Molecular structure and reentrant phases in polar liquid crystals

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Molecular structure and reentrant phases in polar liquid crystals

J.O. Indekeu, A. Nihat Berker

To cite this version:

J.O. Indekeu, A. Nihat Berker. Molecular structure and reentrant phases in polar liquid crystals.

Journal de Physique, 1988, 49 (2), pp.353-362. �10.1051/jphys:01988004902035300�. �jpa-00210702�

353

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-3030} Leuven, Belgium

(Requ ^{le 11} septembre 1987, accepte le 23 octobre 1987)

Résumé.

On emploie le modèle de gaz de spins des cristaux liquides pour décrire ^et expliquer les réentrées

simples, ^{doubles et} quadruples ^de phases nématiques ^et smectiques, ainsi que les réentrées ^sous la phase

monocouche smectique A1, qui sont toutes observées expérimentalement. ^On prédit également de nouvelles réentrées sextuples ^et octuples ^le long ^{de la} séquence nématique-smectique Ad-nématique-smectique Ad-nématique-smectique Acnématique-smectique Ad(-nématique-smectique A1). Les réentrées multiples ^sont

très sensibles à la structure moléculaire et, ^en particulier, ^{à la} longueur de la chaîne de la molécule. On examine l’information microscopique qu’on peut tirer de la statistique ^des configurations microscopiques ^en comparant ^les poids statistiques des différentes classes de configurations. Il y ^a deux mécanismes distincts pour la réentrée qui ^font appel, ^d’une part, ^{à la} frustration de dipôles microscopiques ^{et, d’autre} part, ^{à une} compétition 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 ^la

limite du continuum de la perméation librationnelle. On examine aussi le rôle subtil joué par les dimères de molécules, ainsi que le concept ^d’une longueur ^de paire dipolaire.

Abstract.

The spin-gas ^{model of} liquid crystals ^is employed ^to describe and explain single, double and

quadruple reentrances of nematic and smectic phases, ^{as well as} reentrances below the monolayer ^smectic A1 phase, all of which are observed experimentally. ^New sextuple (and octuple) reentrances are predicted ⁱⁿ

the 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, in

particular ^to the molecular tail length. ^The microscopic information available from the statistics of microscopic configurations is scrutinized. Statistical weights of classes of positional configurations ^are monitored. Two distinct mechanisms of nematic reentrance feature : microscopic frustration of dipoles ^and competing ^local

antiferroelectric and ferroelectric orders. Frustration is relieved by atomic and librational permeations, causing

distinct smectic reentrances. The continuum limit of librational permeation is taken. The subtle role played by

molecular 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 ^Abstracts

61.30B - 64.70M - 77.80B - 05.70F

1. Introduction.

In condensed matter physics, phenomena ^{of « reen-}

trance » of more ordered phases ^when cooling ^or compressing from less ordered phases ^can generally

be argued ^{to occur from} competition of tendencies towards different orderings. ^{At a} phenomenological

level ^a Landau theory ^{and the} associated free-energy optimization program ^are analysed, featuring ^{a cho-}

sen set of order parameters. ^{At a} microscopic ^level

one 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 monitoring

the 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} observations of multiply ^reentrant nematic and smectic phases ⁱⁿ polar liquid crystals [7-16].

The thermodynamic phases ^and phase transitions in these systems ^are very sensitive ^to details of the molecular structure. In some liquid crystals ^of

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

molecules with strongly dipolar ^heads (-CN ^or -N02) ^nematic phases (orientational order) ^and

smectic phases (orientational ^and partial positional order) ^reenter. Experimentally ^observed reentrances

typically 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 theoretical side all of these reentrances have been obtained with the spin-gas ^{model of} liquid crystals [1-6]. ^In figures

1-3 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, ^these

have 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 and}

8

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 for} data 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 ^of

concentration ^x.

355

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 by

f 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 the concentration of a terminal nonpolar compound ^{in a} ternary mixture. The value x

0 corresponds ^{to an} equimolar

mixture 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 the first time the microscopic information which the

spin-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 ^molecules

under consideration is given ⁱⁿ figure 4. The molecu- lar length ^f is of the order of 30 A and the width is about 5 A. A longitudinal dipolar head, ^a rigid

aromatic _core, and ^a semi-flexible aliphatic ^{tail are} distinguished. The intermolecular pair potential ^em-

bodies steric hindrance, ^{van der Waals} attraction,

and dipole-dipole forces. The latter are important

since reentrant phases ^{are almost} exclusively ^{seen in} polar 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 indicate oxygen atoms, and closed circles mark the positions ^of

carbon atoms in an all-trans configuration ^{of the} aliphatic

tail. The molecular pairs ^have dominant tail-tail attraction

(a) ^or steric hindrance (b).

