Phase diagrams of polymer dispersed liquid crystalsAkihiko Matsuyama and Tadaya Kato

Citation: The Journal of Chemical Physics 108, 2067 (1998); doi: 10.1063/1.475585 View online: http://dx.doi.org/10.1063/1.475585 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/108/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Solvent effect on phase transition of lyotropic rigid-chain liquid crystal polymer studied by dissipative particledynamics J. Chem. Phys. 138, 024910 (2013); 10.1063/1.4774372 Phase diagrams of binary mixtures of liquid crystals and rodlike polymers in the presence of an external field J. Chem. Phys. 136, 224904 (2012); 10.1063/1.4728337 Phase behavior of lyotropic rigid-chain polymer liquid crystal studied by dissipative particle dynamics J. Chem. Phys. 135, 244901 (2011); 10.1063/1.3671451 Crystal-liquid crystal binary phase diagrams J. Chem. Phys. 124, 224902 (2006); 10.1063/1.2200688 Orientations and phase transitions in liquid crystals consisting of short linear polymer chains J. Chem. Phys. 114, 2466 (2001); 10.1063/1.1329882

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Phase diagrams of polymer dispersed liquid crystalsAkihiko Matsuyamaa) and Tadaya KatoDepartment of Chemistry for Materials, Faculty of Engineering, Mie University, Tsu Mie 514, Japan

~Received 2 September 1997; accepted 22 October 1997!

A simple model is introduced to describe liquid crystal transitions and phase separations in binarymixtures of a flexible polymer and a liquid crystal. By combining the McMillan theory for thesmectic A phase of liquid crystals with the Flory–Huggins theory for the isotropic mixing of twocomponents, we examine binodal and spinodal lines on the temperature-concentration plane. Wepredict the appearance of phase separations such as the smectic A-nematic, smectic A-isotropic, andsmectic A-nematic-isotropic phase separations. We also find a tricritical point caused by theinterference between a second-order liquid crystal transition and a phase separation. ©1998American Institute of Physics.@S0021-9606~98!50905-1#

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I. INTRODUCTION

Binary mixtures of a solute and a solvent molecule ehibit phase separations, critical phenomena, and spinodacompositions depending on temperature and concentraIn the case of mixtures of a polymer and a liquid crys~so-calledpolymer dispersed liquid crystals!,1–3 we can ex-pect the co-occurrence of macroscopic phase separationliquid crystalline ordering such as a smectic and a nemphase. In these systems, the orientational order and trational order parameters of liquid crystals as well as the ccentration are closely related to the phase separation.example, in binary mixtures of a flexible polymer and a liuid crystal which exhibits a first-order nematic-isotropphase transition~NIT!, the nematic-isotropic phase separtion and the isotropic-isotropic one with an upper criticsolution point have been observed.4–7 These nematic-isotropic phase separations have been investigatedexperimentally4–9 and theoretically.10–23

Generally, pure liquid crystalline materials have an istropic phase at high temperature. At lower temperature somaterials exhibit a nematic phase where the center of mastill randomly placed and the long axes line up parallel tpreferred direction. Further decreasing temperature we ha smectic A phase in which the long axes line up parallea preferred direction and the center of mass sits on a pperpendicular to the preferred direction. Some years aKobayashi17 and McMillan18,19 predicted, on the basis ofmean field theory, that the smectic A-nematic phase tration ~SNT! in pure liquid crystals should be second orderTSN

+ /TNI+ ,0.87 and first order for larger values ofTSN

+ /TNI+ ,

whereTSN+ andTNI

+ show the SNT and NIT temperature ofpure liquid crystal, respectively. Their theories can qualtively describe the observed phase transitions in pure cponents of liquid crystals such as the alkylcyanobiphe~nCB! homologous series.24,25 For example, pure 8CB itselhas the second-order SNT and pure 10CB exhibits the fiorder smectic A-isotropic phase transition~SIT! as tempera-ture is decreased.24 When such a liquid crystal is mixed wita flexible polymer, we can expect phase separations and

a!Author to whom correspondence should be addressed.

