A no el o ien a ion-dependen po en ial model o p ola e mesogens
B. Ma ı
´nez-Haya,a) A. Cue os, and S. Lago
Depa amen o de Ciencias Ambien ales, Uni e sidad Pablo de Ola ide, 41013 Se ille, Spain
L. 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,
Apa ado Pos al 1065, 41080 Se ille, Spain
共Recei ed 1 Decembe 2003; accep ed 19 Oc obe 2004; published online 22 Decembe 2004兲
An in e molecula po en ial is in oduced o he s udy o molecula mesogenic luids. The model
combines dis inc ea u es o he well-known Gay-Be ne and Kiha a po en ials by inco po a ing
dispe si e in e ac ions dependen on he ela i e pai o ien a ion o a sphe ocylinde molecula co e.
Resul s o a Mon e Ca lo simula ion s udy ocused on he liquid c ys al phases exhibi ed by he
model luid a e p esen ed. Fo he chosen po en ial pa ame e s, molecula aspec a io L*⫽5 and
empe a u es T*⫽2, 3, and 5, iso opic, nema ic, smec ic-A, and hexa ic phases a e ound. The
loca ion o he phase bounda ies as well as he equa ion o s a e o he luid and u he
he modynamical and s uc u al pa ame e s a e discussed and con as ed o he Kiha a luid. In
compa ison o his la e luid, he model induces he o ma ion o o de ed liquid c ys alline phases
a lowe packing ac ions and i a o s, in pa icula , he appea ance o laye ed hexa ic o de ing as
a consequence o he g ea e a ac i e in e ac ion assigned o he pa allel side- o-side molecula pai
con igu a ions. The esul s con ibu e o he e alua ion o he ole o speci ic in e ac ion ene gies in
he mesogenic beha io o p ola e molecula liquids in dense en i onmen s. © 2005 Ame ican
Ins i u e o Physics. 关DOI: 10.1063/1.1830429兴
I. INTRODUCTION
The use o simple models o ep esen he o e all pai
in e ac ions in molecula luids has p o en o be a success ul
s a egy o s udy he beha io o mesogens, as well as o
p edic and cha ac e ize hei liquid c ys al phases. The aim
o such models is o cap u e he essen ial aspec s o he phys-
ics unde laying he mesogenic beha io o he eal sys ems,
such as excluded olume e ec s and dispe si e in e ac ions,
while keeping easonable analy ical and compu a ional e i-
ciency o heo e ical and simula ion s udies.
Fluids o elonga ed o odlike molecules a e an impo -
an class o mesogens wi h ele an echnological and bio-
logical applica ions and, he e o e, di e en models ha e
been in oduced in o de o explo e hei p ope ies. Fo in-
s ance, a amily o igid molecula models o ellipsoidal
symme y has been p oposed, among which he Gay-Be ne
共GB兲 luid anks as one o he mos ex ensi ely s udied.1
This model ex ended he pionee ing s udies o F enkel and
co-wo ke s on he ha d ellipsoid luid,2and was speci ically
in oduced as an imp o emen o he Gaussian o e lap
model.3The main ea u e o he GB model is a ou -
pa ame e unc ionali y ha con ols he aspec a io o he
ellipsoidal co e and he aniso opy o he a ac i e in e ac-
ions. The phase diag am o he GB luid has been cha ac e -
ized o a b oad ange o aspec a ios and in e ac ion pa am-
e e s and, in pa icula , nema ic and laye ed smec ic and
hexa ic liquid c ys al phases ha e been epo ed.4–6
In spi e o he success o he Gay-Be ne model, mo e
de ailed in e ac ion app oaches, such as si e-si e Lenna d-
Jones chain models, indica e ha he ac ual co e o p ola e
molecules is signi ican ly be e ep oduced by a sphe ocyl-
inde co e 共i.e., a cylinde o heigh /diame e aspec a io
L*⫽L/
, capped a bo h ends wi h a hemisphe e o he
same diame e
兲.5,7 In ac , a numbe o luid models o his
la e symme y ha e been in oduced in he pas decades,
especially a e e icien algo i hms we e de eloped o com-
pu e he minimum dis ance be ween he cen al ods o such
molecules.8Examples o his amily o models a e he ha d
sphe ocylinde 共HSC兲 luid and i s squa e-well 共SWSC兲o
so epulsi e 共SRS兲 a ian s,9–14 and he Kiha a luid.15 The
Kiha a model was in oduced as a gene aliza ion o he
Lenna d-Jones luid o aniso opic molecules, and has been
employed in nume ous in es iga ions o he modynamic,
s uc u al, and anspo p ope ies o luids o linea
molecules.16 Pe haps su p ising, i has no been un il ecen ly
ha he abili y o he Kiha a luid o o m liquid c ys als
phases has been sys ema ically in es iga ed.17
One d awback ha he Kiha a model sha es wi h he
HSC, SWSC, and SRS models is ha i assigns he same
in e ac ion ene gy o all pai o ien a ions, as long as he
minimum dis ance be ween he molecules emains cons an .
This is in con as wi h he in e ac ions o eal sys ems whe e
he dispe si e o ces a e o ien a ion dependen and, unless
speci ic in e ac ions come in o play, in he mo opic luids
hey usually end o be g ea e o aligned molecula con igu-
a ions 共e.g., side- o-side pa allel pai s兲 han o he mis-
aligned ones 共e.g., head- o- ail o T-shaped pai s兲.18,19 This
beha io o he in e ac ion ene gy is quali a i ely ep oduced
by he igid chain models wi h mul iple in e ac ion si es.
Howe e , al hough mul iple-si e models o his la e ype
ha e been employed o s udy liquid c ys al phases,7,20 hei
a兲Elec onic mail: [email p o ec ed]
THE JOURNAL OF CHEMICAL PHYSICS 122, 024908 共2005兲
122, 024908-10021-9606/2005/122(2)/024908/8/$22.50 © 2005 Ame ican Ins i u e o Physics
use is limi ed by he compu a ional cos associa ed o he
la ge numbe o si es equi ed in o de o mimic ealis ic
mesogenic molecules.
