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Molecular structure and reentrant phases in polar liquid crystals

J. O. Indekeu (1,2) and A. Nihat Berker (1)

(1) Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.

(2, *) Laboratorium voor Molekuulfysika, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3030Leuven, Belgium

(Requ le 11 septembre 1987, accepte le 23 octobre 1987)

Résumé. 2014 On emploie le modèle de gaz de spins des cristaux liquides pour décrire et expliquer les réentréessimples, doubles et quadruples de phases nématiques et smectiques, ainsi que les réentrées sous la phasemonocouche smectique A1, qui sont toutes observées expérimentalement. On prédit également de nouvellesréentrées sextuples et octuples le long de la séquence nématique-smectique Ad-nématique-smectiqueAd-nématique-smectique Acnématique-smectique Ad(-nématique-smectique A1). Les réentrées multiples sonttrès sensibles à la structure moléculaire et, en particulier, à la longueur de la chaîne de la molécule. Onexamine l’information microscopique qu’on peut tirer de la statistique des configurations microscopiques encomparant les poids statistiques des différentes classes de configurations. Il y a deux mécanismes distincts pourla réentrée qui font appel, d’une part, à la frustration de dipôles microscopiques et, d’autre part, à unecompétition d’ordres locaux ferroélectriques et antiferroélectriques. La frustration est relâchée par des

perméations atomiques et librationnelles qui aboutissent à des réentrées smectiques distinctes. On prend lalimite du continuum de la perméation librationnelle. On examine aussi le rôle subtil joué par les dimères demolécules, ainsi que le concept d’une longueur de paire dipolaire.

Abstract. 2014 The spin-gas model of liquid crystals is employed to describe and explain single, double andquadruple reentrances of nematic and smectic phases, as well as reentrances below the monolayer smecticA1 phase, all of which are observed experimentally. New sextuple (and octuple) reentrances are predicted inthe sequence nematic-smectic Ad-nematic-smectic Ad-nematic-smectic A1-nematic-smectic Ad(-nematic-smec-tic A1), as temperature is lowered. The multiple reentrances are very sensitive to molecular structure, inparticular to the molecular tail length. The microscopic information available from the statistics of microscopicconfigurations is scrutinized. Statistical weights of classes of positional configurations are monitored. Twodistinct mechanisms of nematic reentrance feature : microscopic frustration of dipoles and competing localantiferroelectric and ferroelectric orders. Frustration is relieved by atomic and librational permeations, causingdistinct smectic reentrances. The continuum limit of librational permeation is taken. The subtle role played bymolecular dimers is outlined and the concept of a dipolar-pair length is critically examined.

J. Phys. France 49 (1988) 353-362 FTVRIER 1988,

Classification

Physics Abstracts61.30B - 64.70M - 77.80B - 05.70F

1. Introduction.

In condensed matter physics, phenomena of « reen-trance » of more ordered phases when cooling orcompressing from less ordered phases can generallybe argued to occur from competition of tendenciestowards different orderings. At a phenomenologicallevel a Landau theory and the associated free-energyoptimization program are analysed, featuring a cho-sen set of order parameters. At a microscopic levelone attempts to incorporate relevant molecular

(*) Permanent address.

characteristics and intermolecular interactions in a

statistical mechanical model. From there a classifi-

cation of thermodynamic phases as well as an

explanation of their stability is derived by monitoringthe statistical probabilities of sets of configurations.Recently, a molecular theory has been developed [1-6] which exemplifies the potentialities of a micro-scopic approach in accounting for the observationsof multiply reentrant nematic and smectic phases inpolar liquid crystals [7-16].The thermodynamic phases and phase transitions

in these systems are very sensitive to details of themolecular structure. In some liquid crystals of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004902035300

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molecules with strongly dipolar heads (-CN or

