Local dipolar order induced by steric effects in smectic B phases

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Local dipolar order induced by steric effects in smectic B phases

G. Coulon

To cite this version:

G. Coulon. Local dipolar order induced by steric effects in smectic B phases. Journal de Physique,

1982, 43 (7), pp.1173-1178. �10.1051/jphys:019820043070117300�. �jpa-00209492�

Local dipolar ^{order induced} by steric effects in smectic B phases

G. Coulon

Equipe ^de dynamique ^des cristaux moléculaires (*), Université de Lille I, ⁵⁹⁶⁵⁵ Villeneuve d’Ascq, ^France (Reçu le 14 décembre 1981, révisé le 5 mars 1982, accepté ^{le 16 mars} 1982)

Résumé. 2014 Il est bien connu que, dans les phases smectiques B, ^les répulsions stériques ^entravent la rotation des molécules autour de leur grand ^axe. L’objet ^{de cet article est de} déterminer l’influence de ces effets stériques

sur l’ordre local des dipôles ^{dans les} plans smectiques. Aussi, ^{en utilisant une} technique de renormalisation de

graphes, nous avons pu obtenir ^une valeur très précise du facteur de corrélation diélectrique G~. ^{Nous montrons}

que, quelle que soit la force des répulsions stériques, G~ ^est supérieur ^ou égal à 1 : ceci correspond ^{à un ordre}

local des dipôles ^de type ferroélectrique.

Abstract.

It is well-known that, in the smectic B phases, the rotation of the molecules around their axis is geared

because of steric hindrance. The aim of this paper is ^to determine the influence of these steric effects on the local

dipolar order in the smectic planes. Also, by using ^a graphical renormalization technique, ^{we have been able to}

obtain a very ^accurate value of the dielectric correlation factor G~. Whatever ^{is the} strength ^{of the steric} repulsion

we show that G~ ^is greater ^{than or} equal ^{to 1 : it} corresponds ^{to a local} dipolar ^order of ferroelectric type.

Classification

Physics ^Abstracts

61. 30C - 77. 20 - 05. 50

1. Introduction.

Up ^{to now, because of} experi-

mental difficulties, the dielectric properties of smectic B

phases ^{have been} scarcely studied in comparison

with those of the nematic and smectic A phases.

In these uniaxial compounds, ^the dielectric _per-

mittivity differs in value along the molecular axis (ell)

and perpendicular ^{to this axis} (e1.). ^{In non} polar compounds, ^the dielectric anisotropy s 11 -,61

is always positive in the nematic phase; while in the nematic phase ^of polar substances, ^{AE can be} positive

or negative according ^{to the} angle between the permanent dipole p and the molecular axis [1].

At the nematic-smectic A transition, experimental

studies have shown that, generally, ^for polar ^com- pounds ⁱⁿ which tt ^is nearly parallel ^to the molecular axis [2, 3], e 1. decreases slowly.

On the other hand, ^other experimental studies, performed ^{on non} polar compounds ^{or on} polar

ones in which p ^{is not close to} the molecular axis

[3, 4, 5, 6], have revealed that, ^at the nematic- smectic A transition, e 1. increases while ell decreases;

furthermore, ^{when As is} positive in the nematic phase,

there is a reversal of Ae at the transition [7].

(*) ^{Associee au C.N.R.S.} (ERA 465).

This increase of El ^versus _{8 II} has been related to the

occurrence of smectic planes ^{and has been} explained ⁱⁿ

terms of short-range dipolar interactions in the smectic planes [8].

Among these last studies, only ^some of them have

investigated the smectic B phase. Apart ^{from one of}

them where _El is found to decrease _very strongly [3],

in all the others, El tends to increase while E decreases

[3, 6] and the difference between s-L ^and 811 ^{is more}

important in the smectic B phase than in the high temperature phases.

It is now well-known that, in smectic B phases, ^the

molecules are arranged ^{in a} two-dimensional hexa-

gonal (or pseudo-hexagonal) array and that the lattice parameter ^is often smaller than the size of the molecules [9]. Incoherent quasi-elastic neutron scat-

tering ^data [10] ^have revealed that the molecules rotate around their molecular axis. But X-ray ^scat- tering experiments [11] I ^and theoretical calcula- tions [12, 13] ^have shown that the molecular rotation is geared because of steric effects between neighbouring

molecules.

The _purpose of this paper is ^to study ^{the influence}

of these steric interactions, characteristic of smectic B

phases, ^on the dielectric properties ^{and more} espe-

cially ^{on the local} dipolar order in the smectic planes :

this can be deduced from the value of the microscopic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043070117300

1174

dielectric correlation factor G 1. in the smectic planes

which is defined as :

where N is the total number of molecules in the _crys- tal and (pj_)j represents ^the component ^{of the} perma- nent dipole ^{of the} molecule j in the smectic plane.