The main idea behind he p esen wo k is o co ec o
he de iciency o he Kiha a model commen ed abo e by
inco po a ing one he mos dis inc ea u es o he Gay-Be ne
luid, namely, i s pa ame ic modula ion o he pai o ien a-
ion dependence o dispe si e o ces. The model is p esen ed
in Sec. II and he emaining o he pape is hen de o ed o
explo e, by means o Mon e Ca lo simula ions, he quali a-
i e e ec s ha his ea u e in oduces in he liquid c ys al
phase diag am o he luid.
II. INTERACTION MODEL
In his wo k, we in oduce an in e ac ion model ha in-
co po a es he pai o ien a ion dependence o he dispe si e
in e ac ions o he Gay-Be ne po en ial o he sphe ocylinde
molecula co e o he Kiha a luid. In o de o achie e his in
a s aigh o wa d and easily ecognizable way, we ha e buil
a ‘‘hyb id’’ in e ac ion ene gy unc ional by mul iplying he
Kiha a po en ial by he same o ien a ional p e ac o o he
Gay-Be ne po en ial. These la e ac o depends explici ly
on he h ee- ec o co ela ions be ween he di ec o s o he
gi en pai o pa icles (uˆi,uˆj) and a uni ec o in he di ec-
ion o he cen e -o -mass in e molecula dis ance ec o
( ˆij), in con as o he Kiha a in e ac ion which depends on
he ela i e o ien a ion o he molecula pai s only h ough
i s in luence on he minimum dis ance be ween he molecula
co es dm( ij,uˆi,uˆj). We will e e o his model as he Gay-
Be ne-Kiha a po en ial o , in sho , GB-K po en ial. The
model is hus de ined by he ollowing exp essions:
UGB-K共 ij,uˆi,uˆj兲⫽
⑀
GB共 ˆij,uˆi,uˆj兲UK共dm兲,共1兲
UK共dm兲⫽4
⑀
关共
/dm兲12⫺共
/dm兲6兴,共2兲
⑀
GB共 ˆij,uˆi,uˆj兲⫽
⑀
GO
共uˆi,uˆj兲
⑀
⬘
共 ˆij,uˆi,uˆj兲,共3兲
⑀
GO共uˆi,uˆj兲⫽关1⫺
2共uˆi•uˆj兲2兴⫺1/2,共4兲
⑀
⬘共 ˆij,uˆi,uˆj兲⫽1⫺
⬘
2
冋
共 ˆij•uˆi⫹ ˆij•uˆj兲2
1⫹
⬘共uˆi•uˆj兲
⫹共 ˆij•uˆi⫺ ˆij•uˆj兲2
1⫺
⬘共uˆi•uˆj兲
册
.共5兲
The p e ac o
⑀
GB is cha ac e ized by he usual Gay-
Be ne ou -pa ame e se 共
,
⬘,
, and
兲, wi h he no a ion
⫽(
2⫺1)/(
2⫹1) and
⬘⫽(
⬘1/
⫺1)/(
⬘1/
⫹1).
Simple geome ical a gumen s show ha , in ou model, he
pa ame e associa ed o he ellipsoidal aspec a io
is e-
la ed o he sphe ocylinde aspec a io h ough
⫽L*⫹1
共no e ha L*is he heigh o he cen al cylinde , whe eas
he o al leng h o he sphe ocylinde , including he end caps,
is L*⫹1, always in uni s o he diame e
兲. On he o he
hand, he aniso opy o he dispe si e in e ac ion is con-
olled by
⬘; o ins ance, he a ac i e ene gy well o a
pa allel pai o molecules is
⬘ imes deepe o a side- o-
side con igu a ion han o a head- o- ail one. Fo he p esen
s udy, we ha e aken he se o pa ame e s L*⫽5,
⫽6,
⬘⫽5,
⫽2, and
⫽1, so ha he model he e employed
will be hence o h deno ed GB-K共6,5,2,1兲acco ding o he
no a ion GB-K(
,
⬘,
,
) which is simila o he one used
p e iously o he Gay-Be ne luid.14 The sphe ocylinde as-
pec a io L*⫽5共and, hence,
⫽6) was chosen in o de o
compa e he p esen esul s o ou ecen s udy o he Kiha a
luid,17 whe eas
⬘⫽5,
⫽2,
⫽1, a e he alues o iginally
sugges ed by Gay and Be ne1also employed in se e al p e-
ious s udies o he liquid phase diag am o luids o sho e
aspec a ios
⭐4.4–6 I mus be no ed ha he choice o
⬘,
,
is ac ually no a i ial ask, especially when compa ing
wi h eal sys ems, and o he se s o alues ha e been p o-
posed o speci ic he mo opes.14,21
Figu e 1 illus a es some o he main ea u es o he
GB-K po en ial model in oduced in he p e ious pa ag aph.
The op panel 关con ou plo 共A兲兴 shows he equipo en ial
con ou s o he in e ac ion o wo pe ec ly pa allel GB-
K共6,5,2,1兲pa icles 共i.e., uˆi⫽uˆj) whose ela i e posi ion is
de ined by he in e molecula dis ance ec o in pola coo -
dina es ( ij,
) in he plane o he igu e. As can be eadily
seen in he igu e, he a ac i e well o he pa allel side- o-
side con igu a ion (
⫽0°) is signi ican ly enhanced wi h e-
spec o he head- o- ail one (
⫽90°). The same ep esen a-
FIG. 1. Top panel: Equipo en ial ene gy su aces o he in e ac ion o wo
pa allel pa icles in e ac ing h ough 共a兲 he GB-K共6,5,2,1兲po en ial in o-
duced in his wo k 共wi h pa ame e s L*⫽5,
⫽6,
⬘⫽5,
⫽2, and
⫽1, see ex o de ails兲,共b兲 he Kiha a po en ial 共wi h L*⫽5), and 共c兲 he
Gay-Be ne po en ial GB共6,5,2,1兲. The posi ion o he pai o pa icles is
desc ibed by he pola coo dina es ( ij ,
), wi h
⫽0° o he side- o-side
con igu a ion and
⫽90° o he head- o- ail one. Bo om panel: GB-
K共6,5,2,1兲pai in e ac ion ene gy as a unc ion o he minimum dis ance
be ween he molecula co es dm o pai s o pa allel molecules in side- o-
side, c ossed, head- o- ail, and T-shaped con igu a ions.