-N02) nematic phases (orientational order) and

smectic phases (orientational and partial positionalorder) reenter. Experimentally observed reentrancestypically involve nematic (N), monolayer smectic A(A1 ), and interdigitated partial bilayer smectic A(Aa ) phases. The reentrance sequences include« single reentrance » [7] (N-Ad-N), « double reen-trance » [8] (N-Ad-N-Al), « quadruple reentrance »[9-12] (N-Ad-N-Ad-N-Al), and « reentrance below

Al» [13-16] (N-A1-N-Ad(-A1». On the theoreticalside all of these reentrances have been obtained with

the spin-gas model of liquid crystals [1-6]. In figures1-3 representative theoretical phase diagrams are

shown together with their experimental counter-

parts. The theory further predicts sextuple and

octuple reentrances for a very narrow range of

molecular constants [6]. To our knowledge, thesehave not yet been seen experimentally.

Fig. 1. - (a) Doubly reentrant phase diagram obtained in the spin-gas model for B/A = 1.5, n = 5, m = 3 and8 = 0.01 f /n. (b) Experimental doubly reentrant pressure-temperature phase diagram for the compound 9 OBCAB,taken from reference [30].

Fig. 2. - (a) Quadruply reentrant phase diagram obtained in the spin-gas model for B/A = 1.455, n = 5,m = 3, 5 = 0.015 f In. For comparison with experiment, a tentative pressure scale is added. (b) Experimental pressure-temperature phase diagram of DB90NO2, displaying quadruple reentrance. The pressure scale marked at the top is fordata points shown as triangles. The bottom scale is for data shown as circles. This figure is taken from reference [11].(c) Experimentally observed quadruple reentrance, from reference [10], of DB90NO2 and its mixtures with homologs ofconcentration x.

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Fig. 3. - (a) Reentrances below A, obtained in the spin-gas model for B/A = 1.451, n = 5, m = 5 and 5 =

0.0087 f In. For comparison with experiment, the pressure and/or concentration variable is represented byf la instead of a If. At low temperatures the nematic phase becomes exceedingly narrow before terminating at a zero-temperature bicritical point. (b) Experimental reentrances below At observed in mixtures. The variable x denotes theconcentration of a terminal nonpolar compound in a ternary mixture. The value x = 0 corresponds to an equimolarmixture of the polar compounds DB8NO2 and DBgN02. This figure is taken from references [14] and [15].

The purpose of this paper is to scrutinize for thefirst time the microscopic information which thespin-gas model provides for tracing reentrance

mechanisms and explaining the intricacies of the

phase diagrams.

2. The spin-gas model.

A schematic representation of the polar moleculesunder consideration is given in figure 4. The molecu-lar length f is of the order of 30 A and the width isabout 5 A. A longitudinal dipolar head, a rigidaromatic core, and a semi-flexible aliphatic tail aredistinguished. The intermolecular pair potential em-bodies steric hindrance, van der Waals attraction,and dipole-dipole forces. The latter are importantsince reentrant phases are almost exclusively seen inpolar compounds. In these systems dipolar frus-tration plays a key role.

Fig. 4. - Schematic representation of molecules of, e.g.,the compound DB90NO,. Arrows denote dipolar heads,rectangles represent aromatic cores, open circles indicateoxygen atoms, and closed circles mark the positions ofcarbon atoms in an all-trans configuration of the aliphatictail. The molecular pairs have dominant tail-tail attraction(a) or steric hindrance (b).

The pair potential for nearest-neighbour moleculesis taken as

where ri is the position of the dipole of molecule i,91 is the unit vector describing the dipolar orien-tation, and rl2 = rl - r2 and r12 = r121 I r121. . For

purely dipolar forces, A = B. The alternate possibili-ties of dominant tail-tail attraction, figure 4(a), orsteric repulsion, figure 4(b), are incorporated in theratio B/A : B A for net hindrance (favoring theantiferroelectric term) and B > A for net entangle-ment. (The inherent dipolar potential can be au-gmented more realistically to reflect other molecularinteractions, leading to quantitative changes in theresults.) Note that in this context « symmetric »molecules with a central dipole between two identicalmolecular parts on either side are described byA = B. Screening suppresses interactions betweenfurther-than-nearest neighbours. Fluctuations tow-ard the isotropic phase are ignored, namely themolecules are taken aligned along the z direction :91 = ± i, or simply si = ± 1.

