Liquid c ys al beha io o he Kiha a luid
A. Cue os, B. Ma ı
´nez-Haya, 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 1065,
41080 Se ille, Spain
共Recei ed 28 Feb ua y 2003; published 15 July 2003兲
The liquid c ys al phases o he Kiha a luid ha e been s udied in compu e simula ions. The wo k ocuses
on he iso opic–nema ic–smec ic-A iple poin egion, especially ele an o he unde s anding o he p op-
e ies and he design o eal mesogens wi h speci ic phase diag ams. The Kiha a in e ac ion esembles mo e
app op ia ely han o he ela ed models, he shape o elonga ed polyme s and biomolecules, and a close
asse ion is p o ided o he ole o he con igu a ional en opy and he dispe si e in e ac ions in he beha io
o such molecules in dense phases o unde mac omolecula c owding condi ions.
DOI: 10.1103/PhysRe E.68.011704 PACS numbe 共s兲: 64.70.Md, 61.30.Cz, 87.15.Aa
I. INTRODUCTION
A numbe o compu a ionally e icien pai in e ac ion
models ha e been p oposed o e he pas decades o he
s udy o liquid c ys alline mesogens. In pa icula , luids
composed o elonga ed 共p ola e兲molecules cons i u e an im-
po an class o liquid c ys als wi h ele an indus ial and
biological applica ions. The e o e, di e en models ha e
been used o explo e hei beha io such as he ha d ellipsoid
共HE兲 luid, he Gay-Be ne 共GB兲 luid, o he ha d sphe ocyl-
inde 共HSC兲 luid and i s squa e–well 共SWSC兲o so epul-
si e 共SRS兲 a ian s 关1–9兴.
The Kiha a luid model was in oduced in he 1960s 关10兴
as a gene aliza ion, o aniso opic molecules, o he sphe i-
cally symme ic Lenna d-Jones luid 关see Eq. 共1兲兴 and has
been ho oughly employed o in es iga e he he modynamic,
s uc u al, and anspo p ope ies o luids o linea mol-
ecules. In pa icula , his model has been ound o be app o-
p ia e o esemble he liquid- apo equilib ium o di e en
sol en s and alkanes o mode a e elonga ion 关6兴. The Kiha a
in e ac ion po en ial inco po a es a so 共Lenna d-Jones ype兲
a ac i e well a ound he molecula co e, which is absen o
o e simpli ied in o he models in es iga ed p e iously, such
as he SRS o SWSC luids. The la squa e-well a ac ion o
he SWSC luid, o ins ance, allows o small comp essions
o expansions o he sys em wi hou change o ene gy. In
addi ion, he sphe ocylind ical shape o he molecules in he
Kiha a model is mo e ealis ic o many 共bio兲chemical sys-
ems han he ellipsoidal shape imposed by he well-known
HE o GB models. Fo ins ance, ecen in es iga ions ha e
explo ed he ele ance o he sphe ocylind ical co e in he
s abili y o mic o ubules in solu ion 关11兴.
We p esen he e a compu a ional s udy o he Kiha a luid
ocused on he s abili y o he iso opic 共diso de ed兲, nema ic
共o ien a ional o de 兲, and smec ic 共laye ing o de 兲liquid
c ys al phases a di e en empe a u es. To ou knowledge,
his is he i s ime ha he liquid c ys al beha io o his
model luid has been explo ed, since all o he p e ious in-
es iga ions ha e been es ic ed o mode a e molecula elon-
ga ions and/o packing ac ions co esponding o he mody-
namic s a es deep wi hin he iso opic phase o his sys em.
We hope ha he p esen esul s will aid in he assessmen o
he oles o en opy and o he dispe si e in e ac ions in he
s abili y o o de ed phases o eal mesogens and in o he
app op ia e s a egy o model hem.