024908-2 Ma ı
´nez-Haya
e al.
J. Chem. Phys. 122, 024908 (2005)
ion o he Kiha a luid 关plo 共B兲兴 ob iously yields a uni o m
well a ound he molecula co e. On he o he hand, he simi-
la diag am o he Gay-Be ne po en ial wi h he same pa-
ame e s, GB共6,5,2,1兲关plo 共C兲兴, illus a es i s cha ac e is ic
aniso opic well and ellipsoidal co e.
A close quan i a i e ep esen a ion o he aniso opy o
he dispe si e in e ac ions in he GB-K共6,5,2,1兲model is
p o ided in he bo om panel o Fig. 1, whe e he po en ial
ene gy as a unc ion o he minimum dis ance dmis ep e-
sen ed o pai s o molecules in pa allel, c ossed, head- o-
ail, and T-shaped con igu a ions. The i e old deepe well
o he pa allel con igu a ion wi h espec o he head- o- ail,
as co esponds o
⬘⫽5, is appa en in he igu e. I mus be
no iced ha ou o mula ion o he GB-K model assigns a
well dep h o uni y 共in educed uni s兲 o he c ossed con igu-
a ion, which mus be kep in mind when compa ing, a a
gi en empe a u e, he phase diag am o his luid o ha o
he Kiha a luid 共 o which a well dep h o uni y applies o
all molecula o ien a ions兲.
III. SIMULATION DETAILS
We ha e ca ied ou iso he mal-isoba ic (N-P-T) en-
semble Mon e Ca lo 共MC兲simula ions o s udy he liquid
c ys al phase diag am a di e en empe a u es o he GB-
K共6,5,2,1兲 luid 共i.e., wi h pa ame e s
⫽6, and hence L*
⫽5,
⬘⫽5,
⫽2, and
⫽1). Fo compu ing e iciency, in
he p esen simula ions he in e ac ion was unca ed a a
dis ance o dC⫽3
, which co esponds o cen e -o -mass
dis ances anging om ij⫽3
o ij⫽8
, depending on he
ela i e pai o ien a ion, and shi ed so ha he po en ial en-
e gy UGB-K anishes a he unca ion bounda y. Hence, he
po en ial ac ually employed is gi en by
UGB-K共 ij,uˆi,uˆj兲
⫽
再
⑀
GB共 ˆij,uˆi,uˆj兲关UK共dm兲⫺UK共dC兲兴,dm⭐3
0, dm⬎3
.共6兲
The simula ions we e un o a sys em o Np⫽1080 mol-
ecules a h ee educed empe a u es, T*⫽kT/
⑀
⫽2, 3, and 5,
whe e kdeno es he Bol zmann cons an and
⑀
he well dep h
o he c ossed con igu a ion 共see Fig. 1兲. As discussed
h oughou he ollowing sec ions o he pape , a hese em-
pe a u es he GB-K共6,5,2,1兲 luid p esen s s able iso opic
(I), nema ic (N), smec ic-A(Sm-A), and hexa ic 共Hex兲
phases. The calcula ion was s a ed a each empe a u e by
comp essing he luid o a s a e o high densi y well inside
he hexa ic egion o he phase diag am (P*⫽P
3/kT
⫽2.4, 2.6, and 3.1, espec i ely, o T*⫽2, 3, and 5兲. Such
s a e was p ope ly equilib a ed o e some 106MC cycles
be o e a sys ema ic iso he mal expansion o he luid was
pe o med down o he iso opic phase. Such app oxima e
p ocedu e o he es ima ion o liquid c ys al phase bound-
a ies is eliable o ou pu poses and has been ex ensi ely
employed in he pas ,9,11,22 in pa icula , o simila luids o
p ola e molecules. I mus be s essed, howe e , ha a p ope
s udy o he phase diag am should include he accu a e cal-
cula ion o he ee ene gy in each o he phases, which is
beyond he scope o he p esen s udy.
Each s a e was ypically equilib a ed o e 106MC
cycles and ensemble a e ages o he he modynamic and
s uc u al p ope ies o he sys em we e compu ed o e 3
⫻105cycles. The las molecula a angemen was hen used
as ini ial con igu a ion o he subsequen un wi h a smalle
sys em p essu e. Each MC cycle consis s o Npa emp s o
andom displacemen s and/o eo ien a ions o he pa icles
plus a ial change o he box olume. The usual pe iodic
bounda y condi ions a e employed and he accep ance a ios
we e kep wi hin 30%–40% o he il and displacemen o
he pa icles, and wi hin 20%–30% o he box olume
change. Box olume changes we e a emp ed by andomly
changing he leng h o each side o he box independen ly,
wi h he es ic ion ha none o hem could become sho e
han wice he ange o he in e ac ion po en ial, o 8
in he
p esen case. In e es ingly, we ound a endency o he box o
spon aneously adop an aniso opic geome y wi h side
leng hs ul illing Lx⬇Ly⬍Lz共 he subsc ip x deno es he
sho es side兲, which is consis en wi h he geome y chosen
o he simula ions wi h ixed box geome y (Lx⬇Ly
⬇Lz/2) o ou ea lie s udies o he SWSC, SRS, and Kiha a
luids.11,17
In spi e o his la e inding, in o de o pe o m a p ope
compa ison be ween he GB-K共6,5,2,1兲and he Kiha a lu-
ids, we ha e also ecompu ed ull simula ions o he ou
iso he ms T*⫽kT/
⑀
⫽1.5, 2, 3, and 5 o he Kiha a luid
s udied in a p e ious wo k17 employing he same me hodol-
ogy desc ibed abo e. The esul o hese new simula ions,
which essen ially ep oduce he da a o he ea lie wo k,17
a e p esen ed below and include an ex ended egion a low
and high densi y wi h espec o ou p e ious wo k in o de
o o e lap wi h he p esen s udy o he GB-K model.
