces ac ing
be ween aniso opic molecules. Depending on he posi ion
and o ien a ion o he dipole in he molecula ame, he
nema ic o smec ic phases can be p omo ed a he expense o
9530 J. Chem. Phys., Vol. 109, No. 21, 1 Decembe 1998 Houssa, Rull, and McG o he
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he o he . The mos e iden ea u e o all p e ious s udies is
ha he ine de ail o he smec ic s uc u e is sensi i e o he
dipole. In e digi a ion, bilaye s, s iped domains, and e en
il ed phases a e epo ed o a ious sys ems.
In he nex sec ion we desc ibe he po en ial model, and
in Sec. III we gi e de ails o he simula ion me hodologies
ha ha e been employed. The esul s o he simula ion s ud-
ies a e p esen ed and discussed in Sec. IV, and we d aw ou
conclusions in he inal sec ion.
II. POTENTIAL MODEL
In ou simula ions, he in e ac ion ene gy o wo mol-
ecules iand jis gi en by
Uij5UGB~
ˆij,u
ˆi,u
ˆj!1U
mm
~
ˆij,u
ˆi,u
ˆj!,~1!
whe e UGB(
ˆij,u
ˆi,u
ˆj) is he GB po en ial21
UGB~
ˆij,u
ˆi,u
ˆj!54«~
ˆij,u
ˆi,u
ˆj!
3
F
S
s
0
ij2
s
~
ˆij,u
ˆi,u
ˆj!1
s
0
D
12
2
S
s
0
ij2
s
~
ˆij,u
ˆi,u
ˆj!1
s
0
D
6
G
,~2!
whe e u
ˆiis he axial ec o o molecule iand ij is he
dis ance be ween he cen e s o mass o molecules iand j,
ˆij5 ij/ ij is a uni ec o along he in e molecula ec o
ij5
u
i2 j
u
, whe e iand ja e he posi ions o he cen e s
o mass o molecules i and j, espec i ely. He e
s
(
ˆij,u
ˆi,u
ˆj)
and «(
ˆij,u
ˆi,u
ˆj) a e he ange and s eng h pa ame e s, e-
spec i ely ~see Re . 21 o explici exp essions!,
s
and «
depend on he aniso opy pa ame e s
k
~molecula elonga-
ion!and
k
8~ene ge ic aniso opy!.
s
0and «0~ he ange and
ene gy alues in he side-by-side a angemen !a e used o
de ine he p essu e and empe a u e scales in ou simula ions.
Thus P*5P
s
0
3/«0and T*5kT/«0. The dipole–dipole in-
e ac ion is gi en by
Udd~
ˆij,u
ˆi,u
ˆj!5@
m
i
m
j23~
m
i•
ˆij!~
m
j•
ˆij!#
ij
3,~3!
whe e
m
i5
m
u
ˆiand
m
j5
m
u
ˆjdeno e he dipole momen s o
molecules iand j, espec i ely. The dipoles a e longi udinal,
i.e., pa allel o he uni ec o s which ep esen he p incipal
molecula axes u
ˆio u
ˆj.
While he GB po en ial UGB(
ˆij,u
ˆi,u
ˆj) can be unca ed
and shi ed a some dis ance, c, less han L/2, Lbeing he
leng h o he simula ion box, he same is no ue o he
dipola in e ac ion. The long- anged na u e o he dipola
po en ial is aken in o accoun wi h he eac ion ield me hod.
This scheme22 assumes ha pa icles beyond a cu o dis-
ance, c, ac as a dielec ic con inuum o dielec ic cons an
«RF .23 The in e ac ion ene gy o a pai o dipoles can hen
be w i en
U
mm
~
ˆij,u
ˆi,u
ˆj!5Udd~
ˆij,u
ˆi,u
ˆj!22~«RF21!
2«RF11
m
i•
m
j
c
3,
~4!
o ij, cand U
mm
(
ˆij,u
ˆi,u
ˆj)50 o ij. c.