In the close-packing of a liquid, the potential inequation (1) inherently causes frustration due to

substantial near-cancellations of forces between amolecule and its neighbours. For example, when twomolecules form an antiparallel pair, figure 4(b),constituting a molecular dimer, an almost zero netforce is felt by a neighbouring third molecule becauseof net dipolar frustration. The third molecule then

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almost freely permeates (i.e. positionally fluctuatesalong the z direction), as molecules do in nematicphases. On the other hand, three molecules canform a correlated triplet with net dipolar short-rangeantiferroelectric or ferroelectric order. These tripletsare energetically stable against small mutual permea-tions and may or may not, depending on entropicqualities, conglutinate laterally to form a percolatingnetwork or molecular polymer. If a polymer is

formed, enhanced longitudinal viscosity hinders per-meation, and molecules tend to « lock in » intosmectic layers. In the spin-gas model the statistics ofthese molecular configurations are gathered andtheir dependences on temperature, pressure andmolecular constants are derived.

Molecular tails play a crucial role. Apart from thefree energy of their entanglement embodied in

B/A, their lengthwise corrugation is essential, creat-ing n energetically preferred positions (« notches »)of mutual permeation for a nearest-neighbour pair(Fig. 5). Additionally to these discrete « atomic »

Fig. 5. - Examples of configurations of a triplet of

molecules. (a) The atomic permeation positions of thedipolar heads. Librational permeation positions are illus-trated only in the upper right-hand corner. (b) Configu-ration with all three dipoles at the same notch. If all threedipoles are at the same subnotch (bl) frustration results asa zero net force is felt by either of dipoles 1 or 3.

Librational permeation (b2) relieves frustration and allowslocal antiferroelectric order. (c) If dipoles 1 and 2 are atthe same subnotch (cl) frustration results : a zero net forceis felt by dipole 3. Frustration is relieved by librationalpermeation (c2) allowing local antiferroelectric order.

Frustration is relieved by atomic permeation allowing localantiferroelectric order (d) or local ferroelectric order (e).

permeations on a length scale f In (a few A), smalloscillations occur on a length scale 4 « f In. These« librational » permeations can be calculationallyapproximated to occur in m discrete subnotches,separated by 8, within each notch (Fig. 5). Animportant issue addressed in detail in this paper isthe physical continuum limit m - oo with A constant.

3. The prefacing transformation.

Within a three-dimensional system, a reference layerof molecules can be identified by the coordinateszi being within the interval zo ± f/2. The smallestunit of possible frustration is a triplet of molecules inthe reference layer. First, the strengths are obtainedfor effective orientational couplings between the

spins si by thermally averaging over the positionalfluctuations ri. We found that the positional fluctua-tions that underly reentrance are permeations, themolecular motions in the z direction. Possible perme-ations are shown in figure 5(a). Lateral displace-ments (normal to z) do not change the results

qualitatively [1] and have generally been left out forcalculational ease. The theory thus predicts that thedegree of in-plane (xy ) order is not directly related tosmectic order along z. This physical characteristic hasbeen confirmed by subsequent X-ray scatteringexperiments [17]. Accordingly the calculations pro-ceed by considering three nearest-neighbourmolecules on a triangular prism of side a (Fig. 5),where a is the average lateral separation of a

nearest-neighbour pair. After one molecule is

positionally restricted to notch 3 for n = 5, for

example, there are n2 m3 positional configurationsper orientational configuration {Si}’With a prefacing transformation, the average-