The liquid c ys alline ansi ions obse ed in he expan-
sion o he luid a e cha ac e ized by discon inui ies in he
densi y and he nema ic o de pa ame e , as well as by sud-
den quali a i e changes obse ed in he di e en co ela ion
unc ions g
储
(
储
), g⬜( ⬜), g⬜
(n)( ⬜) de ined in p e ious
wo ks.6,9,11,12,23 Fo ins ance, he unc ion g
储
(
储
) ep esen s
he p ojec ion o he pai dis ibu ion unc ion along he nem-
a ic di ec o o he luid and de elops a cha ac e is ic oscil-
la o y s uc u e when laye ed phases a e o med. On he
o he hand, g⬜
(0)( ⬜) accoun s o he co ela ion be ween
pa icles wi hin he same laye and p esen s long- ange s uc-
u e when solidlike o de is p esen . An illus a i e insigh
in o hese co ela ion unc ions is p o ided below wi hin he
amewo k o he luidlike smec ic-Ao hexagonally packed
hexa ic phases ound in his wo k o he GB-K共6,5,2,1兲
luid. The o ma ion o hese la e hexa ic phases was in-
spec ed, in addi ion, h ough he calcula ion o he bond hex-
agonal o de pa ame e :22,24
H6⫽
冓
冏
1
Np兺
j
1
nj兺
(kl)exp共6i
kl兲
冏
冔
,共7兲
whe e njis he numbe o pai s o neighbo s wi hin he i s
in-laye coo dina ion shell o pa icle j, which is de ined as a
cylind ical olume o adius 1.6
and heigh 1.0
cen e ed a
he cen e o mass o he molecule. The sum o e (kl) ap-
plies o all possible pai s o such i s coo dina ion shell pa -
icles and
kl is he angle be ween he p ojec ion o he in-
024908-3 No el po en ial model o p ola e mesogens J. Chem. Phys. 122, 024908 (2005)
e molecula ec o s jk and jl on o he plane pe pendicula
o he di ec o o molecule j. Hence, H6is so de ined as o
app oach uni y o pe ec hexa ic o de wi hin he smec ic
laye s 共wi h nj⫽6 in-laye nea es neighbo s a ound each
pa icle o ming a hexagon兲and end o ze o o diso de ed
luidlike laye s. We employ he gene ic denomina ion hexa ic
h oughou he pape , since he limi ed numbe o pa icles o
ou simula ions does no allow o ully cha ac e ize his
phase as a luid smec ic-Bphase o a solid c ys al Bphase,
as will be discussed below.
IV. RESULTS AND DISCUSSION
Figu e 2 共 op panel兲 ep esen s he liquid phase diag am
o he GB-K共6,5,2,1兲 luid o he h ee iso he ms T*⫽2, 3,
and 5. The p essu es and densi ies o he bounda y s a es a
each o he phase ansi ions obse ed a e lis ed in Table I. A
mo e de ailed in o ma ion abou he iso he ms is p o ided in
Figs. 3, 4, and 5 which depic , espec i ely, he equa ions o
s a e o he luid, he a e age ene gy pe pa icle, and he
nema ic and hexa ic o de pa ame e s o each o he s a es
sampled in he p esen simula ions. The phase diag am and
equa ions o s a e o he Kiha a luid, ecalcula ed he e em-
ploying he a iable box geome y simula ion me hodology
FIG. 2. Summa y o he densi ies o he bounda y s a es a he iso opic-
nema ic (I-N), iso opic–smec ic-A(I-Sm-A), nema ic–smec ic-A
(N–Sm-A), o he smec ic-A–hexa ic (Sm-A–Hex) ansi ions ob ained in
he MC–N-P-Tsimula ions o he GB-K共6,5,2,1兲 luid o he p esen wo k.
The bo om panel shows he phase diag am o he Kiha a luid wi h he
same molecula aspec a io. These la e da a a e e ised wi h espec o
hose epo ed in a p e ious wo k 共Re . 17兲. See also Tables I and II.
TABLE I. Reduced p essu es and densi ies o he bounda y s a es o he liquid c ys al phase ansi ions o he
GB-K共6,5,2,1兲 luid conside ed in his wo k a empe a u es T*⫽2, 3, and 5. No e he educ ion o p essu e
wi h empe a u e (P*⫽P
3/kT) when compa ing he da a.
I–Sm-ASm-A–Hex
pI
*
I
*pSmA
*
SmA
*pSm-A
*
Sm-A
*pHex
*
Hex
*
T*⫽2.0 0.70 0.086共1兲0.75 0.104共1兲1.80 0.130共1兲1.90 0.137共1兲
I–NN–Sm-ASm-A–Hex
pI
*
I
*pN
*
N
*pN
*
N
*pSm-A
*
Sm-A
*pSm-A
*
Sm-A
*pHex
*
Hex
*
T*⫽3.0 1.10 0.100共1兲1.15 0.104共1兲1.40 0.112共1兲1.45 0.123共1兲2.25 0.140共1兲2.30 0.146共1兲
T*⫽5.0 1.30 0.107共1兲1.35 0.110共1兲2.35 0.137共1兲2.40 0.141共1兲2.95 0.151共1兲3.00 0.158共1兲
FIG. 3. Equa ions o s a e o he iso he ms T*⫽2, 3, and 5 o he GB-
K共6,5,2,1兲and he Kiha a luids wi h sphe ocylinde aspec a io L*⫽5. The
illed ci cles co espond o he GB-K luid whe eas he open squa es a e o
he Kiha a luid. The co esponding liquid c ys al phases (I,N, Sm-A,o
Hex兲a e indica ed nex o he da a. In he egions whe e he iso he ms o he
wo luids o e lap bu he phases a e di e en , he phase co esponding o
he Kiha a luid is indica ed in pa en heses. The s a is ical e o ba s asso-
cia ed o he densi y alues a e o he size o he symbols o smalle . No e
he educ ion o p essu e wi h empe a u e (P*⫽P
3/kT) when compa ing
he da a.
024908-4 Ma ı
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e al.