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cal phenomena, depending on temperatureconcentration.20

In this paper, we propose a molecular theory to descrphase behaviors in the binary mixtures of a flexible polymand a liquid crystal which exhibits the smectic A, nematand isotropic phases. The current understanding of the pbehaviors is based on the co-occurrence of liquid crytransitions and phase separations.2,3 By combining the mo-lecular theory17–19,26for the smectic A phase of liquid crystals with the Flory–Huggins theory27 for the mixing of iso-tropic liquids, we calculate the binodal and spinodal linesthe temperature-concentration plane. Our approach issame as that of Kyuet al.21–23 They calculated the phasdiagrams in mixtures of polymers and liquid crystals whiexhibit a first-order SNT and SIT. We predict the appearaof spinodal lines in a smectic A phase and new critical pnomena, such as a tricritical point and a triple point, cauby the interference between phase separations and licrystal phase transitions.

II. THEORY

Consider binary mixtures of a flexible polymer andliquid crystal. We here assume that the polymer chainssufficiently flexible. LetNr and N be the number of liquidcrystals and polymer chains, respectively. The free energour system is given by

F5Fmix1Fani , ~2.1!

where the first term shows the free energy of mixing of istropic phases. According to the Flory–Huggins theorypolymer blends, it is given by27

bFmix /Nt5~12f!

nrln~12f!1

f

nln f1x~12f!f,

~2.2!

whereb[1/kBT; T is the absolute temperature andkB is theBoltzmann constant. Then is the number of segments onpolymer chain andnr the axial ratio of a liquid crystal. Thevolume fraction f of polymer segments is given bf5nN/Nt , whereNt5nrNr1nN is the total number of lat-

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2068 A. Matsuyama and T. Kato: Polymer dispersed liquid crystals

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tice sites of our systems. Thex(}1/kBT) is the isotropicinteraction~Flory–Huggins! parameter between polymer anliquid crystal molecules.

The second term in Eq.~2.1! shows the free energy fothe anisotropic ordering of liquid crystals. To describesmectic A-nematic-isotropic phase transition, we useKobayashi–McMillan model for the smectic A phase of themotropic liquid crystals.17,18The combination of the free energy of Flory–Huggins and the Maier–Saupe free energy28,29

has been used to describe the nematic-isotropic phaseration in polymer dispersed liquid crystals.10,14,15On the ba-sis of the Kobayashi–McMillan theory,17,18 the free energyFani of the anisotroic ordering is given by

bFani /Nt512f

nrE

0

1E0

1

f ~r ,cosu!ln f ~r ,cosu!drd~cosu!

21

2xa~12f!2~S21gh21ds2!, ~2.3!

where u is the angle between the long axis of the liqucrystals and the preferred axis~thez direction!, r[z/ d̃ , d̃ isthe average distance between smectic layers, andf (r ,cosu)is the distribution function of liquid crystals. Thexa

(}1/kBT) shows the orientational dependent~Maier–Saupe!interaction between liquid crystals and determinesnematic-isotropic transition. Theg shows the dimensionlesinteraction strength for a smectic A phase. According toMcMillan theory the parameter g is given byg52 exp@2(r0 /d̃)2#, which can vary between 0 and 2, anincreases with increasing chain length of alkyl end-chainsliquid crystal molecules.18 The smectic condensation is mofavored for larger values ofg. Theg andd are the numericaparameters. Equation~2.3! has three order parameters;S, h,and s. The order parameterS describes the orientationaorder and is given by

S5E E P2~cosu! f ~r ,cosu!drd~cosu!, ~2.4!

where P2(cosu)[(3/2)(cos2 u21/3). The h describes thecoupled order parameter between the orientational and trlational orders:

h5E E P2~cosu!cos~2pr ! f ~r ,cosu!drd~cosu!

~2.5!

and the translational order parameters is given by

s5E E cos~2pr ! f ~r ,cosu!drd~cosu!. ~2.6!

Equations~2.4!, ~2.5!, and~2.6! exhibit three phases of liquidcrystals: an isotropic phase withS5h5s50; a nematicphase withSÞ0, h50, s50; and a smectic A phase witSÞ0, hÞ0, sÞ0.

In the thermal equilibrium, the distribution functiof (r ,cosu) is determined by minimizing the free energy~2.3!with respect to this function: (]F/] f (r ,u))T,f50. We thenobtain

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f

s-

f ~r ,cosu!51

Zexp@nrxa~12f!A~r ,u!#, ~2.7!

whereA(r ,u) is defined as

A~r ,u![SP2~cosu!1ghP2~cosu!cos~2pr !