strongest (KS ), average-intermediate (KI ), and av-erage-weakest (Kw) antiferroelectric couplings areextracted (Ks -- KI , Kw) :

where the molecule labels (12), (23), and (31)respectively span the strongest, intermediate, andweakest antiferroelectric couplings specific to eachpositional configuration. Precisely this conditional

labeling guarantees that the couplings Ka reflect thevariance of the orientational pair couplings. Thisvariance provides the important information pertain-ing to the ordering or disordering capacities of thetriplets inside a molecular layer. The positionalensemble average taken in equation (2) thus corres-ponds to a spatial average over the layer at onemoment in time. By contrast, with absolute labeling(the same in every configuration) the sum in

equation (2) naturally yields isotropic couplings pro-

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portional to the mean pair coupling (KS = KI =Kw) and the physically relevant information is

averaged away.The resulting couplings {Ks, KI, Kw) depend

sensitively and highly non-linearly on temperature,which is given as £3 kT /A in dimensionless units, onpressure and concentration in a mixture, whichaffects the ratio a /f of the average lateral separationto the effective molecular length, and on molecularproperties, which are embodied in B/A, n and A.(The temperature scale in our calculation does notdirectly correspond to the experimental temperaturescale. Our present calculation ignores fluctuationstowards the isotropic, smectic-C and crystallinephases. As a consequence, the nematic and smectic-A phases artificially span the entire temperaturerange. Moreover, energy can be stored in a varietyof degrees of freedom, such as intramolecular tailflexing, which are not needed in our theory, butwhich will distort the « metric » of the temperaturescale via characteristic energy levels.) The next stepin the treatment is to gauge, approximately, whethersmectic order is supported, by referring to Houtap-pel’s ordering condition [18] of a uniformly distortedIsing model with the couplings {Ks, KI, Kw) :

where Ka are Ka with or without any pairwisechange of signs. The zero-temperature location ofthe bicritical point [1] in our results is dictated by ourcurrent approximation.

Thus, the couplings Ka are tested on their capacityto sustain long-range ferroelectric or antiferroelectricorder in a planar Ising model. For the three-dimen-sional liquid crystal, such ordering in the xy planewould not imply bulk ferro- or antiferroelectricorder. Indeed, the local order can change from layerto layer (along z), or, even within one layer,domains of different orders can be induced by theannealed random fields exerted by domains in

neighbouring layers.When equation (3) is satisfied and the two coup-

lings with the largest two moduli are both positive,local in-plane ferroelectric order is sustained. This

stabilizes the monolayer smectic A, phase with

predominant parallel dipolar correlation

( (SI S2) == 1) [2]. When equation (3) is satisfied andthe couplings with largest moduli are not both

positive, local in-plane antiferroelectric order ob-

tains, which stabilizes the interdigitated partialbilayer smectic Ad phase with antiparallel dipolarcorrelation ( (SI S2) «0) [2]. Note that distinct

permeational correlations, as discussed in the nextsection, stabilize the orientational couplings. In a

freely permeating and unlayered system, the

orientational coupling between given molecules

would average out to zero. Average layer thicknessesd in A, and Ad have been computed [2] via

The theoretical ratio of the layer thickness in

Ad to that in A, is found to be around 1.3, whichagrees with experimental determinations [9]. Finally,when equation (3) is not satisfied, smectic fluctua-tions are not supported and the nematic phaseobtains.