The pair potential ^for nearest-neighbour ^molecules

is 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), ^or

steric repulsion, figure 4(b), ^are incorporated ^{in the}

ratio B/A : ^{B A for net hindrance} (favoring ^the

antiferroelectric term) ^{and B} > A for net entangle-

ment. (The ^inherent dipolar potential ^{can be au-} gmented ^more realistically ^to reflect other molecular

interactions, leading ^to quantitative changes ^{in the} results.) Note that in this context « symmetric »

molecules with a central dipole ^{between two identical}

molecular parts ^on either side are described by

A

B. Screening suppresses interactions between further-than-nearest neighbours. Fluctuations tow- ard the isotropic phase ^are ignored, namely ^the

molecules are taken aligned along ^{the z} direction :

91

± i, ^or simply si ^{= ± 1.}

In the close-packing ^{of a} liquid, ^the potential ⁱⁿ equation (1) inherently ^causes frustration ^{due to}

substantial near-cancellations of forces between a

molecule and its neighbours. ^For example, ^{when two}

molecules form an antiparallel pair, figure 4(b), constituting ^{a molecular} dimer, ^{an almost zero net}

force is felt by ^a neighbouring third molecule because

of net dipolar frustration. The third molecule then

almost freely permeates (i.e. positionally ^fluctuates along ^{the z} direction), ^as molecules do in nematic

phases. On the other hand, three molecules can

form a correlated triplet ^{with net} dipolar short-range

antiferroelectric or ferroelectric order. These triplets

are energetically ^stable against small mutual permea- tions and may or may not, depending ^on entropic qualities, conglutinate laterally ^{to form a} percolating

network or molecular polymer. ^{If a} polymer ^is formed, ^enhanced longitudinal viscosity hinders per-

meation, and molecules tend to « lock in » into smectic layers. ^{In the} spin-gas model the statistics of these molecular configurations ^are gathered ^and

their dependences ^on temperature, pressure and molecular constants are derived.

Molecular tails play a crucial role. Apart ^{from the}

free 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 the} dipolar 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 three

dipoles ^{are at the same subnotch} (bl) frustration results as a zero net force is felt by either of dipoles ^{1 or 3.}

Librational permeation (b2) relieves frustration and allows local antiferroelectric order. (c) ^If dipoles ^{1 and 2 are at}

the same subnotch (cl) frustration results : a zero net force is felt by dipole 3. Frustration is relieved by librational

permeation (c2) allowing local antiferroelectric order.

Frustration is relieved by ^atomic permeation allowing ^local

antiferroelectric order (d) ^{or local} ferroelectric order (e).

permeations ^{on a} length ^scale f In (a ^few A), ^small

oscillations occur on a length ^{scale 4 «} f In. ^These

« librational » permeations ^{can be} calculationally approximated ^{to occur in m discrete} subnotches, separated by ^8, within each notch (Fig. 5). ^An important issue addressed in detail in this paper is the physical ^{continuum limit m - oo with A constant.}

3. The prefacing transformation.

Within a three-dimensional system, ^{a reference} layer

of molecules ^can be identified by the coordinates zi being within the interval zo ± f/2. The smallest unit of possible frustration is ^a triplet of molecules in the reference layer. First, ^the strengths ^{are obtained}

for effective orientational couplings between the

spins si by thermally averaging ^{over the} positional

fluctuations ri. We found that the positional ^fluctua-

tions that underly reentrance are permeations, ^the

molecular 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 for}

calculational ease. The theory ^thus predicts ^{that the} degree of in-plane (xy ) ^{order is not} directly ^{related to}

smectic order along ^{z. This} physical characteristic has been confirmed by subsequent X-ray scattering experiments [17]. Accordingly the calculations pro- ceed by considering ^three nearest-neighbour

molecules ^{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 configurations

per orientational configuration {Si}’

With a prefacing transformation, the average- strongest (KS ), average-intermediate (KI ), ^{and av-} erage-weakest (Kw) antiferroelectric couplings ^are

extracted (Ks -- KI , Kw) :

where the molecule labels (12), (23), ^and (31) respectively span the strongest, intermediate, ^and

weakest antiferroelectric couplings specific ^{to each} positional configuration. Precisely this conditional

labeling guarantees ^{that the} couplings Ka reflect the variance of the orientational pair couplings. ^This

variance provides ^the important information pertain- ing ^{to the} ordering ^or disordering capacities ^{of the} triplets ^{inside a molecular} layer. ^The positional

ensemble average taken in equation (2) ^{thus corres-} ponds ^{to a} spatial average ^over the layer ^{at one}

moment in time. By contrast, with absolute labeling

(the ^same in every configuration) ^{the sum in}

equation (2) naturally yields isotropic couplings pro-

357

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, ^on

pressure and concentration in ^a mixture, ^which

affects the ratio a /f of the average lateral separation

to the effective molecular length, ^{and on molecular} properties, ^{which are} embodied in B/A, n and ^A.