II. METHOD
We ha e applied he iso he mal-isoba ic (NPT) ensemble
Mon e Ca lo 共MC兲 echnique o e alua e he modynamic and
s uc u al p ope ies o a Kiha a luid wi h elonga ion L*
⫽L/
␴
⫽5. Wi hin his model, he molecules a e ep esen ed
by igid cylinde s o diame e
␴
wi h semisphe ical caps
共i.e., a so-called sphe ocylinde 兲and he pai in e ac ion po-
en ial is gi en by
U共 ,⍀兲⫽
再
4
⑀
关共
␴
/dm兲12⫺共
␴
/dm兲6兴,dm⭐3
␴
0, dm⬎3
␴
.共1兲
The minimum dis ance be ween he cen al axes o he
wo p ola e molecules, dm⫽dm( ,⍀), is a unc ion o hei
ela i e o ien a ion ⍀, de e mined by h ee independen
angles, and o he dis ance be ween hei cen e s o mass, .
Fo he p esen simula ions, he in e ac ion was unca ed a a
dis ance dC⫽3
␴
, which co esponds o cen e -o -mass dis-
ances anging om ⫽3
␴
o ⫽8
␴
, depending on he ela-
i e pai o ien a ion.
The MC simula ions we e un o e a sys em o Np
⫽768 molecules a ou educed empe a u es, T*⫽kT/
⑀
⫽1.5, 2, 3, and 5 (kdeno es he Bol zmann cons an and
⑀
deno es he well dep h兲and s a ed om an equilib a ed high
p essu e s a e well inside he smec ic A egion o he phase
diag am. A e equilib a ion (1⫻106MC cycles兲and a e -
aging (3⫻105MC cycles兲, he las molecula a angemen is
s o ed and 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 an a emp o change he box ol-
ume. The usual pe iodic bounda y condi ions a e employed
and he simula ion box is aniso opic wi h ixed a ios
PHYSICAL REVIEW E 68, 011704 共2003兲
1063-651X/2003/68共1兲/011704共4兲/$20.00 ©2003 The Ame ican Physical Socie y68 011704-1
Ly/Lx⫽1.15 and Lz/Lx⫽1.94. As in ou p e ious s udy o
he SWSC and SRS luids 关5兴, he longe dimension along
he zdi ec ion is mean o inc ease he numbe o possible
smec ic laye s wi h espec o he cubic box o a gi en num-
be o molecules. Fu he me hodological de ails can be
ound in Re . 关5兴. The 共 i s -o de 兲liquid c ys alline ansi-
ions obse ed in he expansion o he luid a e cha ac e ized
by discon inui ies in he densi y, he nema ic o de pa am-
e e , he bond hexagonal o de pa ame e , and by sudden
quali a i e changes obse ed in he di e en co ela ion
unc ions 关3,5,12兴. No ace o he hexagonal o de ing o
long- ange co ela ions cha ac e is ic o smec ic-Bo solid-
like phases we e ound wi hin he scope o he p esen wo k.
III. RESULTS
The equa ions o s a e in educed uni s (P*⫽P
␴
3/kT s
␳
*⫽
␳␴
3) o he L*⫽5 Kiha a luid along he ou iso-
he ms in es iga ed a e ep esen ed in Fig. 1. In addi ion,
Fig. 2 and Table I summa ize he coexis ence p essu es and
densi ies o each o he liquid c ys al phase ansi ions ob-
se ed.
A he highes empe a u e in es iga ed, T*⫽5, he Ki-
ha a luid p esen s a s able smec ic-Aphase ha unde goes a
i s -o de ansi ion o a nema ic phase, as he sys em is
expanded o he p essu es below P*⫽1.8. A u he expan-
sion o he sys em induces a ansi ion o an iso opic phase
a a p essu e jus below P*⫽1.35. A quali a i ely simila
beha io is ound in he expansion a T*⫽3. Howe e , a
his lowe empe a u e he nema ic phase p esen s a educed
ange o s abili y, as he ansi ion om smec ic-A o nema ic
is delayed o lowe p essu es and densi ies in compa ison o
T*⫽5. Also no iceable is he inc eased di e ence in he
densi y o bo h phases a coexis ence. On he o he hand,
al hough he subsequen ansi ion o he iso opic phase oc-
cu s a a g ea e educed p essu e in he expansion along he
T*⫽3 iso he m han along he T*⫽5 one, he nema ic-
iso opic coexis ence densi ies a e e y simila a bo h em-
pe a u es.
