Molecular properties of some nematic liquids. II. Refractive index and order parameter

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Molecular properties of some nematic liquids. II.

Refractive index and order parameter

I.H. Ibrahim, W. Haase

To cite this version:

I.H. Ibrahim, W. Haase. Molecular properties of some nematic liquids. II. Refractive index and order parameter. Journal de Physique, 1979, 40 (2), pp.191-198. �10.1051/jphys:01979004002019100�.

�jpa-00208898�

Molecular properties ^{of some nematic} liquids. ^{II. Refractive}

index and order parameter

I. H. Ibrahim and W. Haase

Institut für Physikalische Chemie, Technische Hochschule Darmstadt, 61 Darmstadt, Petersenstr. 20, ^R.F.A.

(Reçu ^le 1 er août 1978, révisé le 23 octobre 1978, accepté le 30 octobre 1978)

Résumé.

On mesure les densités et les indices de réfraction d’une série de phényl benzoates disubstitués 4,4’, de deux isomères de phényl thiobenzoates disubstitués 4,4’, ^des produits Merck ZLI 1052 et ZLI 389, ^{en fonction} de la température ^{dans les} phases nématiques ^et isotropes. ^On compare les modèles de Vuks ^et de de Jeu-Bordewijk.

On calcule les polarisabilités moléculaires sur la base de ces modèles et les paramètres ^{d’ordre en sont déduits}

suivant le modèle de Vuks. On discute les effets de la nature des substituants terminaux des groupes ^{centraux sur} le

paramètre d’ordre. Les composés ^{isomères ont} à peu près ^{le même} paramètre d’ordre. Les composés ^{nitriles et} alkyls ^{ont des} paramètres ^d’ordre supérieurs ^{à ceux des} composés alkoxys. ^{Esters et} thioesters ont des effets à _peu

près identiques.

Abstract.

The refractive indices and densities of a number of 4,4’-disubstituted phenyl benzoates and the Merck nematic phases ZLI 1052 and ZLI 389 are reported ^as functions of temperature in the nematic and isotropic phases.

Also two isomers of 4,4’-disubstituted phenyl thiobenzoates are investigated. Comparison of Vuks model with that of de Jeu-Bordewijk is made. Based on these models and some assumptions, the molecular polarizability anisotropies ^are calculated. Then the order parameters ^{are deduced} using Vuks model. The effect of the terminal substituents and central groups ^on the order parameter is discussed. The isomeric compounds ^have nearly ^{the same}

order parameter. ^For compounds with similar chemical framework the order parameters ^{in the case} of nitriles

are higher than in the case of alkoxides. Also it is higher ^{in the case of} alkyls than in the case of alkoxides. The effect of the esters and thioesters on the order parameter ^is nearly ^identical.

Classification Physics ^Abstracts 61.30-62.10-78.20

1. Introduction.

This work is part ^{of a} study ^of

the temperature dependence of the order parameter in nematic liquid crystals. ^{As a} consequence of the

long-range orientational order of the nematic phase [1], ^{most of the} physical properties ^are anisotropic enabling ^{a choice} among several parameters ^for describing ^the degree ^of ordering. Herein, the refrac- tive index has been chosen as a macroscopic property

to represent ^the degree ^of ordering ^{in nematic} liquids.

Measurements of the refractive indices of liquid crystals, using ^different methods, ^{have been} reported [2-6].

We present ^here measurements of the refractive indices and densities of four compounds ^of 4,4’- disubstituted phenyl benzoates,

the Merck mixtures, ^nematic phase ZLI 1052 and nematic phase ^ZLI 389, ^{and two} isomers of 4,4’- disubstituted phenyl thiobenzoates

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004002019100

192

We notice that the 4-methoxy-benzoic ^acid- [4’-n- pentyl-phenylester], 4-n-hexyloxy-benzoic ^acid- [4’-n- pentyl-phenylester] ^and 4-n-butyl-benzoic ^acid- [4’-n- hexyloxy-phenylester] ^were investigated ^earlier [4].

We shall study ^the 4-n-pentyloxy-benzoic ^acid- [4’- cyano-phenylester] ^and 4-n-pentyl-benzoic ^acid- [4’- cyano-phenylester] using ^the reported refractive index and density ^data [5].

2. Theoretical background.

^{It is} customary ^to

assume that in the nematic phase ^the constituent molecules can be represented by simple rigid ^rods.

The order parameter ^{S has been} defined from ^a

microscopic point ^{of view as} [7]

be treating ^{it as a} statistical _average of individual molecular behaviour. 0 is the angle between the long

molecular axis and the optical ^axis (director).