Gj_ ^{is the sum} of correlation functions over all mole- cules in the infinite system. ^If Gl > 1, ^{there is a ten-} dency ^{for the} perpendicular components ^{of the} molecular dipoles ^{to form} parallel short-range ^order- ing ; ^if Gl ¹ the local order is antiparallel ^{and if} Gl

^{1 there is no} correlations between the dipoles.

The theoretical calculation of Gi, given by (1),

needs to perform ^a statistical _average over all the different molecular configurations ^on the basis of a

structure and interaction model.

According ^{to the} X-ray ^results [9, II], ^{we assume}

that the molecules, ^{located on a} hexagonal array,

can perform rotational jumps ^of H/3 about the hexa-

gonal ^axes.

In liquid crystals, the molecules are not rigid; ^DMR experiments [14] ^have revealed the existence of intra- molecular motions in the molecular aromatic core

itself.

Therefore, because of the multitude of possible conformers, ^we consider, here, only ^{two schematic}

interaction models :

(i) Very restrictive correlations induced by ^steric

effects which ^occur simultaneously ^{on both} sides of the molecular axis.

(ii) Softer correlations induced by steric effects which occur on only ^one side of the molecular axis.

These two models, ^{which are} different from a steric

point ^of view, ^are equivalent ^with respect ^{to the} dipolar disorder : during the molecular rotation the

dipole takes, ^{in both} cases, six discrete equilibrium positions ^{as is shown on} figure ^1.

Because of steric effects, ^two neighbouring ^mole-

cules (i) ^and (j) ^cannot occupy simultaneously ^the edge ij ) ^{and we assume that} configurations (I) ^are sterically impossible ^while configurations (C1) ^and

Fig. ^1.

Orientational disorder of a molecule and of its

dipole : (i) A molecular state is unchanged by ^{a n} rotation,

but its dipole ^is changed ^{to its} opposite : 3 molecular states

=

6 dipolar orientations. (ii) ^{There are} 6 discrete molecular states, thus 6 discrete dipolar orientations.

Fig. ^2.

The different molecular configurations ^{for the}

models (i) ^and (ii) (projection in the smectic plane).

(C2) ^are sterically possible (of ^zero repulsion energy) (Fig. 2).

For such ^« all or nothing » interaction models it is

possible ^{to obtain a} very ^accurate value of Gl by ^use

of a closed weak graph renormalization technique [15].

We have evaluated Gl ^{in the case} of the interaction model (ii). ^{The value} of Gj_, ^{for the} interaction model (i),

is foreseeable : for each ^« steric » molecular state, ^the

dipole ^{can take} randomly ^two opposite directions ;

so, the dipolar orientation does not depend ^{on steric}

effects and Gl ^{must be} equal ^{to 1.}

2. Calculation of Gl.

^{In such a} bidimensional

isotropic dielectric the correlation factor Gl ^{is related}

to the electric susceptibility ^xl by [13] :

where fl ⁼ 1/kT ^{and V} is the volume of the crystal.

The electric susceptibility ^xl is defined by :

where Pl ^and El ^are respectively ^the components, ⁱⁿ the smectic planes, ^{of the} polarization per unit volume P and of the electric field E; ^{and e =} E/E.

Performing ^the average ^over all the possible ^confi- gurations ^{of the} crystal, ^{xl can be} expressed ^{as the}

second derivative of the partition function of the

crystal [13] ; ^thus Gl ^is given by :

with

and ZN(E1.) ^{is the} partition ^{function of the} crystal

in the presence of the electric field.

The starting point of the calculation of G 1. ^{is to}

express Z’(E_L); ^the description of the molecular orientational disorder described above enables us to write the partition function in the following ^suitable

form :

where si is the orientational state of the ith molecule,

the summation ^is performed ^{over all the} configura-

tional states { s I ^{of the} crystal, ^{the first} product ^is

over all pairs ij > ^of nearest-neighbour ^molecules;

the second one over all the sites i of the lattice.

The definition of the compatibility ^function A(si, Sj)

is :

Because of the presence of the electric field the orientational states si ^{of a molecule i are not} equi- probable ^and g(si) ⁱⁿ (6) represents ^{the «} activity » ^of

the orientational state sj, defined by :

For sake of simplicity, ^{we shall assume, in the} following, ^{that p,} is contained in the molecular plane;

besides, because the permittivity ^{tensor is} isotropic

in the smectic planes, ^we shall suppose the electric field to point along the bissectrix of the angle ^between

the dipolar ^{states 1 and 2} (Fig. 3).

The orientational activities g(si) ^{are thus} equal ^{to :}

Fig. ^3.