4. Statistics of microscopic configurations.

For exposing the details of the molecular statistics anidentification scheme is introduced for the micro-

scopic configurations. Consider n = 5, i.e., withnotches 1, 2, 3, 4, 5 and m = 3, i.e., with subnotchesa, b, c at each notch. The configuration in

figure 5(b) is denoted by (3, 3, 3 ). The overbar

signifies si = - 1. From figure 5(b) the occupationof the notches is apparent. Two possible occupationsof subnotches are shown in figures 5(bl) and 5(b2).The respective identifications are (3b, 3b, 3b ) and

(3a, 3b, 3c). In figure 5(c) we have (3, 3, 4), with

(3b, 3b, 4b ) in figure 5(cl) and (3a, 3c, 4b ) in

figure 5(c2). Figure 5(d) represents (3, 2, 4 ) and

figure 5(e) illustrates (3, 5,1 ).The spin-gas model explains multiple reentrances

of nematic and smectic phases on the basis of anexamination of the distinct classes of microscopicconfigurations which stabilize a particular phase.The configurations are classified according to theircapacity to promote short-range ferroelectric or

antiferroelectric order, which stabilizes smectic

layering.By examining couplings La in each positional

configuration {ri}’ defined by

five classes can be distinguished. In « Ferroelectric »configurations the La satisfy equation (3) and thecouplings with largest moduli are both positive. Atypical configuration is shown in figure 5(e). In

atomically permeated « Antiferroelectric, AP » andlibrationally permeated « Antiferroelectric, LP »

configurations, equation (3) is also satisfied but the

couplings with largest moduli are not both positive.The difference between AP and LP is that all three

dipoles in a triplet of AP occupy different notches,whereas at least two dipoles are at the same notch(but different subnotches) in LP. Figure 5(d) repre-sents a configuration belonging to AP, whereas

figures 5(b2) and 5(c2) exemplify members of the LPclass. In « Frustrated » configurations Ls Lj LW 0

and the two couplings with smallest moduli have

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equal moduli. Equation (3) is not satisfied. Examplesare presented by figures 5(bl) and 5(cl). Finally, allremaining configurations are thermally disordered(not frustrated) and belong to the « Disordered »

class (Eq. (3) is not satisfied). As temperature islowered configurations gradually move from the« Disordered » class into the three ordered classes.Frustrated configurations are frustrated at all tem-peratures. Note that the representations in figure 5do not distinguish between disordered and orderedconfigurations. This distinction is made only on thebasis of equation (3).The statistical weight wk of class k is computed as

the thermal average 8 k( {La} ) > ’ where

The average is taken over positional {ri} as well as

orientational (si ) freedoms. Note the normalization5

L Wk = 1. The weights wk depend sensitively on thek=1 1

thermodynamic variables and on the molecular con-stants. As configurations move from the disorderedclass into the ordered classes as temperature is

lowered, the weights display discontinuities. Theseoccur at relatively high temperatures and are notrelated to phase transitions.

In the generic topology of the phase diagrams, forn = 5, 6 or 7 and B/A around 1.5, exemplified infigure 6(a), the dominant configurations are iden-tified and the statistical weights wk are computed.At low temperatures the dominant configurations

are (3a, 3b, 3c) for all -- 0.1 and (3c, 2a, 2c) fora/f m 0.15, in the high-pressure Ad phase ; (3,1, 2) in the high-pressure Al phase ; (3a, 2a, 2c) fora/Q = 0.3, in the Ad phase ; (3, 1, 5) for all ==0.4, in the Al phase ; and (3,1, 2 ) in the low-

pressure Ad phase. Note that the medium- and high-pressure Ad phases are based (at low T) on configur-ations with at least two dipoles at the same notch.There, frustration is relieved by librational permea-tions. If libration is forbidden (m = 1) these

Ad phases disappear and the nematic phase obtainswith abundant frustrated configurations such as

(3a,la,la).Figure 6(b) shows the statistical weights w of the

five different configurational classes versus a Ie at

fixed temperature f3 kT/A = 0.6 (dashed line in

Fig. 6(a)). Their correspondence with the ther-

modynamic phases is seen from a comparison withfigure 6(a). Note that, as pressure is increased

(a /f decreased) from the low-pressure Ad phase atalf =--- 0.6, the nematic phase reenters before theA, phase is reached. Note also the subtle role playedby frustrated configurations, providing a continuousbackground with, e.g., a maximum in the nematicphase at alf =-= 0.5.