A he lowe empe a u es, T*⫽1.5 and T*⫽2, he phase
diag am o he Kiha a luid changes quali a i ely. The mos
ele an ea u e is he absence o he nema ic phase and,
hence, he appea ance o a smec ic-A–iso opic ansi ion.
This kind o phase ansi ion ea u es a g ea e densi y asym-
me y in he coexis en phases and he e o e a s onge i s -
o de cha ac e han he smec ic-A–nema ic o iso opic-
nema ic ansi ions. In e es ingly, he densi y di e ence
be ween he smec ic-Aand iso opic phases a coexis ence is
g ea e a T*⫽1.5 han a T*⫽2. This beha io has been
also ound o he Gay-Be ne luid 关2,12,13兴and appea s o
be cha ac e is ic o sys ems wi h phase diag ams in which
he empe a u e o he I–N–Sm-A iple poin is g ea e han
he c i ical empe a u e, which o his luid is signi ican ly
smalle han T*⫽1关14兴. The coexis ence s a es de i ed
om ou simula ions o he L*⫽5 Kiha a luid allow o an
es ima ion o he iple-poin empe a u e o oughly T*
⬇2.5, as en a i ely indica ed in Fig. 2.
Figu e 1 compa es he equa ions o s a e and phase an-
si ions a T*⫽5 o he Kiha a luid and i s epulsi e coun-
e pa , he SRS luid 关5兴. The pu ely epulsi e SRS in e ac-
FIG. 1. Equa ions o s a e o he iso he ms T*⫽1.5, 2, 3, and 5
o he Kiha a luid wi h elonga ion L*⫽5. The e ical lines deno e
he coexis ence densi ies a he iso opic-nema ic (I-N), iso opic–
smec ic-A(I–Sm-A) o he nema ic–smec ic-A(N–Sm-A) ansi-
ions o he luid. The op panel shows he equa ion o s a e o he
SRS luid o same elonga ion a T*⫽5共Re . 关5兴兲 o compa ison.
FIG. 2. Summa y o he coexis ence densi ies a he iso opic-
nema ic (I-N), iso opic–smec ic-A(I–Sm-A), o nema ic–
smec ic-A(N–Sm-A) ansi ions ob ained in he p esen MC-NPT
simula ions o he Kiha a luid. The open squa e ep esen s a en a-
i e ex apola ion o he esul s o he I-N–Sm-A iple poin .
CUETOS e al. PHYSICAL REVIEW E 68, 011704 共2003兲
011704-2
ion po en ial cons i u es a gene aliza ion o he Weeks-
Chandle -Ande sen po en ial 关15兴and is buil om a shi ed
Kiha a po en ial unca ed a he well minimum. Due o he
upwa d shi o he epulsi e wall wi h espec o he Kiha a
one, he e ec i e size o he molecules a a gi en empe a-
u e is la ge in he SRS model 关4,5兴. Hence, he SRS luid is
expec ed o be less comp essible and o be a ec ed by en-
opic excluded olume e ec s o a g ea e ex en han he
Kiha a luid, which explains he shi o bo h he iso opic-
nema ic and he nema ic–smec ic-A ansi ions owa d lowe
densi ies in he SRS sys em wi h espec o he Kiha a one a
T*⫽5共see Fig. 1兲. The a gumen a ion is as well consis en
wi h he loca ion o hese liquid c ys al ansi ions a e en
lowe densi ies in he igid HSC and SWSC luids 关5兴,ascan
be seen in Table I.