(The temperature ^{scale in our} calculation does not

directly correspond ^{to the} experimental temperature scale. Our present calculation ignores fluctuations towards the isotropic, smectic-C and crystalline phases. ^{As a} consequence, the nematic and smectic- A phases artificially span the entire temperature range. Moreover, energy ^can be stored in a variety

of degrees ^of freedom, ^{such as} intramolecular tail

flexing, ^{which are not needed in our} theory, ^but

which will distort the ^« metric » of the temperature scale via characteristic _energy levels.) ^{The next} step in the treatment is to gauge, approximately, ^whether

smectic order is supported, by referring ^to Houtap- pel’s ordering ^condition [18] ^{of a} uniformly ^distorted Ising model with the couplings {Ks, KI, Kw) :

where Ka ^are Ka ^{with or} without any pairwise change ^of signs. ^The zero-temperature location of the bicritical point [1] ^{in our} results is dictated by ^our

current approximation.

Thus, ^the couplings Ka ^{are tested on their} capacity

to sustain long-range ferroelectric or antiferroelectric order in a planar Ising model. For the three-dimen- sional liquid crystal, ^such ordering in the xy plane

would not imply bulk ferro- or antiferroelectric order. Indeed, the local order can change ^from layer

to layer (along z), ^{or, even within one} layer,

domains of different orders can be induced by ^the

annealed 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 and the couplings ^with largest ^{moduli are not both} positive, ^local in-plane antiferroelectric order ob-

tains, ^which stabilizes the interdigitated partial bilayer ^smectic Ad phase ^with antiparallel dipolar

correlation ( (SI S2) ^«0) [2]. ^Note that distinct

permeational correlations, ^as discussed in the next

section, stabilize the orientational couplings. ^{In a} freely permeating ^and unlayered system, ^the orientational coupling ^between given ^molecules

would average ^{out to zero.} Average layer thicknesses d 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, ^which

agrees with experimental determinations [9]. Finally,

when equation (3) ^{is not} satisfied, smectic fluctua- tions are not supported and the nematic phase

obtains.

4. Statistics of microscopic configurations.

For exposing ^the details of the molecular statistics an

identification scheme is introduced for the micro-

scopic configurations. Consider n

5, i.e., ^with

notches 1, 2, 3, 4, ^{5 and m}

3, i.e., with subnotches a, 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 occupation

of the notches is apparent. ^Two possible occupations

of 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 an

examination of the distinct classes of microscopic configurations ^{which stabilize a} particular phase.

The configurations ^are classified according ^{to their} capacity ^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 the} couplings ^with largest ^{moduli are both} positive. ^A typical configuration is shown in figure 5(e). ^In atomically permeated ^« Antiferroelectric, ^{AP » and} librationally 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 LP class. In ^« Frustrated ^» configurations Ls Lj LW ⁰

and the two couplings ^{with smallest moduli have}

equal ^moduli. Equation (3) ^{is not} satisfied. Examples

are presented by figures 5(bl) ^and 5(cl). Finally, ^all remaining configurations ^are thermally ^disordered (not frustrated) ^and belong ^{to the «} Disordered »

class (Eq. (3) ^{is not} satisfied). ^As temperature ^is lowered 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 ⁵

do not distinguish between disordered and ordered

configurations. This distinction is made only ^{on the}

basis 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 normalization

5

L Wk ^{= 1. The weights wk} depend sensitively ^{on the}

k=1 1

thermodynamic variables and on the molecular con- stants. As configurations ^move from the disordered class into the ordered classes as temperature ^is lowered, ^the weights display discontinuities. These

occur at relatively high temperatures ^{and are not} related to phase transitions.