J. Chem. Phys. 122, 024908 (2005)
desc ibed in he p eceding sec ion, a e also included in he
same igu es o di ec compa ison wi h he GB-K da a. I
was ound ha essen ially he same esul s a e ob ained o
he Kiha a luid wi h espec o ou ea lie wo k,17 excep o
a⫾0.05–0.10 di e ence in he educed p essu e o he sys-
em in he smec ic s a es o same densi y. This la e e ec is
likely o be ela ed o aniso opic s ess e ec s25 induced by
he ixed box geome y app oach o ou p e ious wo k ha
should no be a ec ing he p esen esul s. O he han ha ,
he liquid c ys al ansi ions ele an o he p esen wo k oc-
cu in he expansion o he Kiha a luid a he same packing
ac ions, wi hin s a is ical unce ain y, as ound in he ixed
box geome y simula ions o ou ea lie wo k.17
We ocus now on he esul s o he GB-K model. A he
h ee empe a u es in es iga ed, he GB-K共6,5,2,1兲 luid p e-
sen s a s able hexa ic phase a su icien ly high densi ies in
which he luid is in e nally o de ed in smec ic laye s o hex-
agonally packed molecules. We lea e he cha ac e iza ion o
his liquid c ys alline phase o he las pa o his Sec ion.
When dec easing he sys em p essu e, and hence expanding
he luid, he hexa ic phase e en ually mel s o a smec ic-A
phase in which he laye s become luidlike and he wo-
dimensional posi ional o de wi hin hem is los . A u he
expansion o he sys em leads o a ansi ion o ei he a nem-
a ic phase 共 o he iso he ms T*⫽3 and 5兲o di ec ly o an
iso opic phase 共 o T*⫽2). This la e phase ansi ion
shows a s onge i s -o de cha ac e han he es o ansi-
ions obse ed, as he luid ea anges om a ela i ely dense
laye ed s uc u e o a ully diso de ed iso opic phase wi h a
much smalle packing ac ion.
One no iceable aspec o he phase diag am o he GB-
K共6,5,2,1兲 luid is he subs an ial educ ion in he ange o
s abili y o he nema ic phase when cooling down he sys em
om T*⫽5 oT*⫽3, as he ansi ion om smec ic-A o
nema ic shi s o lowe p essu es and densi ies. In ac , he
nema ic phase disappea s al oge he a T*⫽2, as al eady
poin ed ou abo e. Hence, he esul o he MC simula ions
sugges s he p esence o an iso opic–nema ic–smec ic-A
iple poin a a empe a u e o oughly T*⬇2.5 and a den-
si y close o
*⬇0.1 共see Fig. 2兲.
The compa ison o he phase diag am o he GB-
K共6,5,2,1兲 luid o ha o he Kiha a luid in he same ange
o empe a u es should p o ide ele an in o ma ion abou
he ela i e impo ance o ene gy and en opy as ‘‘d i ing
o ces’’ o he liquid c ys al ansi ions. As can be seen in
Fig. 2, he o e all ea u es o he phase diag am o bo h
luids p esen many simila i ies, al hough i is in e es ing o
ind ha he I–N–Sm-A iple poin o he Kiha a luid is
loca ed a oughly simila empe a u e bu a a highe densi y
FIG. 4. A e age pai po en ial ene gy (U*⫽U/
⑀
) along he iso he ms s ud-
ied in his wo k o he GB-K共6,5,2,1兲and he Kiha a luids. The illed
ci cles co espond o he GB-K luid and he open squa es o he Kiha a
luid. The e ical e o ba s co espond o one s anda d de ia ion o he
compu ed ene gies.
FIG. 5. Top panel: Nema ic o de pa ame e S2and bond hexagonal o de
pa ame e H6as a unc ion o numbe densi y o he GB-K共6,5,2,1兲 luid a
educed empe a u e T*⫽3. The e ical e o ba s co espond o one s an-
da d de ia ion o he compu ed o de pa ame e s. Bo om panel: In-laye
dis ibu ion unc ion in he di ec ion pe pendicula o he di ec o g⬜
(0)( ⬜
*)
in he smec ic-A共sho -dash cu e兲and hexa ic 共long-dash and solid cu es兲
phases o he same luid. The a iable ⬜
*⫽ ⬜/
ep esen s he p ojec ion o
he pai in e molecula dis ance ec o on o he plane pe pendicula o he
di ec o . Hence, his dis ibu ion shows he spa ial pai co ela ion be ween
he pa icles in he same smec ic laye . The educed p essu e o each s a e is
indica ed in he legend, whe eas he empe a u e is T*⫽3 in all cases.
024908-5 No el po en ial model o p ola e mesogens J. Chem. Phys. 122, 024908 (2005)
han in he GB-K共6,5,2,1兲 luid. The e a e u he signi ican
di e ences be ween bo h phase diag ams which ha e o be
a ibu ed o he aniso opy o he dispe si e in e ac ions in
he GB-K po en ial a ound he molecula co e in compa ison
o he iso opic in e ac ions o he Kiha a luid. A i s el-
e an aspec is ha he smec ic phase o he Kiha a luid does
no p esen any ace o in-laye hexagonal packing o de
wi hin he ange o densi ies scoped in he p esen s udy. In
o de o co obo a e his obse a ion, he densi y ange co -
e ed in ou ea lie wo k o he Kiha a luid17 was ex ended
o o e lap wi h he p esen simula ions. In ac , he hexa ic
s a es ound o he GB-K共6,5,2,1兲 luid mel ed sys ema i-
cally o smec ic-As a es when he GB-K po en ial was ex-
changed by he Kiha a po en ial.