These la e conside a ions indica e ha , in gene al e ms,
he main ‘‘d i ing o ce’’ o he iso opic-nema ic-smec ic
ansi ions in he luids s udied in his wo k is o en opic
na u e, he sys em ying o maximize he olume o phase
space accessible o he molecules. The ole o he ene ge ic
con ibu ion o he ee ene gy is a mo e sub le, ye ele an
one. The mos signi ican e ec o he pai in e ac ion po en-
ial ene gy is linked o en opy, since i is he na u e o he
sho - ange epulsi e in e ac ion ha imposes he e ec i e
size o he molecules and, hus, he olume excluded by he
co e o he molecules. Ano he impo an quali a i e aspec is
he ole played by he p esence o he a ac i e well in he
pai in e ac ion. A i s app oach o his ques ion is gi en in
ou p e ious wo k on he SWSC luid 关5兴, whe e i was
shown ha he squa e-well in e ac ion induces a delay in he
I–N–Sm-A ansi ions wi h espec o he pu ely epulsi e
HSC luid 共Table I兲. Thus, he e ec o he a ac i e well
a ec s he ansi ions in a simila way as a educ ion o he
e ec i e molecula size would do. Fu he mo e, he in lu-
ence o he a ac i e well should be mo e ele an a low
empe a u e, bu his aspec has no been much explo ed o
luids o sphe ocylind ical symme y 关5兴, e en hough i is
well cha ac e ized o he Gay-Be ne luid 关2,9,12,13兴.A
glance a he ene ge ic cons ains on he liquid c ys al phase
ansi ions as a unc ion o empe a u e is p o ided in Fig. 3,
which shows he e olu ion o he a e age po en ial ene gy
pe pa icle along he ou iso he ms o he Kiha a luid s ud-
ied in his wo k. In e es ingly, he quali a i e beha io o he
ene gy a T*⫽5 is qui e simila o he one ound o he SRS
luid a he same empe a u e 关5兴共also shown in Fig. 3 o
di ec compa ison兲. In bo h o hese cases, he ene gy g ows
mono onously wi h densi y, excep a he nema ic–smec ic-A
ansi ion whe e a sligh dec ease in ene gy is obse ed. This
quali a i e end changes a he lowe empe a u es, o
which a dec ease in ene gy is e en ually obse ed as he
densi y g ows wi hin he smec ic phase. Hence, i appea s
ha a low empe a u e he smec ic phase is a o ed by bo h
en opic and ene ge ic cons ains. In ac , he inc eased s a-
biliza ion o he smec ic phase opposes ha o he nema ic
phase, which con ibu es o he disappea ance o his la e
phase a T*⭐2.
TABLE I. Coexis ence p essu es and densi ies o he liquid c ys al phase ansi ions o he Kiha a luid and empe a u es in es iga ed in
his wo k. The abula ed densi ies a e hose o he consecu i e s a es in he MC simula ion belonging o di e en phases, whe eas he
coexis ence p essu e is ob ained om he mean alue o he p essu es o he wo bounda y s a es. Simila esul s o he HSC 关3兴, he SWSC,
and he SRS luids 关5兴a e also included o compa ison.
I–Sm-AI-NN–Sm-A
P*
␳
I
*
␳
SmA
*P*
␳
I
*
␳
N
*P*
␳
N
*
␳
SmA
*
T*⫽1.5 1.675 0.1117 0.1326
T*⫽2 1.675 0.1126 0.1288
T*⫽3 1.425 0.1069 0.1101 1.725 0.1193 0.1294
T*⫽5 1.325 0.1067 0.1101 1.775 0.1260 0.1348
SRS T*⫽5 1.175 0.0990 0.1027 1.625 0.1191 0.1250
SWSC T*⫽5 1.325 0.0968 0.0993 1.675 0.1099 0.1155
HSC 1.190 0.0914 0.0932 1.540 0.1061 0.1095
FIG. 3. Pai po en ial ene gy along he ou iso he ms s udied in
his wo k o he Kiha a luid. The pai ene gies o he T*⫽5
iso he m o he SRS luid om Re . 关5兴a e also included o com-
pa ison.
LIQUID CRYSTAL BEHAVIOR OF THE KIHARA FLUID PHYSICAL REVIEW E 68, 011704 共2003兲
011704-3
A u he end eadily no ed in Fig. 3 o he Kiha a luid
is he global shi o he ene gy pe pa icle owa d mo e
nega i e alues as he empe a u e is educed; ha is, he
a ac i e con ibu ion o he pai in e ac ion becomes p o-
g essi ely dominan when cooling down he sys em. The in-
spec ion o he adial dis ibu ion unc ions 共no shown兲in-
dica es ha , in ac , he a e age con inemen o neighbo ing
pa icles wi hin he a ac i e well is mo e e ec i e a low
empe a u e, whe eas a highe empe a u es he pai s show a
g ea e deg ee o pene a ion in o he sho - ange epulsi e
egion o he po en ial.