In the generic topology ^{of the} phase diagrams, ^for

n

5, ^{6 or 7 and} B/A ^{around 1.5,} exemplified ⁱⁿ figure 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) ^for

a/f m ^{0.15, in the} high-pressure Ad phase ; (3, 1, 2) ^{in the} high-pressure Al phase ; (3a, 2a, 2c) ^for a/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 ^obtains

with 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 ^with figure 6(a). ^{Note that, as} pressure is increased

(a /f decreased) ^{from the} low-pressure Ad phase ^at

alf =--- ^0.6, the nematic phase ^{reenters before the} A, phase is reached. Note also the subtle role played by frustrated configurations, providing ^{a continuous} background ^with, e.g., ^a maximum in the nematic

phase ^at alf =-= ^0.5.

Fig. 6(a).

^Generic phase diagram topology ^{over wide}

ranges 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 five classes 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 ^of figure 7(a). ^At high temperatures, the nematic phase

occurs through ordinary thermal disorder. Lowering

temperature ^{at an} appropriate ^fixed a/f, ^two seg- ments of smectic Aa phase ^are encountered, sepa- rated by ^a segment ^of (reentrant) ^nematic phase. ^On

the one hand, if the local-antiferroelectric configur-

ations that relieve frustration through atomic perme- ation (Antiferro, ^AP class), ^{such as} (3, 2, 4 ), ^are

eliminated from the prefacing transformation of

equation (2), the upper Ad segment disappears. ^This

is shown in figure 7(b). On the other hand, ^{if the} local-antiferroelectric configurations that relieve frustration through librational permeation (Antifer-

ro, LP class) ^{such as} (3a, 3b, 3c), ^{are eliminated}

from the prefacing transformation, the lower Ad segment disappears. ^This is illustrated in figure 7(c),

where librations have been ^« frozen ^» (m

1 ).

Evidently, the upper and lower smectic Ad phases

are due to local-antiferroelectric configurations ^that

relieve frustration by, respectively, atomic and librational permeations, whereas in the intervening

nematic phase, frustrated configurations ^{such as}

359

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 from the 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 of the statistical weights ^{of the} configurational ^classes

are sufficient to bring ^{about the} N-Ad-N-Ad phase

transitions. This ^can be seen from figure ^8.

The smectic phases encountered so far are charac- terized by antiparallel molecular orientation and are

therefore composed ^of interdigitated partial bilayers.

This is also reflected in the quantitative calculations of layer thicknesses [2], ⁱⁿ agreement ^with experi-

ments.

Upon lowering temperature below the two Ad segments, ^{a second reentrant nematic} segment ^is encountered. From figure 8 it is evident that the frustrated configurations ^have declined in import-

ance in this region ^{of the} phase diagram. ^This

nematic phase ^occurs through ^the simultaneous

importance, competition and mutual cancellation of local-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 classes of positional configurations ^versus temperature ^{in a scan at}

ale

^{0.463 in} figure 7(a).

ance upon further lowering ^the temperature, causing

the smectic Al phase. ^This phase is characterized by parallel molecular orientation.

Figure 2(a) resembles the experimental phase diagram (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, quadruple

reentrance is hard to find both experimentally ^{and in}

the spin-gas model where it occurs for the limited

cases and narrow intervals of n

4 and 1.856

B/A ^{1.896, n} = 5 and 1.445 B/A ^{1.466, or}

n

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} length

of the molecule. Moreover, n

^{4 or 5 are indeed}

the most reasonable notch numbers that can be deduced from considering DB90N02 ^{with its tail of}

nine carbon atoms (see Fig. 4).

6. Continuum limit of librational permeation.

The continuous librational permeations ^{on a} length

scale a .l/n ^are calculationally approximated ^to

occur in m discrete subnotches, separated by 8,

within each notch. In this section calculational

stability ^with respect ^to taking the continuum limit

m -> oc with A constant is investigated.

The length A ^{is defined such that} A2 equals ^six

times the variance of mutual librational permeation

of a nearest-neighbour dipolar pair ^{at the same} notch,

Figures 9 and 10 show the phase diagram ^{which is}

found 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

directly ^from computations ⁱⁿ the continuum limit

(m

oo ). ^The prefacing transformation then reads

where _p are the ^z coordinates of the dipoles ^relative

to 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 and}

A

0.0424 f In. ^{The different} N-Ad phase ^boundaries correspond ^{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 of} taking the continuum limit, ^{for m} > 3, ^a qualitatively

new topology emerges : ^reentrant N and Ad phases

appear below the A, phase [6]. Figure ¹⁰ presents

reentrances below Al ^{which are} stable in the con-

tinuum limit. This indicates that the ^new sextuple (and octuple) reentrances (apparent ^from combining Figs. 9 ^and 10) ^{are stable.} Experimentally, ^reen-

trances below Al ^{have been} reported [13-15]. ^A comparison ^of theory ^and experiment is illustrated in

figure ^{3. The} newly predicted sextuple (and octuple)

reentrances N-Ad-N-Ad-N-Al-N-Ad (-N-Al) ^{have to}

our knowledge ^not yet been observed.