1ds cos~2pr !. ~2.8!

The constantZ is determined by the normalization conditio

E E f ~r ,cosu!drd~cosu!51 ~2.9!

as

Z5E E exp@nrxa~12f!A~r ,u!#drd~cosu!. ~2.10!

Substituting Eq.~2.7! into Eq. ~2.3!, we obtain the free en-ergy of our systems:

bF/Nt5~12f!

nrln~12f!1

f

nln f1x~12f!f1

1

2xa

3~12f!2~S21gh21ds2!2~12f!

nrln Z.

~2.11!

When f50, Eq. ~2.11! results in the mean field theory fopure liquid crystals.17–19,26From Eqs.~2.4! and~2.5!, we cansolve self-consistently for the three order parametersS, h,and s as a function of temperature and concentration. Tchemical potentials are given bymp(f)5(]F/]N)Nr ,T forthe polymer molecules andm r(f)5(]F/]Nr)N,T for the liq-uid crystals as follows:

bmp~f!5 ln f1S 12n

nrD ~12f!1nx~12f!2,

1n

2xa~12f!2~S21gh21ds2!, ~2.12!

bm r~f!5 ln~12f!1S 12nr

n Df1nrxf2

1nr

2xa~12f!2~S21gh21ds2!2 ln Z.

~2.13!

In the isotropic phase, the chemical potentials are given

bmpi 5 ln f1S 12

n

nrD ~12f!1nx~12f!2, ~2.14!

bm ri 5 ln~12f!1S 12

nr

n Df1nrxf2. ~2.15!

The spinodal line, which separates the metastable frthe unstable state, is obtained by the condition]m r /]f50.The coexistence~binodal! curves of the phase separations aderived by the coupled equations,m r(f8)5m r(f9) andmp(f8)5mp(f9), wheref8 and f9 are the polymer con-centrations in the lower and the higher concentration pharespectively.

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2069A. Matsuyama and T. Kato: Polymer dispersed liquid crystals

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III. PHASE DIAGRAMS

The smectic A-nematic-isotropic phase transitionspure liquid crystals have been discussed by several authMcMillan calculated the phase diagram of pure liquid crytals for various values ofg with d50.18 Lee et al. also cal-culated the phase diagrams withdÞ0.26 The results arequalitatively similar to those of McMillan.18 In the followingcalculations, we neglect for simplicity the translational ordparameters (d50) and so the phase diagrams for pure luid crystals (f50) result in those of the McMillan model.18

We then have five parameters characterizing our stems:n, the number of segments on a flexible polymer;nr ,the axial ratio of a liquid crystal;xa5Ua /kBT, theorientational-dependent interaction parameter between liqcrystals; g, the smectic interaction parameter; ax5U0 /kBT, the isotropic interaction~Flory–Huggins! pa-rameter between a polymer and a liquid crystal. For ourmerical calculations, we here define the dimensionless natic interaction parametera[xa /x.

The nematic phase appears atnrxa(12f)54.54.29 Wethen obtain the nematic-isotropic transition~NIT! tempera-ture TNI(f) as a function of the polymer concentrationf:

TNI~f!5nrUa~12f!

4.54kB. ~3.1!

When the temperature is belowTNI(f), the nematic phase istable and whenT.TNI(f) the isotropic phase is stable. Foa pure liquid crystal, the NIT temperatureTNI+ is given by

TNI+ 5

nrUa

4.54kB, ~3.2!

and so the NIT line is given byTNI(f)/TNI+ 512f. The

smectic A-nematic transition~SNT! line on the temperatureconcentration plane can be derived by numerically solvthe self-consistent equations forS and h. The smectic A-nematic-isotropic transitions of pure liquid crystals habeen discussed by McMillan for various values ofg.18 In thefollowing, we have fixedn54, nr52 for a typical example.