^{The 6} dipolar ^{states and the direction} of the electric field E 1. in the model (ii) (projection ⁱⁿ the smectic plane).

with

According ^{to the} general weak-graph ^formula-

tion [16], A(si, Sj) is written as :

where :

In the general ^{case of} any ^« activity » b(si) ^the weak-graph ^{condition [17] is :}

By applying this condition to the « activities »

g(si) ^{we obtain :}

These three equations impose relations between x

and y ^{which are not} compatible, ^so that closed weak-

graph expansion ^is impossible [18]. ^The way ^to eli- minate all the _open graphs ^{is to} perform ^a graphical

renormalization : i.e., ^to define three independent pairs ^of { x, y I coefficients and to introduce a new

« activity » b(si) which takes the values a, b ^{and 1} corresponding respectively ^to the three types ^of

configurations given ⁱⁿ (9) ; these activities are tempo- rarily arbitrary ^ones.

Then, ^we impose ^the closed-weak graph ^condi-

tion (13) ^{to the} b(si) ^{and to the three} pairs f x, y

this leads to the relations :

According ^to the direction of the ( ij ) bond, the steric incompatibilities (A(si, Sj)

^{0) lead to the two}

following values of K :

1176

There are two types ^of compatibilities :

-

compatibility Cl : (/ ) ^{bond non} occupied

-

compatibility C2 : ij ) ^bond occupied by ^{one molecule}

Let be N the total number of molecules and _suppose that, ^{for one} configuration, ^{there are} (a+ N) ^molecules

the dipole of which is parallel ^to the electric field and (a- N) molecules the dipole of which is antiparallel; thus,

there are :

The partition function of the crystal (6) ^can be rewritten as :

where the g(si) ^are still those defined in (9).

a + and cx - vary with each { s } configuration, they ^are implicit functions of the g (si) ^and any ^term depending

on them cannot be factorized.

The graph renormalization technique consists of eliminating ^{these terms} by incorporating ^{them in a new}

« activity » b(si) ^{defined as :}

So that :

Imposing ^{the condition} (13) ^{to the} b(si) defined in (20) the values of a, b, x 11, y 11, x-u., y4, ^{xl and yl are defini-} tively ^fixed by (15) and it is then possible ^to develop ZN(E.L) ⁱⁿ closed weak graph ^series.

6

Since Y ^b(Si)

^{2 a + 2 b + 2, ZN(El) can be written :}

s,=i

I.e. :

where N Ge is the number of closed graphs ^{with e} edges ^{and the} weight of which is W Ge"

For the hexagonal lattice the first closed graphs ^{are the 3 N} triangles, they ^{are all of the same} weight :

From the relations (4) ^and (10) the correlation factor is equal ^{to :}

Since ( JlI >LO ⁼ Jli, ^{the first terms of the} Gl expansion ^{are :}

knowing ^{that :}

We obtain :

This leads to :

3. Conclusion.

The numerical expression ^of G, given by (28), ^{shows the} efficiency of the closed weak

graph renormalization method : the series expansion

of Gl converges very rapidly and the third order term is much weaker that the ^zero order term which is the preponderant ^one.

Concerning the value of Gl, we can see that the dielectric correlation factor Gl ^is greater ^or equal ^{to 1 :}

its value _goes from 1 for _very restrictive steric effects to 1.25 for softer ones.

So, ^{we can state} that the correlations of steric

origin ^{tend to induce a} local order of dipoles ^{in the}

smectic planes of ferroelectric type.

We must point ^out that this calculation of Gl

is valid for an uniaxial compound, characterized

by ^a pure hexagonal lattice in the smectic planes :

i.e. a non-tilted smectic B phase.

It can be assumed that for ^a tilted smectic B phase,

in which the hexagonal lattice is only slightly ^dis- torted, ^this calculation of Gl remains valid.

When the correlations are long-range the dielectric correlation factor G 1. defined in (1) ^is different from the Kirkwood correlation factor _{g 1.} which is deter- mined by ^the experimentalists ^{from the} measurement of the static permittivity [19]. Indeed, contrary ^to Gl, g 1. does not depend ^{on the} shape ^{of the} sample [20]. However it has been shown that, ^{in the case of} short-range correlations, G 1. = g_L [21].

In our problem, the correlations of steric origin

are short-ranged [11, 13]; thus, any comparison

with the experimental ^value g 1. should be immediate.

Such a comparison ^{would be} very interesting, particularly ^{in order to test} if, compared ^{with the} dipolar interaction, ^the correlations of steric origin

between phenyl rings play a prominent part ^{in the} dielectric properties of smectic B phases.

Indeed, if steric repulsions ^were preponderant, they should lead to an athermal dielectric correlation factor G 1..

Only ^one experimental study [6], performed ^{on the}

1178

507 compound ^{of the} p-pentoxy-benzylidene-alkyl-

aniline series has attempted ^to obtain, ^{in the smectic}

B phase, the values of gl and g ^{I, from the experimen-}

tal _8.1 and 811. Unfortunately, ^this calculation which is an approximation of the static dielectric theory ^of Bordewijk [1], ^{does not lead to realistic} expressions

of the dielectric correlation factors and makes _any

comparison with the theoretical value impossible.

Acknowledgments.

The author would like to thank Dr. M. Descamps for his collaboration and D. Druon and J. M. Wacrenier for helpful discussions.

References

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