7. Molecular dimers and dipolar-pair lengths.

Landau-Ginzburg ^theories [19-23] ^{discuss reentrance}

in polar liquid crystals ^{in terms of two} competing lengths : the molecular length ^{f and the} dipolar-pair length ^d, wavelengths, respectively, ^{of the} density

Fig. ^10.

Low-temperature details of the quadruply

reentrant phase diagrams ^of figure 9, ^{for m}

2, 3, ⁵ (partly), ⁹ (partly) ^{and oo} (continuum limit). The nematic

phase ^between Al ^and Ad ^becomes exceedingly ^{narrow as}

the zero-temperature bicritical point ^is approached. ^For

m > 3 reentrances below Al ^{are obtained.} Sextuple ^and octuple 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 energy considerations and lies at a/Q

^0.4542.

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

scopic origin ^{of d.}

It has become fashionable in molecular theories to discuss reentrance in terms of monomers (freely permeating molecules) and molecular dimers (anti- parallel associations of two molecules with dipoles compensated ^{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 associations of at least three molecules are very important, especially in view of their capacity ^{to stabilize}

smectic layering. ^{In the} spin-gas ^{model a disordered} triplet contains three monomers. A frustrated triplet

consists 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 monomers}

and dimers do not. This picture ^{contrasts with those}

of alternative molecular theories [24-26].

The calculated concentrations of _monomers, di-

mers and ordered triplets, ^{which are} directly ^related

to 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 in}

reentrances of Ad and N. This is compatible ^{with the} temperature 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 a}

calculation from experimental dielectric anisotropy

361

Fig. 11. - Concentration of molecular dimers _XD versus

temperature. ^{Shown are} (i) ^{in the} spin-gas model, ^the statistical weight ^{of the} frustrated class along ^{a scan at}

a/Q

0.481 in the N-Ad-N region ^of figure l(a). ^{Note that}

at low temperature ^the weight dips down upon approach

of the A¡ phase (not shown) ; (ii) the calculated ^« exper- imental ^» dimer concentration, ^as obtained from dielectric

anisotropy measurements in an 8 OCB-6 OCB mixture

(Refs. [27] ^and [28]) ; (iii) in the model of Longa ^{and de}

Jeu (LdJ), ^the input ^curve of dimer concentration. Of exclusive importance ^{in this} figure ^are the derivatives of the 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 the onset of the smectic Al phase ^as correlated local- ferroelectric triplets form. This causes the relatively large 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 length d, associated with the wavelength of the « incom- mensurate » smectic fluctuations responsible ^{for the} Ad phase. ^Our study indicates that there is no unique pair length. ^{It is one} length ^{if the} pair ^{is a dimer,}

e.g., figures 5(b) ^and 5(c), but different lengths

result 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 the different classes of microscopic configurations. ^The

average pair length, ^for example, affects the average

layer ^{thickness as} defined in equation (4). ^This

statistical nature of the dipolar-pair length ^{is borne}

out 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 local} intralayer ferroelectric order is identified with the

monolayer ^smectic Al phase. ^As already suggested

in section 3, within the present development ^{of the}

model a distinction cannot yet be made between

Al ^and, e.g., A2 which is the bilayer ^{smectic A} phase

with alternating local-ferroelectric order in succes-

sive layers along ^z [21]. Also, ^the phase A ^is possible, ^which locally ^resembles Al ^but possesses

transverse (xy ) modulations of polarization (a periodic array of kinks) [21].

Three modes of disorder feature in the spin-gas

model: thermal noise (high-temperature nematic), microscopic frustration of dipoles (reentrance

mechanism for a nematic phase ^between Ad phases),

and competing short-range ^orders (reentrance

mechanism for ^a nematic phase ^between Ad ^and Al phases). The latter mode consists of coexisting

ferroelectric and antiferroelectric smectic fluctua- tions with incommensurate wavelengths. ^{We have}

obtained from this a nematic reentrance. Another

possible phenomenon would, however, ^{be a smectic} phase ^{with two collinear} incommensurate fluctua- tions : the phase Aic occurring ^{in the} phenomenology [23] ^{and found} experimentally [29] in the sequence

Ad-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 under Contract No. DMR-84-18718 and by ^{the National}

Fund for Scientific Research of Belgium.

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