Ano he pa icula i y o he GB-K共6,5,2,1兲 luid eadily
obse ed when compa ing he wo panels o Fig. 2 is he
sys ema ic dec ease o he educed p essu es and densi ies o
he bounda y s a es sepa a ing he phase ansi ions ha akes
place as he empe a u e o he luid cools down om high
(T*⫽5) o low (T*⫽2) empe a u e. This end, which is
especially p onounced o he N–Sm-A ansi ion, as com-
men ed abo e, is as well appa en o he Sm-A–Hex and
I-N ansi ions o he GB-K luid. In con as o his beha -
io , he phase bounda ies o he Kiha a luid 共Table II兲a e
conside ably less sensi i e o empe a u e 共wi h he excep-
ion o he N–Sm-A ansi ion, due o he ins abili y o he
nema ic phase a low empe a u e兲. A i s o e all conclusion
ha can be d awn om hese indings is ha he o ien a ion
dependence in oduced in he GB–K po en ial and, in pa -
icula , he enhanced in e ac ion o aligned pai con igu a-
ions 共see Fig. 1兲, a o s he o ma ion o he liquid c ys al
phases explo ed in he p esen wo k. In ac , he iso opic
phase becomes signi ican ly des abilized in compa ison o
he Kiha a luid, especially a low empe a u e. A T*⫽2, o
ins ance, he densi y o he bounda y iso opic s a e a he
I–Sm-A ansi ion is
*⫽0.086 in he GB-K共6,5,2,1兲 luid,
in compa ison o he much highe alue o
*⫽0.112 in he
Kiha a luid 共see Table I兲.
A somewha unexpec ed inding ha o some ex en
seems o con adic his la e conclusion is ha he nema ic
phase o he GB-K共6,5,2,1兲 luid a he highes empe a u e
in es iga ed, T*⫽5, p esen s a g ea e ange o s abili y han
in he Kiha a luid. In o he wo ds, he N–Sm-A ansi ion is
delayed owa d highe densi ies wi hin he nema ic phase in
he o me luid in compa ison o he la e one. In addi ion,
i is also obse ed ha he change in densi y a he ansi ion
is la gely educed in he GB-K共6,5,2,1兲 luid. A possible ex-
plana ion o hese ea u es is ha he GB-K nema ic s a es
become ac ually mo e comp essible han he Kiha a ones as
a consequence o he slowe ise o he sho - ange epulsi e
o ces o a la ge ac ion o pai o ien a ions. In Fig. 1 i can
be app ecia ed ha he epulsi e pa o he in e ac ion a
dis ances sho e han dm/
⫽1共 he ze o o he po en ial o
all pai o ien a ions兲becomes weake o molecula con igu-
a ions wi h a p e ac o
⑀
GB smalle han uni y. A he same
ime, he co e epulsion is enhanced in he GB-K共6,5,2,1兲
po en ial wi h espec o he Kiha a po en ial o he side- o-
side pa allel con igu a ions ypical o he smec ic phase. As a
consequence o his, he e ec i e olume excluded by he
molecules in he smec ic laye s is g ea e in he GB-K luid,
so ha he en opic cons ains wo k in a o o he mo e
comp essible nema ic phase 共in which many pai o ien a ions
a om he side- o-side one a e ele an 兲, hence delaying
he N–Sm-A ansi ion. Since he sho pai dis ances a
which he epulsi e pa o he po en ial domina es a e mo e
e icien ly accessed by he molecules o he luid a high
empe a u e, i is no su p ising ha his e ec is only ob-
se ed in ou simula ions a T*⫽5, whe eas a he lowe
empe a u es, mo e sensi i e o he a ac i e pa o he in-
e ac ion, he smec ic-Aphase is s able down o signi ican ly
lowe densi ies in he GB-K共6,5,2,1兲 luid han in he Kiha a
one.Figu e 3 compa es he equa ions o s a e (P*
⫽P
3/kT e sus
*⫽
3) along he iso he ms T*⫽2, 3,
and 5 o he GB-K共6,5,2,1兲and he Kiha a luids. We i s o
all ema k ha in all cases he iso he ms o bo h models
consis en ly con e ge a low densi y. Wi h inc easing em-
pe a u e he beha io o he luids is expec ed o become
p og essi ely less sensi i e o he de ails o he pai in e ac-
ion po en ial 共especially a low densi y whe e he a ac i e
o ces a e dominan 兲and, in ac , a T*⫽5 he good o e lap
be ween he equa ions o s a e o he GB-K共6,5,2,1兲and he
Kiha a luids ex ends o a subs an ial egion o he nema ic
TABLE II. Reduced p essu es and densi ies o he bounda y s a es o he liquid c ys al phase ansi ions o he
Kiha a luid conside ed in his wo k a empe a u es T*⫽1.5, 2.0, 3.0, and 5.0. No e he educ ion o p essu e
wi h empe a u e (P*⫽P
3/kT) when compa ing he da a. The abula ed alues di e sligh ly wi h espec o
hose gi en in a ecen wo k 共Re . 17兲due o he g ea e sys em size and he imp o ed p essu e equilib a ion
me hodology o he p esen simula ions 共see ex o de ails兲.
I–Sm-A
pI
*
I
*pSMA
*
SMA
*
T*⫽1.5 1.75 0.114共1兲1.80 0.132共1兲
T*⫽2.0 1.65 0.112共1兲1.70 0.125共1兲
I–NN–Sm-A
pI
*
I
*pN
*
N
*pN
*
N
*pSM-A
*
SM-A
*
T*⫽3.0 1.40 0.108共1兲1.45 0.111共1兲1.70 0.119共1兲1.75 0.130共1兲
T*⫽5.0 1.30 0.107共1兲1.35 0.110共1兲1.75 0.128共1兲1.80 0.135共1兲
024908-6 Ma ı
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J. Chem. Phys. 122, 024908 (2005)
phase. A he su icien ly high densi y, howe e , signi ican
di e ences be ween bo h sys ems e en ually a ise. A T*
⫽5, o smec ic-As a es o simila densi y, he p essu e is
signi ican ly g ea e o he GB-K luid han o he Kiha a
luid. In e es ingly, his end ends o e e se a he lowe
empe a u es, so ha a T*⫽2 and 3 he p essu e o he
Kiha a s a es becomes compa able o ha o he GB-K s a es
wi h he same densi y. A hese lowe empe a u es, howe e ,
he compa ison be ween he equa ions o s a e o he GB-
K共6,5,2,1兲and he Kiha a luids a high densi y is no as
s aigh o wa d as o T*⫽5, since he o e lap be ween he
liquid c ys al phases is g ea ly educed. Fo ins ance, he
acciden al coincidence o he equa ions o s a e o bo h luids
is no iceable a T*⫽2 in he high densi y ange (
*
⬎0.137) whe e he GB-K共6,5,2,1兲 luid has en e ed he
hexa ic phase, whe eas he Kiha a luid emains in he Sm-A
phase. The eason o inding g ea e p essu es in he GB-K
luid a high empe a u e is again ela ed o he la ge p e -
ac o in he GB-K in e ac ion po en ial which enhances he
sho - ange epulsion o he side- o-side pai con igu a ions
共a he same ime as i inc eases he dep h o he well a
la ge dis ances兲. The epulsi e wall is el mo e e icien ly
by he molecules a high empe a u es wi h he ne e ec o
inc easing he p essu e o he sys em wi h espec o he Ki-
ha a luid. A lowe empe a u es, he g ea e a e age a ac-
ions educe he p essu e o he GB-K sys em, especially in
he smec ic-Aphase.