Figure 1 shows the order parametersS and h plottedagainst the polymer concentrationf for various values ofgwith a58: ~a! g50.7, T/TNI

+ 50.83; ~b! g50.85, T/TNI+

50.9; and~c! g51.1, T/TSI+ 50.96. In Fig. 1~a!, the smectic

order parameterh decreases continuously to 0, as the pomer concentration increases and the SNT is second-oThe nematic order parameterS drops discontinuously to 0, athe concentration increases and the NIT is first-order. In F1~b!, the smectic order parameterh drops discontinuously to0, as the concentration increases and the SNT is first-orIn Fig. 1~c!, the nematic phase (SÞ0, h50) disappears andwe only have the first-order SIT. The NIT and SNT linesthe temperature-concentration plane are depicted in Fig.

Figure 2 shows the phase diagram on the temperatconcentration plane fora58. The smectic interaction parameter g is changed from Figs. 2~a!–2~c!: ~a! g50.7; ~b!g50.85; and~c! g51.1. The solid curve refers to the binodal and the dashed-and dotted line shows the spinodal.short-dashed line shows the first-order NIT and SNT lin

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and the dotted line corresponds to the second-order SThe shaded area shows two- or three-phase regions. Thephase region of an isotropic phase is denoted by I, thatnematic phase N, and that of a smectic S. The nematic sodal line merges to both the NIT and SNT lines. The covergence of the nematic spinodal and the NIT lines has bfounded by Shen and Kyu.14 As shown in Fig. 3, in the phasdiagram, we have three different metastable regions: antropic metastable~Im!; a nematic metastable~Nm!; and asmectic metastable~Sm!. We also have two unstable regiona nematic unstable~Nu! and a smectic unstable~Su!. Theseregions play important roles in the dynamics of phase serations and pattern formations. In Figs. 2~a! and 2~b! thetemperature is normalized by the NIT temperatureTNI

+ of thepure liquid crystal. In Fig. 2~c! the temperature is normalizeby the smectic-isotropic transition temperatureTSI

+ of thepure liquid crystal.

In Fig. 2~a! we find a triple point~three-phase equilib-rium! where the smectic A, nematic, and isotropic phasessimultaneously coexist. At the higher temperature side oftriple point, we have the two-phase coexistence betweennematic and isotropic phases~N1I!, which was observed inthe mixtures of polystylene~PS! and (p-ethoxybenzylidene!-p-n-butylaniline~EBBA!.4,5 We also find the two-phase coexistence between the smectic A and nematic pha~S1N!.22 At the lower temperature side of the triple pointhe broad biphasic region between the smectic A and isopic phases~S1I! appears. Wheng50.7, the SNT is secondorder. Therefore the point where the second-order SNTmeets the binodal at the top of the biphasic region~S1N! isa tricritical point ~TCP!. This TCP resembles thel-point inthe mixture of liquid3He and liquid4He.30

FIG. 1. Order parametersS andh plotted against the polymer concentratiof for various values ofg: ~a! g50.7, T/TNI

+ 50.83; ~b! g50.85, T/TNI+

50.9; ~c! g51.1, T/TSI+ 50.96.

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2070 A. Matsuyama and T. Kato: Polymer dispersed liquid crystals

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As the smectic parameterg increases, the SNT temperature TSN

+ of the pure liquid crystal is shifted to the highetemperature side as shown in Fig. 2~b!. The TCP movesupwards and disappears. The SNT becomes first order.@seeFig. 1~b!#. At the higher temperature side of the triple poinwe have two-phase regions~N1I! and ~S1N!. At the lowertemperature side of the triple point, the broad biphasic reg~S1I! appears. Further increasing the smectic interactionrameterg, the nematic phase disappears and we only hthe biphasic region~S1I! as shown in Fig. 2~c!. The smectic

FIG. 2. Phase diagram on the temperature-concentration plane fora58.The smectic interaction parameterg is changed from~a! to ~c!: ~a! g50.7;~b! g50.85; ~c! g51.1. The horizontal lines show the regions of two-three-phase coexistence. The one-phase region of an isotropic phasenoted by I, that of a nematic phase N, and that of a smectic S. See texdetails.

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phase consists almost entirely of liquid crystals and theexisting isotropic phase consists of liquid crystals and somolecules. The regions of the Sm, Su, and Im appear onphase diagram.