Fig. 6(a). - Generic phase diagram topology over wideranges of temperature and pressure, for B I A == 1.5,n = 5, 6 or 7, m = 2 or 3 and 6 == 0.01 Q /n. In this figure,B/A= 1.5, n=5, m=3 and 8 = 0.01 fin.

Fig. 6(b). - The isothermal statistical weights of the fiveclasses of positional configurations versus a / f in a scan

along the dashed line at f3 kT/A = 0.6 in figure 6(a).

5. Quadruple reentrance and molecular tail length.

Consider the quadruply reentrant phase diagram offigure 7(a). At high temperatures, the nematic phaseoccurs through ordinary thermal disorder. Loweringtemperature at an appropriate fixed a/f, two seg-ments of smectic Aa phase are encountered, sepa-rated by a segment of (reentrant) nematic phase. Onthe one hand, if the local-antiferroelectric configur-ations that relieve frustration through atomic perme-ation (Antiferro, AP class), such as (3, 2, 4 ), areeliminated from the prefacing transformation of

equation (2), the upper Ad segment disappears. Thisis shown in figure 7(b). On the other hand, if the

local-antiferroelectric configurations that relievefrustration through librational permeation (Antifer-ro, LP class) such as (3a, 3b, 3c), are eliminatedfrom the prefacing transformation, the lower Adsegment disappears. This is illustrated in figure 7(c),where librations have been « frozen » (m = 1 ).Evidently, the upper and lower smectic Ad phasesare due to local-antiferroelectric configurations thatrelieve frustration by, respectively, atomic and

librational permeations, whereas in the interveningnematic phase, frustrated configurations such as

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Fig. 7. - (a) Quadruply reentrant phase diagram obtained for B/A = 1.451, n = 5, m = 3 and 5 = 0.015 fln.(b) Phase diagram for B/A = 1.451, n = 5, m = 3 and 8 = 0.015 e In, after elimination of the antiferro AP class fromthe prefacing transformation : the upper Ad segment has disappeared. (c) Phase diagram for B/A = 1.451,n = 5 and m = 1. Librational permeations have been eliminated : the lower Ad segment has disappeared.

(3b, 3b, ... ) prevail. However, subtle variations ofthe statistical weights of the configurational classesare sufficient to bring about the N-Ad-N-Ad phasetransitions. This can be seen from figure 8.The smectic phases encountered so far are charac-

terized by antiparallel molecular orientation and aretherefore composed of interdigitated partial bilayers.This is also reflected in the quantitative calculationsof layer thicknesses [2], in agreement with experi-ments.

Upon lowering temperature below the two Adsegments, a second reentrant nematic segment isencountered. From figure 8 it is evident that thefrustrated configurations have declined in import-ance in this region of the phase diagram. Thisnematic phase occurs through the simultaneous

importance, competition and mutual cancellation oflocal-antiferroelectric (LP class), and local-fer-roelectric (Ferro class), such as (3, 5, 1), configur-ations. The latter configurations dominate in import-

Fig. 8. - The isobaric statistical weights of the five classesof positional configurations versus temperature in a scan at

ale = 0.463 in figure 7(a).

ance upon further lowering the temperature, causingthe smectic Al phase. This phase is characterized byparallel molecular orientation.

Figure 2(a) resembles the experimental phasediagram (Fig.2(b)) of pure DB90NO2 [11].Figure 7(a) agrees with the experimental phase diag-ram obtained with mixtures of DB90NO2 and its

homologs (Fig. 2(c)) [10]. Interestingly, quadruplereentrance is hard to find both experimentally and inthe spin-gas model where it occurs for the limitedcases and narrow intervals of n = 4 and 1.856

B/A 1.896, n = 5 and 1.445 B/A 1.466, orn = 6 and 1.468 B/A 1.479. We have not found

quadruple reentrance for n = 3, 7, 8, 9. This is

equivalent to the experimental finding that the

phenomenon depends sensitively on the tail lengthof the molecule. Moreover, n = 4 or 5 are indeedthe most reasonable notch numbers that can bededuced from considering DB90N02 with its tail ofnine carbon atoms (see Fig. 4).