IV. SUMMARY AND CONCLUSIONS
I can be concluded ha he Kiha a po en ial cons i u es a
alid model o s udy he phenomenology o mesogenic lu-
ids. The po en ial in e es o he p esen esul s is wo old:
hey aid o he de elopmen o molecula modeling s a egies
o mesogens wi h speci ic phase diag ams and p o ide a
close asse ion o he ole played by he con igu a ional en-
opy and he dispe si e in e ac ions in he beha io o mol-
ecules in dense en i onmen s, ypical o liquid c ys als bu
also o he mac omolecula c owding condi ions o many
biological sys ems.
As al eady men ioned, he ad an age o he Kiha a model
is ha i inco po a es a mo e ealis ic desc ip ion o he mo-
lecula shape and o he dispe si e pai in e ac ions in com-
pa ison o o he ela ed models. As shown abo e, he pai
in e ac ion in he Kiha a luid b ings he liquid c ys al phases
o signi ican ly highe densi ies in compa ison o he HSC,
he SWSC, and he SRS luids, due o he combined e ec o
ene ge ic and excluded olume en opic cons ains 关5兴.In
pa icula , he a ac i e well has a bene icial in luence upon
he s abiliza ion o he smec ic phase wi h espec o he
nema ic one and con ibu es o he des abiliza ion o his
la e phase a low empe a u e.
The p esen esul s ha e se ed o es ima e, o he L*
⫽5 luid, he iso opic-nema ic–smec ic-A iple poin , a
he modynamical pa ame e ela i ely sca ce in he li e a u e
o any o he model luids commonly employed. A physical
in e p e a ion o he supe c i ical loca ion o he iple poin
ound he e o he Kiha a luid can be g asped om he be-
ha io o pep ide chains, which a e o en modelled by igid
long ods. Recen simula ions o polyalanine show ha he
c i ical empe a u e o his pep ide is well abo e i s decom-
posi ion empe a u e 关16兴. Since his is a common si ua ion
o p o eins 共wi h a e excep ions, such as collagen兲,i may
seem ha a supe c i ical iple poin , al hough ealis ic o
ideal s able molecules, would ake place a empe a u es un-
a ainable o pu e sys ems. Howe e , elonga ed pep ides
and, in pa icula , polyalanine unde go i s -o de ansi ions
in solu ion 关16兴. Ou simula ions a e hen ele an wi hin his
con ex , whe e he e ec o solu e concen a ion is analogous
o ha o an in e se empe a u e.
The mo e comple e pic u e o he in e play o en opy and
ene gy in he Kiha a luid would be desi able om an ex en-
sion o he p esen wo k o luids o a ying elonga ion. In
addi ion, one d awback o he Kiha a model 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 con-
s an . In eal sys ems, in con as , he dispe si e in e ac ions
a e ypically g ea e o aligned con igu a ions han o mis-
aligned con igu a ions o he molecula pai s. This kind o
beha io is quali a i ely ep oduced by models composed o
igid chains wi h Lenna d-Jones in e ac ion si es. Al hough
mul iple-si e models o his la e ype ha e been employed
o s udy liquid c ys al phases 关2,17兴, hei 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.
Ou g oup is a p esen wo king on an al e na i e app oach
based on a e inemen o he Kiha a model ha inco po a es
an o ien a ional dependence o he a ac i e well wi h he
aim o explo e he quali a i e e ec s ha his ea u e may
in oduce in he phase diag am o he luid.
ACKNOWLEDGMENT
We acknowledge suppo om he Spanish Di eccio
´n
Gene al de In es igacio
´n Cien ı
´ ica y Te
´cnica 共G an No.
BQU2001-3615-C02兲, Ins i u o de Salud Ca los III 共G an
No. 01/1664兲, and Plan Andaluz de In es igacio
´n共g oups
FQM-205, FQM-319兲.
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