The macroscopic ^{tensor order} parameter Q ^{can be} defined through ^the polarizability anisotropy ^as [8]

where aa is the anisotropic part ^{of the} polarizability

tensor a and Aoe the polarizability anisotropy ^{of an}

isolated molecule.

OC 11, a22 and _X33 which are the principal polarizabilities

of a molecule, may be derived from refractive index measurements on a single crystal. al is the longitudinal

molecular polarizability (along ^the long ^molecular axis) ^and at the transverse molecular polarizability (across ^the long ^molecular axis). ^{The mean} polari- zability ^{has the value}

The subscripts Il ^{and 1 refer to the} parallel ^and per-

pendicular directions to the director.

It can be _seen, for axially symmetric ^molecules,

that [8]

Combining eqs. (4) ^and (5) ^{we obtain}

and

It has been shown by Chandrasekhar and Madhu- sudana [9] ^{that the} Vuks formula [10] ^{works reasona-} bly ^{well for} liquid crystals. The formula is written as

where nj ^and oej ^{are the} refractive index and the effec- tive molecular polarizability along ^the principal j axis, respectively, ^{N is the} number of molecules _per unit volume and

In Vuks formula the local field is assumed to be

isotropic. However, ^the application ^of this formula

to liquid crystals ^{leads in} practice ^to satisfactory

results [4, 5, 9, 11, 12]. ^The attempts ^made by Neuge-

’

bauer [13] ^{and Dunmur} [14] ^to calculate the local field corrections have been less successful.

Recently ^{de Jeu and} Bordewijk [15] ^{have found}

that the optical dielectric anisotropy ^is proportional

to the diamagnetic susceptibility anisotropy. ^There-

fore the local field for axially symmetric ^molecules

can be taken as independent ^{of the} anisotropy ^{of the} surroundings ^{of a molecule.} Representing ^{the mole-}

cules by homogeneously polarizable spheroids, the

principal refractive indices of the nematic phase ^are

related to the molecular polarizabilities ^{and the order}

parameter by

Al ^and A, ^{are the} depolarizing ^factors given by

for a prolate spheroid, ^{with w2}

a2/(a2 - b2).

a and b are the long ^{and short axes of the} spheroid.

The length of the all-stretched molecule is taken

equal to a ; ^b is obtained by equating ^{the volume of}

the spheroid ^to the volume available to a molecule in the nematic phase ^{at the} nematic-isotropic tempe-

rature (TNI). Eqs. (8) have been used by ^{the authors}

to calculate the molecular polarizabilities al and _at

using ^order parameters deduced from the magnetic

data [15].

Maier and Saupe [16] ^have formulated a description

of the long-range ^molecular ordering ^{in real nema-} togens. ^Their theory predicts ^a theoretical order parameter ^{which is} proportional ^{to the} parameter

TVN2(T)ITNI VN2(TNI), ^where VN(T) ^{is the molar volume}

in the nematic phase ^at temperature T, ^and gives

S

0.429 at the transition (N-I). ^The Maier-Saupe (MS) theory ignores ^the contributions of the short- range forces ^to the intermolecular potential. ^None-

theless the temperature dependence ^{of the} theoretical

S is in good agreement ^with experimental data, ^but drffers at temperatures ^{close to} TNl [17, 18].

3. Experimental.

^The refractive indices nli, ^{n 1.}

and n; ^{in the} nematic and isotropic phases, respecti- vely, ^{were measured as} functions of temperature ^with

a Leitz-Jelley microrefractometer. The alignment ^of

the nematic liquids along ^the long axis of the prism

is achieved by rubbing the surface of the prism. ^When

a drop ^of anisotropic liquid (LC) ^is placed ^{in the} angle ^{of the} prism, ^the light passing through ^the prism

is doubly refracted and two images are seen on the scale. One of the images ^{is due to} ordinary rays and the other image ^{is due to} extraordinary rays. Thereby

the ordinary refractive index _{no =- n,} and the extraor-

dinary refractive index ne _{=- nl,} ^.can be measured at the same time. The nematic phase ^is positively ^bire- fringent, ^i.e. (n ^Il

^nl) > 0. In the isotropic phase n

^nl

^{n;. The} prism ^was put ^{in a} Mettler FP5

heating stage. ^{The rate of} heating ^or cooling ^was 1°/min. ^The refractive index determination for sodium

light ^{can attain an} accuracy of ± ^0.002 for n and

± ^0.001 for _nl and _n;.