6. Continuum limit of librational permeation.

The continuous librational permeations on a lengthscale a .l/n are calculationally approximated tooccur in m discrete subnotches, separated by 8,within each notch. In this section calculational

stability with respect to taking the continuum limitm -> oc with A constant is investigated.The length A is defined such that A2 equals six

times the variance of mutual librational permeationof a nearest-neighbour dipolar pair at the same

notch,

Figures 9 and 10 show the phase diagram which isfound for B/A = 1.451, n = 5 and A = 0.0424 l /n,when the librations are discretized with m = 2, 3, 5,7 and 9. Also shown is the phase diagram obtained

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directly from computations in the continuum limit(m = oo ). The prefacing transformation then reads

where p are the z coordinates of the dipoles relativeto the centres of the notches, and where conditional

labeling of the molecules is performed as in

equation (2). Figure 9 clearly shows that the quad-

Fig. 9. - Quadruply reentrant phase diagram : conver-

gence under the continuum limit of librational permeation.Phase diagrams are shown at B/A = 1.451, n = 5 andA = 0.0424 f In. The different N-Ad phase boundariescorrespond to m = 2, 3, 9 and oo (continuum limit).

ruply reentrant topology is stable, and in fact

converges fast to its continuum limit. At low tem-

peratures (f 3 kT/A 0.1) the convergence is slower,as figure 10 illustrates. Moreover, in the course oftaking the continuum limit, for m > 3, a qualitativelynew topology emerges : reentrant N and Ad phasesappear below the A, phase [6]. Figure 10 presentsreentrances below Al which are stable in the con-tinuum limit. This indicates that the new sextuple(and octuple) reentrances (apparent from combiningFigs. 9 and 10) are stable. Experimentally, reen-

trances below Al have been reported [13-15]. Acomparison of theory and experiment is illustrated infigure 3. The newly predicted sextuple (and octuple)reentrances N-Ad-N-Ad-N-Al-N-Ad (-N-Al) have toour knowledge not yet been observed.

7. Molecular dimers and dipolar-pair lengths.

Landau-Ginzburg theories [19-23] discuss reentrancein polar liquid crystals in terms of two competinglengths : the molecular length f and the dipolar-pairlength d, wavelengths, respectively, of the density

Fig. 10. - Low-temperature details of the quadruplyreentrant phase diagrams of figure 9, for m = 2, 3, 5

(partly), 9 (partly) and oo (continuum limit). The nematicphase between Al and Ad becomes exceedingly narrow asthe zero-temperature bicritical point is approached. Form > 3 reentrances below Al are obtained. Sextuple andoctuple reentrance is found for m = 5 or 9. The con-

tinuum-limit phase diagram is sextuply reentrant. The

bicritical point for m = oo is easily located from energyconsiderations and lies at a/Q = 0.4542.

and polarization modulations associated with smecticfluctuations. An important problem is the micro-

scopic origin of d.It has become fashionable in molecular theories to

discuss reentrance in terms of monomers (freelypermeating molecules) and molecular dimers (anti-parallel associations of two molecules with dipolescompensated as in Fig. 4(b)) [24-26].Our study with the spin-gas model emphasizes

that monomers and dimers are only part of the

physical picture. Energetically favorable associationsof at least three molecules are very important,especially in view of their capacity to stabilizesmectic layering. In the spin-gas model a disorderedtriplet contains three monomers. A frustrated tripletconsists of a dimer and a monomer. Correlated

triplets (with local antiferro- or ferroelectric order)can associate into an extended network, the molecu-lar polymer, in the xy plane. Thus, correlated

triplets support smectic order, whereas monomersand dimers do not. This picture contrasts with thoseof alternative molecular theories [24-26].The calculated concentrations of monomers, di-

mers and ordered triplets, which are directly relatedto the statistical weights Wk introduced in section 4,are smooth across N-Ad phase transitions. In particu-lar, dimers play a subtle rather than emphatic role inreentrances of Ad and N. This is compatible with thetemperature variation of the measured dielectric

anisotropy [27, 28]. Figure 11 shows typical tempera-ture dependences of dimer concentrations across