Figure 3 shows the chemical potentialm r of the liquidcrystal plotted againstf for g50.7. The reduced temperaturetNI[T/TNI

+ is changed from 0.83 to 0.87. The chemicpotential has a kink at the second-order SNT concentrashown by open triangles and jumps at the first-order Nconcentration. The closed circles show the spinodal pointthe smectic and nematic phases. The region]m r /]f.0(]m r /]f,0) corresponds to the unstable~stable! region. Asthe temperature is increased, the concentration differencetween the smectic spinodal and SNT points decreasesbecomes 0 at the TCP shown by the open circle.@see Fig.2~a!#. These analyses suggest that the interplay ofsecond-order SNT and the smectic A-nematic phase seption in polymer dispersed liquid crystals plays importaroles in the field of phase separations and critical phenoena.

Figure 4 shows the free energy plotted against the ccentrationf for g50.85 anda58. The temperaturetNI ischanged. The corresponding phase diagram is 2~b!. WhentNI50.96, the free energy has a kink at the first-order N

de-for

FIG. 3. Chemical potentialm r of the liquid crystals plotted againstf forg50.7. The reduced temperatureT/TNI

+ is changed from 0.83 to 0.87.

FIG. 4. Free energy plotted against the concentrationf for g50.85 anda58.

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2071A. Matsuyama and T. Kato: Polymer dispersed liquid crystals

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concentration. The two-phase equilibrium can also be eslished by means of a double tangent method where the elibrium volume fractions fall on the same tangent line of tfree energy curve. WhentNI50.92, the free energy curvhas kinks at the first-order SNT and NIT concentration.

Figure 5 shows the phase diagram forg50.85, a58,and n520. As the numbern of segments on the polymechain is increased, the temperature of the triple point~S1N1I! is increased and approaches the SNT temperature opure liquid crystal.@see Fig. 2~b!.# The width of the two-phase region increases with increasingn.

While decreasing the nematic interaction parametera,the unfavorable interaction (x) parameter between a polymeand a liquid crystal becomes dominant in the free ene~2.11!. Figure 6 shows the phase diagram on ttemperature-concentration plane fora53. The smectic inter-action parameterg is changed. We find the coexisting of twisotropic liquid phases (L11L2) with an upper critical solu-tion temperature~UCST! which has been observed for Pwith 7CB. The critical solution point (fc , Tc) in the isotro-pic phase is given by27

fc5Anr

Anr1An, ~3.3!

Tc52nnr

~Anr1An!2S U0

kBD . ~3.4!

With increasing the lengthn, the critical concentrationfc isdecreased and the temperatureTc is increased. The width othe two-phase region increases with increasing the molecweight of the polymer.15 In Fig. 6~a!, we find two triplepoints. The upper triple point corresponds to the smecticnematic-isotropic phase separation and the lower triple pcorresponds to the coexisting of a smectic A phase andisotropic phases. The later triple point also appears in6~b!.

As shown in Figs. 2 and 6, there are two types of phdiagrams. The conditions for the existence of these phdiagrams are obtained by comparing the NIT tempera

FIG. 5. Phase diagram on the temperature-concentration plane forg50.85,a58, andn520. As the lengthn of the polymer chain is increased, thtemperature of the triple point~S1N1I! is increased and approaches to tSNT temperature of the pure liquid crystal.

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(TNI) @Eq. ~3.2!# and the UCST (Tc) @Eq. ~3.4!# in the iso-tropic phase. From Eqs.~3.1! and ~3.4!, the nematic param-etera is given by

a5Ua

U05acrS TNI~fc!

TcD , ~3.5!

where

acr[9.08An

Anr1An. ~3.6!

FIG. 6. Phase diagram on the temperature-concentration plane fora53.The smectic interaction parameterg is changed from~a! to ~b!: ~a! g50.85;~b! g51.1.

FIG. 7. Dependence of the upper critical solution temperatureTc in theisotropic phase on the nematic interaction parametera. Whena.acr , theisotropic-isotropic phase separation disappears.

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2072 A. Matsuyama and T. Kato: Polymer dispersed liquid crystals

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As shown in Fig. 7, fora,acr , or TNI(fc),Tc , we havethe coexisting of two isotropic liquid phases (L11L2) withan UCST. Whena.acr , or TNI(fc).Tc , the (L11L2)phase separation disappears and we only have the coexof a liquid crystalline phase and an isotropic phase. Figurshows the value ofacr plotted against log(n) for nr52. Inregion~1!, we have the phase diagrams as shown in Fig. 2region ~2!, the phase diagrams with an UCST appears~seeFig. 6!. As n increases the value ofacr approaches 9.08. Thtype of phase diagram is characterized by the nematic inaction parametera.