III. SIMULATION DETAILS
The simula ions we e pe o med using Mon e Ca lo in
he iso he mal–isoba ic ensemble ~i.e., cons an numbe o
molecules N, p essu e P, and empe a u e T!wi h he eac-
ion ield me hod. Each Mon e Ca lo cycle consis s o N ial
displacemen s and eo ien a ions o he GB molecules and
app oxima ely one ial olume change. The maximum dis-
placemen , o a ion, and olume change a e al e ed o ensu e
ha a ound 40% o each ype o mo e is accep ed; his alue
should lead o he mos e icien sampling o phase space.
The eac ion ield is a simple bu accu a e me hod o ac-
coun ing, in an a e age way, o he long ange o he dipola
in e ac ion. The echnique is much as e han he Ewald
summa ion me hod, bu yields esul s which a e essen ially
indis inguishable. Fo compa ison o hese wo me hods see
Re s. 9, 24, and 25.
In o de o analyze he o ien a ional o de and he pos-
sible pola iza ion o he mesophases, we ha e calcula ed wo
o de pa ame e s Sand P1; he o me ~ he nema ic o de
pa ame e !is ob ained as he la ges posi i e eigen alue o
he Q enso ,26 he elemen s o which a e de ined as
Q
ab
51
N(
i51
N
1
2~3u
a
iu
b
i2
d
ab
!~5!
~uk
iis he k h componen o he axial o ien a ion u
ˆio he i h
molecule!, wi h he nema ic di ec o nbeing he co espond-
ing eigen ec o . The pola i y P1is ob ained as
P151
N
U
(
i51
N
u
ˆi•n
U
.~6!
The nema ic o de pa ame e , S, is ze o in he iso opic
phase, and has a alue o one in a pe ec ly aligned phase; i
p o ides no in o ma ion as o posi ional o de wi hin he
sys em. In eal sys ems, Sa ains alues a ound 0.4 a he
I–N ansi ion, whe eas in simula ion, highe alues a e e-
quen ly encoun e ed due o ini e size e ec s. The pola i y
P1is also ze o in he iso opic phase, can only a ain i s
maximum alue ~one!in a pe ec ly aligned s a e, and simi-
la ly gi es no in o ma ion on he posi ional o de wi hin he
sys em. The unc ions a e dis inc because P1lacks ‘‘head–
ail’’ symme y, and is only nonze o when mo e o he di-
poles ‘‘poin ’’ in one di ec ion han ano he . As such, P1
measu es he spon aneous pola iza ion o he phase.
We calcula e a ious pai dis ibu ion unc ions, e.g., he
i s - ank o ien a ional co ela ion unc ion de ined by
g1~ !5
^
P1~u
ˆ1,u
ˆ2!
&
5
^
cos
u
12
&
,~7!
and he second- ank o ien a ional co ela ion unc ion,
g2~ !5
^
P2~u
ˆ1,u
ˆ2!
&
5
K
3
2cos2
u
1221
2
L
,~8!
whe e
u
12 is he angle be ween he p incipal molecula axes
o molecules 1 and 2. In addi ion, we also de e mine he
o ien a ionally a e aged pai co ela ion unc ions o wo
molecules whose cen e s lie on a line pa allel, g
i
(
i
), o on a
line pe pendicula , g'( '), o he di ec o . He e he dis-
ances
i
and 'a e measu ed pa allel and pe pendicula o
he di ec o , espec i ely.
9531J. Chem. Phys., Vol. 109, No. 21, 1 Decembe 1998 Houssa, Rull, and McG o he
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These unc ions alone canno unequi ocally de e mine
he na u e o ce ain smec ic phases. To e i y he s uc u e
o hese liquid c ys als, we calcula e he bond-o ien a ional
o de wi hin he smec ic laye s. This unc ion is de ined as
B51
6N
K
(
j51
6
exp~6i
ij!
L
,~9!
whe e
ij is he angle be ween he bond linking pa icles i
and jand a ixed e e ence axis. The sum is o e only
nea es -neighbo bonds. We de ine nea es -neighbo bonds o
be hose wi hin b'1.2
s
0. This o de pa ame e akes al-
ues close o ze o when no in-plane bond o de exis s and
close o one in he p esence o pe ec hexa ic in-plane bond
o de . P incipally his pa ame e is used o dis inguish he
smec ic-A phase om he mo e o de ed smec ic-B s uc u e.