F om he compa ison o he phase diag ams and equa-
ions o s a e o he GB-K共6,5,2,1兲and he Kiha a luids i
becomes appa en ha , e en hough s e ic excluded olume
e ec s cons i u e he main cons ain ha d i es he liquid
c ys al ansi ions o he p ola e molecula luids o he ype
conside ed he e, he ene ge ic con ibu ion o he ee ene gy
has an impo an ole ha canno be neglec ed and, hence,
ha an app op ia e desc ip ion o he dispe si e o ces is
equi ed in o de o model he p ope ies o eal mesogens.
Fu he suppo o hese conside a ions is p o ided in Fig. 4,
which e eals he quali a i ely di e en e olu ion in he
GB-K and Kiha a luids o he a e age in e nal ene gy pe
pa icle along he h ee iso he ms conside ed in ou wo k. In
he Kiha a luid, he ene gy g ows mono onously wi h den-
si y a high empe a u e (T*⫽3 and 5兲, whe eas a weak
dec easing end is obse ed a lowe empe a u es (T*
⫽2) in he low densi y side o he iso opic b anch and o
he smec ic-As a es.17 The ac ha he in e nal ene gy
g ows wi h densi y a high empe a u e can be aced back o
he p og essi e ele ance o he epulsi e pa o he pai
po en ial a he ela i ely high densi ies conside ed in ou
s udy. In he Kiha a luid, he a ac i e well only domina es
a su icien ly low empe a u es o , al e na i ely, in a dilu ed
densi y egime, much close o he ideal gas limi han he e
conside ed. Thus, i u ns ou ha in he ange o densi ies
ele an o liquid c ys al beha io , he iso opic Kiha a a -
ac ion inco po a es li le no el quali a i e ea u es wi h e-
spec o he simila beha io also ound o he pu ely epul-
si e SRS luid 共basically a unca ed Kiha a luid wi hou
a ac i e well兲.11 On he con a y, he speci ic opology o
he a ac i e in e ac ions in he GB-K luid leads o a quali-
a i ely di e en ole o ene gy in he s abili y o he liquid
c ys al phases. Especially no iceable is he d op in ene gy
ha akes place as he sys em en e s he smec ic-Aphase
om ei he he iso opic (T*⫽2) o nema ic (T*⫽3 and 5兲
phase. Such ene gy discon inui y is absen o much weake
in any o he sphe ocylinde model luids p oposed in he
pas , such as he Kiha a17 o he SRS and SWSC luids,11 and
is a di ec consequence o he bias o he GB-K po en ial
a o ing speci ic pai con igu a ions. I ollows ha he s a-
bili y o he smec ic phases is suppo ed no only by s e ic
e ec s, bu also by he bene icial con ibu ion o he dispe -
si e in e ac ions o he ee ene gy. In e es ingly, a signi ican
u he d op in ene gy is also obse ed in he h ee iso he ms
a he Sm-A–hexa ic ansi ion. This ein o ces he idea ha
he appea ance o hexagonal o de in he GB-K共6,5,2,1兲 luid
a densi ies whe e he Kiha a luid main ains a s able
smec ic-Aphase is also closely ela ed o he ene ge ic e -
ec s induced by he aniso opic na u e o he in e ac ion po-
en ial.
In o de o cha ac e ize he hexa ic phases obse ed in
he GB-K共6,5,2,1兲 luid and clea ly disce n hem om he
smec ic-Aphases, se e al diagnos ics we e applied du ing
he simula ions. The ansi ion om he smec ic-A o he
hexa ic phase implies a sudden change o he hexagonal
bond o de pa ame e 关Eq. 共7兲兴, in ou case om oughly
H6⬇0.2 o H6⬇0.5, as can be seen in he uppe panel o Fig.
5 whe e H6is ep esen ed along wi h he nema ic o de pa-
ame e o he iso he m T*⫽3. The o ma ion o he hexa ic
phase can also be isualized h ough he long- ange s uc u e
appea ing ab up ly in he in-laye dis ibu ion unc ion
g⬜
(0)( ⬜)共 he pai dis ibu ion wi hin he plane o he laye 兲.
Figu e 5 illus a es he quali a i e change ha his unc ion
unde goes a he Sm-A–hexa ic ansi ion a T*⫽3, om a
smoo h liquidlike dis ibu ion wi h only sho - ange o de o
a much mo e s uc u ed wo-dimensional hexa ic c ys al-like
co ela ion unc ion wi h well-de ined nea es -neighbo posi-
ions o e se e al coo dina ion shells.