IV. CONCLUSION

We have presented a mean field theory to describe pseparations in polymer dispersed liquid crystals. The thepredicts phase separations; the smectic-nematic, smeisotropic, smectic-nematic-isotropic phase separations,new unstable regions in smectic A coexistence regions whplay important roles in the dynamics of phase separatioWe also find a tricritical point where the second-order Sline meets the binodal of the biphasic region~S1N!. Thesephase diagrams appeared as the result of the co-occurrbetween the two intrinsically different phase transitions; luid crystal transitions and phase separations.

FIG. 8. Value ofacr plotted against log(n) for nr52.

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ACKNOWLEDGMENT

A. M. thanks Professor T. Kyu for sending his recepapers and for helpful suggestions.

1Liquid Crystalline and Mesomorphic Polymers, edited by V. P. Shibaevand L. Lam~Springer-Verlag, New York, 1993!.

2P. S. Drzaic,Liquid Crystal Dispersions~World Scientific, Singapore,1995!.

3Liquid Crystals in Complex Geometries, edited by G. P. Crawford and SZumer ~Taylor & Francis, London, 1996!.

4B. Kronberg, I. Bassignana, and D. Patterson, J. Phys. Chem.82, 1714~1978!.

5A. Dubaut, C. Casagrande, M. Veyssie, and B. Deloche, Phys. Rev.45, 1645~1980!.

6H. Orendi and M. Ballauff, Liq. Cryst.6, 497 ~1989!.7W. Ahn, C. Y. Kim, H. Kim, and S. C. Kim, Macromolecules25, 5002~1992!.

8Y. Dormoy, J. L. Gallani, and P. Martinoty, Mol. Cryst. Liq. Cryst.170,165 ~1989!.

9G. W. Smith, Phys. Rev. Lett.70, 198 ~1993!.10F. Brochard, J. Jouffroy, and P. Levinson, J. Phys.~Paris! 45, 1125~1984!.11M. Ballauff, Mol. Cryst. Liq. Cryst.136, 175 ~1986!.12R. Holyst and M. Schick, J. Chem. Phys.96, 721 ~1992!.13W. K. Kim and T. Kyu, Mol. Cryst. Liq. Cryst.250, 131 ~1994!.14C. Shen and T. Kyu, J. Chem. Phys.102, 556 ~1995!.15A. Matsuyama and T. Kato, J. Chem. Phys.105, 1654~1996!.16P. I. C. Teixeira and B. M. Mulder, J. Chem. Phys.105, 10145~1996!.17K. K. Kobayashi, J. Phys. Soc. Jpn.29, 1010 ~1970!; Mol. Cryst. Liq.

Cryst. 13, 137 ~1971!.18W. L. McMillan, Phys. Rev. A4, 1238~1971!.19W. L. McMillan, Phys. Rev. A6, 936 ~1972!.20E. P. Sokolova, Mol. Cryst. Liq. Cryst.192, 179 ~1990!.21T. Kyu and H. W. Chiu, Phys. Rev. E53, 3618~1996!.22T. Kyu, H. W. Chiu, and T. Kajiyama, Phys. Rev. E55, 7105~1997!.23H. W. Chiu and T. Kyu, J. Chem. Phys.~to be published!.24M. A. Anisimov, Mol. Cryst. Liq. Cryst.162, 1 ~1988!, and references

therein.25H. Marynissen, J. Thoen, and W. V. Dael, Mol. Cryst. Liq. Cryst.124,

195 ~1985!.26F. T. Lee, H. T. Tan, Y. M. Shih, and C. W. Woo, Phys. Rev. Lett.31,

1117 ~1973!.27P. J. Flory,Principles of Polymer Chemistry~Cornell University, Ithaca,

1953!.28W. Maier and A. Saupe, Z. Naturforsch.14, 882 ~1959!.29P. G. de Gennes and J. Prost,The Physics of Liquid Crystals, 2nd ed.

~Oxford, London, 1993!.30C. M. Knobler and R. L. Scott, inPhase Transitions and Critical Phe

nomena, edited by C. Domb and J. L. Lebowitz~Academic, New York,1984!, Vol. 9.

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