Fo GB molecules wi h
k
53, we ha e simula ed N
5256 molecules wi h longi udinal poin dipoles loca ed a
he cen e o he molecules, wi h dipole momen
m
*. In o -
de o minimize he e ec o he simula ion cell on he ob-
se ed phases, a cubic simula ion box was employed wi h
pe iodic bounda y condi ions. Fo mo e elonga ed pa icles
(
k
54) we a e o ced o examine la ge sys ems (N5500),
in a cuboidal cell. Dipole momen s anging om
m
*50.5 o
2.5 ~in s eps o 0.5!we e conside ed. Fo he smalle sys em,
a each alue o
m
*, simula ions we e ini ia ed om a ace-
cen e ed-cubic ~ cc!la ice wi h i e laye s pe pendicula o
he z-axis, which is expanded o a low densi y
*5
s
0
3
50.08 (P*50.5) and quickly loses posi ional and o ien a-
ional o de . The sys em was hen slowly comp essed in e-
duced p essu e s eps o 0.5 o less. Fo he la ge sys em,
s a es in he iso opic, nema ic, and smec ic-A phase we e
gene a ed wi h nonpola GB sys ems. The dipole is in o-
duced and he sys em allowed o equilib a e once mo e.
Typically, be ween 3 and 53105cycles we e pe o med o
each s a e poin , inc easing o 106cycles in he icini y o
phase ansi ions.
By mode n s anda ds hese sys em sizes a e mode a e;
howe e , we pe o med se e al simula ions wi h la ge sys-
em sizes and no ed no sys ema ic di e ence in he esul s.
Fu he mo e, simula ing small sys ems allows us o explo e
phase space much mo e ho oughly. A he liquid c ys alline
phase ansi ions, la ge molecula eo ganiza ions mus occu
and leng hy uns a e c ucial. We a e con iden ha sys em
sizes a e su icien o de e mine he na u e o he phases and
FIG. 1. Phase diag am a T*51.25, o a sys em o 256 Gay–Be ne pa icles ~
k
53,
k
855!wi h embedded cen al, longi udinal poin dipoles. The educed
p essu e P*5P
s
0
3/«0is plo ed as a unc ion o he educed numbe densi y
*5
s
0
3. Each plo depic s a di e en educed dipole momen
m
*
5(
m
2/«0
s
0
3)1/2 ~a!
m
*50.5, ~b!
m
*51.0, ~c!
m
*51.5, and ~d!
m
*52.0. E o ba s deno e one s anda d e o in he densi y. Ho izon al lines deno e he
posi ion o he phase ansi ions.
9532 J. Chem. Phys., Vol. 109, No. 21, 1 Decembe 1998 Houssa, Rull, and McG o he
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accu a ely p edic he he modynamic p ope ies o he sys-
em. This belie is gi en c edence by he excellen ag eemen
no ed be ween ou simula ions and hose o B own e al.28
In his pape we p esen he esul s o a se ies o NPT
Mon e Ca lo ~MC!simula ions on a sys em o molecules
in e ac ing ia he Gay–Be ne dipola po en ial wi h
k
53o
4 and
k
855.
IV. SIMULATIONS RESULTS
A.
k
53
1. T
*51.25
A his empe a u e we ha e cons uc ed iso he ms o
a ious dipola s eng hs. Fo each alue o
m
*, he lowes
p essu e simula ed co esponds o an iso opic luid, and in-
c easing he p essu e e en ually leads o a leas one phase
ansi ion, indica ed by a small discon inui y in he densi y
and a ma ked inc ease in o ien a ional o de .
Focusing ou a en ion on weake dipole momen s
m
*<2.0, he nume ical phase diag ams a e p esen ed in Fig.