A con o e sial opic p esen in se e al ecen wo ks on
he Gay-Be ne luid has been whe he he hexa ic phases
o med in ha sys em a e ac ually liquidlike smec ic-B
phases o solidlike c ys al Bphases on a mac oscopic basis.6
We ad ance ha ou p esen esul s do no se e o close his
a om i ial ques ion. A su icien ly high densi y a s able
solid c ys alline phase is known o o m in sphe ocylinde
co e luids.9In ac , i has been a gued ha he absence o a
well-de ined phase ansi ion when comp essing and/o cool-
ing down he hexa ic Gay-Be ne luid indica es ha he
phase mus be c ys alline om he beginning. Fu he e i-
dence in a o o he c ys alline na u e o he hexa ic phases
o he ype obse ed o he Gay-Be ne sys em a ise om he
calcula ion o shea iscosi ies.23 Howe e , he di e ence
be ween he smec ic-Band he c ys al Bphases is ac ually a
sub le one o he ini e size sys ems employed in compu e
simula ions. In addi ion, ca e mus be aken when dealing
wi h such high densi y s a es whe e he mobili y o he pa -
icles is g ea ly educed and me as able s a es o ine icien
sampling o he phase space may a ec he appa en s uc-
u e o he luid. One u he aspec abou he smec ic-B
phases is ha he long- ange o de wi hin he laye s is un-
ca ed on a mac oscopic scale by he p esence o a la ge num-
024908-7 No el po en ial model o p ola e mesogens J. Chem. Phys. 122, 024908 (2005)
be o punc ual de ec s, in con as wi h he c ys al Bphase
whe e he long- ange o de ex ends o e a mac oscopic
dis ance.6We close by no ing o his espec ha , wi hin he
ange o densi ies and empe a u es explo ed in ou simula-
ion, he hexa ic o de pa ame e emains a alues below 0.6
共see Fig. 5兲, ha is much smalle han he limi ing alue o
uni y, which is an indica ion o a subs an ial p esence o
de ec s wi hin he laye s o he luid, in p inciple compa ible
wi h a smec ic-Bphase. Howe e , he limi ed size o ou
simula ion box p e en s us om making de ini e s a emen s
abou he ange o he in-laye co ela ions and we canno
he e o e d aw conclusions in a o o he smec ic-Bo c ys-
al Bcha ac e o he obse ed hexa ic phase.
V. SUMMARY AND CONCLUSIONS
A igid model po en ial, e e ed o as Gay-Be ne-Kiha a
o GB-K po en ial, has been in oduced which is expec ed o
be eliable o he s udy o molecula mesogenic luids. The
GB-K model ea u es a sphe ocylinde molecula co e
d essed wi h dispe si e in e ac ions dependen on he ela i e
pai o ien a ion. The ma hema ical o mula ion o he model
is compac and combines he unc ionali ies o he well-
known Kiha a and Gay-Be ne po en ials.
The esul s o he Mon e Ca lo simula ion s udy o he
model luid o a speci ic se o pa ame e s a empe a u es
T*⫽2, 3, and 5 p esen ed in his pape show ha i is ca-
pable o ep oduce he iso opic, nema ic, smec ic-A, and
hexa ic liquid c ys al phases obse ed in eal mesogens. A
he h ee empe a u es in es iga ed, he GB-K共6,5,2,1兲 luid
p esen s a s able hexa ic phase a su icien ly high densi ies,
cha ac e ized by laye s o hexagonally packed molecules.
When expanding he luid, he hexa ic phase e en ually mel s
o a smec ic-Aphase s uc u ed in luidlike diso de ed laye s.
A u he expansion o he sys em leads o a ansi ion o
ei he a nema ic phase 共a T*⫽3 and 5兲o di ec ly o an
iso opic phase 共a T*⫽2). In his la e phase change he
luid unde goes a s ongly i s -o de ansi ion om a dense
laye ed s uc u e o a ully diso de ed iso opic phase wi h a
subs an ial change in densi y.
In compa ison o he Kiha a luid, he GB-K共6,5,2,1兲
luid is ound o a o he o ma ion o he o de ed liquid
c ys alline phases and, speci ically, he appea ance o laye ed
hexa ic o de , a lowe packing ac ions. This p ope y can
be in e p e ed as being a di ec consequence o he g ea e
dispe si e in e ac ions assigned in he GB-K luid o speci ic
pai o ien a ions, such as he pa allel side- o-side con igu a-
ion. In ac , he speci ic opology o he GB-K po en ial wi h
espec o he Kiha a po en ial leads o a quali a i ely di e -
en beha io o he in e nal ene gy and o i s in luence on he
s abili y o he liquid c ys al phases. In pa icula , i is ound
ha he en ance o he luid in he smec ic phases (Sm-Ao
hexa ic兲is accompanied by a subs an ial s abiliza ion o he
in e nal ene gy o he luid, in con as o he beha io o
o he sphe ocylinde model luids s udied p e iously, such as
he Kiha a, o he SRS and SWSC luids. Hence, i becomes
appa en ha he ene ge ic con ibu ion o he ee ene gy has
an impo an ole in he mesogenic beha io o p ola e mo-
lecula liquids in dense en i onmen s, and ha an app op i-
a e desc ip ion and ea men o he dispe si e o ces is e-
qui ed in o de o model accu a ely he p ope ies o eal
mesogens, e en a a quali a i e le el. The in luence o he
sho - ange in e ac ions on he in e nal s uc u e o he mo-
lecula luids has as well been s essed ecen ly in a s udy o
sys ems composed o linea dipola molecules.26 The main
ad an age o he GB-K po en ial is ha i combines, wi hin
he in insic limi a ions o he igid models, a quali a i ely
mo e adequa e desc ip ion o bo h, he pai in e ac ions and
he molecula shape o he ypical mesogens, in compa ison
o p e ious models, whe eas i keeps a compa able compac -
ness and nume ical e iciency in i s o mula ion.
The p esen s udy has ocused on he p esen a ion o he
GB-K model and has s essed he quali a i e e ec s in o-
duced by he aniso opic dispe si e in e ac ions o he model
in he equa ion o s a e and in he phase diag am o he luid.
Fu u e wo k in ou g oup will be de o ed o compa e he
beha io o he 共sphe ocylinde 兲GB-K luid o ha o he
共ellipsoidal兲Gay-Be ne luid, so ha he ele ance o he
exac shape o he molecula co e a supe c i ical and sub-
c i ical empe a u es will be exposed. In addi ion, he imple-
men a ion o he GB-K model o obla e 共disk-like兲mesogens
will be explo ed.
ACKNOWLEDGMENTS
The au ho s acknowledge suppo om he Spanish
Di eccio
´n Gene al de In es igacio
´n Cien ı
´ ica y Te
´cnica
共G oups No. BQU2001-3615-C02兲and Plan Andaluz de
In es igacio
´n共G an Nos. FQM-205 and FQM-319兲.
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