N-Ad transitions in the spin-gas model, and in acalculation from experimental dielectric anisotropy

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Fig. 11. - Concentration of molecular dimers XD versus

temperature. Shown are (i) in the spin-gas model, thestatistical weight of the frustrated class along a scan ata/Q = 0.481 in the N-Ad-N region of figure l(a). Note thatat low temperature the weight dips down upon approachof the A¡ phase (not shown) ; (ii) the calculated « exper-imental » dimer concentration, as obtained from dielectricanisotropy measurements in an 8 OCB-6 OCB mixture

(Refs. [27] and [28]) ; (iii) in the model of Longa and deJeu (LdJ), the input curve of dimer concentration. Ofexclusive importance in this figure are the derivatives ofthe curves in the vicinity of the N-Ad and Ad-N transitions.

data [27, 28]. For comparison, also the dimer

,oncentration profile in the molecular model of

Longa and de Jeu [24] is displayed, as in reference[28]. In that model the dimer concentration is an

input and serves to drive the Ad-N transition, as

temperature is lowered.The situation is quite different in the vicinity of

Ad-N-Al transitions. There, dimers break up at theonset of the smectic Al phase as correlated local-ferroelectric triplets form. This causes the relativelylarge transition enthalpy as calculated with the spin-gas model and observed in experiment [3].An important concept in the phenomenology and

in monomer-dimer theories is the dipolar-pair lengthd, associated with the wavelength of the « incom-mensurate » smectic fluctuations responsible for theAd phase. Our study indicates that there is no uniquepair length. It is one length if the pair is a dimer,e.g., figures 5(b) and 5(c), but different lengthsresult if the pair belongs to an ordered triplet(Figs. 5(d) and 5(e)). In conclusion, the pair length

is a statistical object with different identities in thedifferent classes of microscopic configurations. Theaverage pair length, for example, affects the averagelayer thickness as defined in equation (4). Thisstatistical nature of the dipolar-pair length is borneout by experimental studies using high-resolution X-ray scattering of the multiply reentrant polar meso-gen DB90NO2 [12].

8. Final comments.

In the spin-gas model the smectic phase with localintralayer ferroelectric order is identified with the

monolayer smectic Al phase. As already suggestedin section 3, within the present development of themodel a distinction cannot yet be made between

Al and, e.g., A2 which is the bilayer smectic A phasewith alternating local-ferroelectric order in succes-sive layers along z [21]. Also, the phase A is

possible, which locally resembles Al but possessestransverse (xy ) modulations of polarization (aperiodic array of kinks) [21].Three modes of disorder feature in the spin-gas

model: thermal noise (high-temperature nematic),microscopic frustration of dipoles (reentrancemechanism for a nematic phase between Ad phases),and competing short-range orders (reentrancemechanism for a nematic phase between Ad andAl phases). The latter mode consists of coexistingferroelectric and antiferroelectric smectic fluctua-tions with incommensurate wavelengths. We haveobtained from this a nematic reentrance. Another

possible phenomenon would, however, be a smecticphase with two collinear incommensurate fluctua-tions : the phase Aic occurring in the phenomenology[23] and found experimentally [29] in the sequenceAd-Aic-Az.

Acknowledgments.

It is a pleasure to thank C. W. Garland and J. F.Marko for useful discussions. This research was

supported by the National Science Foundation underContract No. DMR-84-18718 and by the NationalFund for Scientific Research of Belgium.

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