1, wi h he p essu e dependence o he o ien a ional o de
pa ame e Sand pola i y P1gi en in Fig. 2. F om hese
igu es we can pe cei e wo dis inc ansi ions. The i s
ansi ion is om S alues a ound ze o ~iso opic phase!up
o nonze o alues ypical o he nema ic phase. Tha he e-
sul an phase is nema ic, can be con i med by inspec ion o
he p ojec ions o he pai dis ibu ion unc ions pa allel
g
i
(
i
) and pe pendicula g'( ') o he di ec o ~Figs. 3 and
4!. These igu es show ha he s uc u e is liquidlike and ha
he e is no signi ican laye ing a he lowe o he wo indi-
ca ed p essu es. The second ansi ion is om mode a e al-
ues o S, up o sa u a ion S'0.94. This high densi y phase is
bes cha ac e ized by g
i
(
i
)~Fig. 3!: he sinusoidal a ia ion
o his unc ion indica es ha he e is one-dimensional lay-
e ing o pa icles, which is he adema k o he smec ic liq-
uid c ys als. The wa eleng h o his pe iodic unc ion is in-
dica i e o he laye spacing o he smec ic s a a and he
ampli ude is ela ed o how well de ined he laye s a e. I is
clea ha he s eng h o he dipole does no signi ican ly
a ec ei he he wa eleng h o he ampli ude o his unc ion
a he nema ic–smec ic ansi ion. This esul is qui e su p is-
ing; p e ious simula ion s udies ha e indica ed he signi i-
can impac o he dipole upon he de ails o he s uc u e,
pa icula ly o he o de ed phases. The s iking simila i y o
hese igu es shows ha he s uc u e o he smec ic liquid
c ys al is dependen p incipally on he GB in e ac ions. The
laye spacing ~see Fig. 3!is sligh ly less han one molecula
leng h, which is a common ea u e o such phases when he
cons i uen molecules ha e an ellipsoidal co e: he ape ing
o he molecule pe mi s a deg ee o in e digi a ion and his
FIG. 2. Va ia ion o he nema ic o de pa ame e S~ iangles!and pola i y P1~ci cles!, as a unc ion o densi y o he dipola Gay–Be ne luid. The ou
igu es co espond o he same dipole momen s as Fig. 1. E o ba s deno e one s anda d e o in he o de pa ame e s.
9533J. Chem. Phys., Vol. 109, No. 21, 1 Decembe 1998 Houssa, Rull, and McG o he
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s uc u e has he mos e icien packing o pa icles. F om
Fig. 4 he N–Sm ansi ion is also accompanied by an in-
c ease in he pe pendicula o de . Signi ican ly, a peak de-
elops a less han one diame e , co esponding o nea es
neighbo s in he nex smec ic s a um. The long- anged na-
u e o his unc ion in he smec ic phase indica es ha he
phase is mo e highly s uc u ed han a simple smec ic-A
phase. In Fig. 5 we show a snapsho o he
m
*51.5 sys em
a a educed p essu e P*59.0, in his smec ic phase. The
smec ic laye s a e clea om Fig. 5~a!, and he posi ional
o de wi hin hose laye s is e iden om Fig. 5~b!. The
N–Sm ansi ion is accompanied by a dis inc inc ease in he
bond-o ien a ional o de . Thus we conclude ha he phase is
smec ic-B. In e es ingly, as he s eng h o he dipole is in-
c eased, he numbe o peaks pe pendicula o he di ec o is
seen o diminish. The o ien a ionally a e aged pai dis ibu-
ion unc ion, Fig. 6, exhibi s he expec ed oscilla o y beha -
io and high alue a long dis ances, o he laye ed smec ic
phases. F om Fig. 6 we see ha he e is signi ican sho -
ange o ien a ional o de in he iso opic phase nea o he
I–N ansi ion. This o de is seen o inc ease sligh ly wi h
inc easing dipole momen . A e he I–N ansi ion hese o i-
en a ional co ela ions no longe ade o ze o wi hin he
simula ion cell. The o ien a ional co ela ion a con ac is an
inc easing unc ion o
m
. In Fig. 7 we display he i s - ank
o ien a ional co ela ion unc ion o s a es in he iso opic
nema ic and smec ic-B phases o hese ou dipole mo-
men s. Nega i e alues indica e ha pa icles a e an ipa allel.
The igu es a e quali a i ely equi alen ; each shows ha :
nea es neighbo s a e an ipa allel; nex nea es neighbo s a e
o ien ed pa allel o he selec ed pa icle; he minima and
maxima inc ease in magni ude as one mo es om iso opic
o nema ic o smec ic phases; he o de dies ou wi hin he
simula ion cell, indica ing no globally pola phases.
Bo h o he ansi ions ~I–N and N–SmB!a e seen o be
weakly i s o de . Nei he phase change a ec s he pola i y
P1, which emains essen ially ze o, o e he en i e p essu e
ange. O e all, hese sys ems ha e he phase sequence I–N–
SmB on comp essing om low densi y a his empe a u e
(T*51.25). This is ue o nonpola GB pa icles and hose
wi h weak, cen al, longi udinal dipoles.
A na u al ques ion is why he sa u a ion alue o Sis less
han one. A pe ec ly aligned s a e is ne e achie ed in simu-
la ions as a esul o se e al pa icles becoming apped a
igh angles o he di ec o . These pa icles a e pa icula ly
no iceable in he smec ic phases, whe e hey p e e o posi-
ion hemsel es in he in e laye egion. This ac is he
sou ce o he cha ac e is ic minimum in he o ien a ional pai
dis ibu ion unc ion o simula ed smec ic liquid c ys als.
FIG. 3. The p ojec ion o he adial pai dis ibu ion unc ion pa allel o he di ec o g
i
(
i
) o s a es on ei he side o he nema ic–smec ic ansi ion o he
same sys ems as Fig. 1.
9534 J. Chem. Phys., Vol. 109, No. 21, 1 Decembe 1998 Houssa, Rull, and McG o he
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The beha io o hese molecules has been s udied in de ail
o ela ed sys ems.27
In Table I we epo ou es ima es o he I–N and N–
SmB ansi ion p essu es and coexis ing densi ies o he di -
e en dipola s eng hs. F om his able, i is clea ha he
in oduc ion o he longi udinal dipole in o he model has
some impac on he I–N ansi ion p essu e, bu he s eng h
o he dipole has li le e ec upon he densi y o he iso opic
phase a he ansi ion. The densi y o he nema ic phase ha
coexis s wi h he iso opic luid is also unchanging wi h
a ia ions in dipole momen . The I–N ansi ion p essu e
(PIN) is seen o all wi h inc easing dipole momen . The
consis ency o he I–N ansi ion densi ies
IN , which a e, o
wi hin expe imen al e o , iden ical o each o he dipole
s eng hs which lead o an I–N ansi ion (
m
*<2.0), is s ik-
ing. A he laye ing ansi ion ~N–SmB!, we now see ha he
densi ies a coexis ence dec ease as he s eng h o he mul-
ipola in e ac ion is inc eased. The ansi ion p essu e shows
a e y clea end, mo ing o lowe alues as he dipole
momen is inc eased. This is in acco dance wi h expec a ion:
longi udinal dipoles a e well known o p omo e laye ed
s uc u es in bo h nume ical and expe imen al s udies, hus
he s abiliza ion o he smec ic phase is o be expec ed. In-
c easing he dipole makes he smec ic-B phase s able a
lowe p essu es.
Con igu a ions a P*57.0 we e used as s a ing poin s
o dec easing he p essu e. A limi ed numbe o expansion
uns ha e been pe o med o gauge he le el o hys e esis a
he ansi ions. Fo hese weak dipoles (
m
*<2.0), we see no
e idence o hys e esis a he I–N ansi ion; howe e , a he
N–SmB phase change, he ansi ion p essu e is e y di e -
en depending on he di ec ion om which he ansi ion is
app oached. The same is ue o he N–SmB ansi ion den-
si y, bu since he phase diag am is e y s eep in his egion
~i.e., la ge changes in p essu e co espond o only small
changes in densi y!, he e ec is less ob ious.
The na u al consequence o he ends no ed o he
weake dipoles in Table I is ha he nema ic phase will dis-
appea a a iple poin as he s eng h o he dipole momen
is inc eased. This is due o he inc easing s abiliza ion o he
smec ic-B phase by he dipole, leading e en ually o he lay-
e ed phase p eemp ing he nema ic phase when he mul ipole
is su icien ly s ong.
Figu e 8 is he phase diag am o
m
*52.5. A s ongly
i s -o de phase ansi ion, di ec ly om he iso opic luid
o he smec ic-B phase, is obse ed. The na u e o he an-
si ion is made clea by a jump in he bond o ien a ional o de
a he I–LC ansi ion. Alignmen o he molecules is accom-
panied no only by laye ing, bu also he de elopmen o
long- anged bond o de wi hin hese laye s. The bond-o de
pa ame e akes a alue o B50.55 a he ansi ion. Fo
hese highly pola GB molecules, he I–SmB ansi ion oc-
FIG. 4. The p ojec ion o he adial pai dis ibu ion unc ion pe pendicula o he di ec o g'( ') o he same s a es as Fig. 3.
9535J. Chem. Phys., Vol. 109, No. 21, 1 Decembe 1998 Houssa, Rull, and McG o he
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FIG. 5. Snapsho o he dipola Gay–Be ne luid a T*51.25, P*59.0,
m
*51.5, in he smec ic-B phase. The same con igu a ion is shown om di e en
pe spec i es: ~a! om he side o show he smec ic laye ing and ~b!exhibi ing he s ong posi ional co ela ion be ween successi e laye s. The colo s indica e
he di ec ion o he dipole. The size o he pa icles is educed o cla i y.
FIG. 6. Second- ank o ien a ional co ela ion unc ion g2( ) o dipola Gay–Be ne luids a he I–N and N–SmB ansi ions. The ou igu es ep esen
di e en dipole momen s ~as o Fig. 1!.
9536 J. Chem. Phys., Vol. 109, No. 21, 1 Decembe 1998 Houssa, Rull, and McG o he
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cu s a a signi ican ly lowe p essu e han he I–N ansi ion
o he weake dipola sys ems. Once again, hough, we no e
om Table I ha he densi y a which he iso opic phase
ceases o be s able is una ec ed
*'0.305.
The posi ion o he phase ansi ion is in e ed om he
p essu e e sus densi y phase diag am, Fig. 8. E en mo e
compelling e idence o he loca ion o he phase ansi ion
is p o ided by he beha io o he nema ic o de pa ame e S
and pola i y P1as a unc ion o densi y ~Fig. 9!. The a ia-
ion o he o de pa ame e s also p o ides in o ma ion as o
he na u e o he esul an phases. The i s nonze o alue o
Soccu s a
*'0.348, bu now we can see no able p e an-
FIG. 7. Fi s - ank o ien a ional co ela ion unc ion g1( ) o dipola Gay–Be ne luids in he iso opic, nema ic, and smec ic-B phases. The ou igu es
ep esen di e en dipole momen s ~as o Fig. 1!.
FIG. 8. Phase diag am a T*51.25, o a sys em o 256 Gay–Be ne pa icles
~
k
53,
k
855!wi h embedded cen al, longi udinal poin dipoles. The e-
duced p essu e P*5P
s
0
3/«0is plo ed as a unc ion o he educed numbe
densi y
*5
s
0
3. The educed dipole momen is
m
*5(
m
2/«0
s
0
3)1/252.5.
E o ba s ep esen one s anda d e o in he densi y.
TABLE I. T ansi ion p essu es and densi ies o dipola GB luid a a e-
duced empe a u e o T*51.25 as ob ained by MCNPT simula ion. GB
pa ame e s a e:
k
53,
k
855.
m
*is he educed dipole momen , PIN is he
iso opic o nema ic ansi ion p essu e, PNSmB is he same p ope y o he
nema ic o smec ic-B ansi ion,
IN a e he densi ies o he coexis ing iso-
opic and nema ic phases, and
NSmB a e he same alues o he laye ing
ansi ion. No e ha he s onges dipole has a di ec iso opic o smec ic-B
ansi ion. The sys em size is N5256.
m
*PIN
IN PNSmB
NSmB
0.5 4.99 0.304, 0.319 9.49 0.368, 0.380
1.0 5.00 0.305, 0.320 9.00 0.364, 0.379
1.5 5.00 0.307, 0.322 7.00 0.346, 0.361
2.0 4.49 0.305, 0.321 4.99 0.321, 0.339
PISmB
ISmB
2.5 3.70 0.300, 0.348
9537J. Chem. Phys., Vol. 109, No. 21, 1 Decembe 1998 Houssa, Rull, and McG o he
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