The densities of all the compounds ^measured except those of the phenyl thiobenzoates, ^{were determined}

as functions of temperature ^{with a} digital density

meter DMA 02 C (Anton Paar, Graz, Austria) ^with

an error of ± ^{1 x} 10-4 g. cm-3. On the other hand, the density measurements for the two isomers of

phenyl thiobenzoates were carried out in a bicapillary

pycnometer ^of nearly ^{0.5 cm3} volume. The accuracy of the measurements with the pycnometer ^was

± ^0.003 g.cm-3.

4. Evaluation of Aoe.

If Ax is not known, ^the order parameter ^cannot be extracted. We shall discuss in the following ^three possibilities ^to evaluate A(x in the absence of the single crystal ^data.

a) For average polarizabilities, ^{it was shown by} Denbigh [19] ^{that an} empirical ^{additive scheme of}

bond polarizabilities ^can give ^the average polariza- bility ^{of a} compound. ^For directional properties ⁱⁿ anisotropic crystals it is necessary ^{to use, not an}

average polarizability ^{for each bond, but an} ellipsoid

of polarizability. ^The polarizability contribution in a

principal direction k ^{of the} polarizability ellipsoid ^of

a polyatomic molecule is given by

(summed for all the bonds), ^{where 0 is the} angle

between a bond and the direction k. _al and _at are the

polarizabilities along ^{and across the} bond, respecti- vely. ^{In the case} of the benzene molecule, the contri- bution along ^a direction k is given by

0i, e2 ^and 63 ^{are the} angles ^between oc 11, a2z and _a33 and the direction in question, respectively. ^{Values of}

the polarizabilities ^{for a} number of individual bonds and the benzene molecule, for sodium light, ^are

summarized in table I.

The polarizability anisotropy ^{da can be calculated}

from the bond polarizability ^{data. We draw a} figure

of the molecule assuming ^{standard values for the bond} lengths ^and angles. The molecular axis is taken to be

along ^{the line} joining ^{the outer} para carbon atoms

of the two benzene rings ^{and assume the molecule}

to be rigid (Fig. 1). Because the polarizability ^aniso-

tropy ^{of the C-H bond is} small, ^the contribution of the -C6H4 group is considered to be as that of the benzene rings. Moreover, the différent _{CH groups} of the terminal substituents are in différent directions with the molecular axis. Their contributions are taken to be equal ^{to their} average polarizabilities. ^Conse- quently ^{the sum of the} polarizability contribution of all the various bonds and _groups parallel ^{to the}

molecular axis gives al for the molecule. Also the

sum of all contributions normal to the molecular axis

gives a,.

b) By aid of the order parameters calculated from the magnetic ^data using the calculated Ax ^{from the}

bond and molecular susceptibilities (part 1) [23], together ^{with the} experimental ^{data and} eq. (5), ^one

can get ^Da.

Table 1.

Bond and molecular polarizability ^data, for the D-line of ^sodium.

(*) ^{ex is} reported ⁱⁿ literature [22]. ^While al is calculated by ^an empirical equation [19] : al ^x 1025

0.98 r6 + 6.0, where r is the

bond length ⁱⁿ Ångströms.

194

Fig. ^1.

^The shape of molecules

c) ^It has been shown that log (S Aoe) appears ^to be

proportional ^to log (- t) ^{in the low} temperature

region [4, 11]. ^{T is a reduced} temperature ;

If one extrapolates ^the straight line section of the

log (S Aa) - log (- r) ^{curve to} the absolute zero, where S

1, ^the intercept ^will yield ^{Da. Haller} [24]

and Horn [25] have discussed similar extrapolation procedures ^{to evaluate Acx.}

5. Results and discussion.

The indices of refrac- tion njj, ^{ni and n; and the densities in} the nematic and isotropic phases ^are reported ^{in table II as func-}

tions of temperature. The relative volume changes

are deduced with an accuracy of ± 0.05 %, ^{are listed}

in table III. These values do not vary ^too much for the various compounds studied. Values of Aoe for the studied compounds, obtained from the three methods discussed above, ^are summarized in table III, ^too.

If the Vuks model is correct then not only ^is

but also

In place of eqs. (10) ^and (11) ^{the de} Jeu-Bordewijk

model suggests

Table II.

Densities and refractive indices, for the D-line of sodium, of ^the compounds investigated.

Table II (continued).

Table III.

Molecular parameters of ^the compounds ^studied, for the D-line of ^sodium.

(*) Quantities ^{are in} principle temperature-dependent ; ^the figures quoted for these apply just ^below TN,.

196

Table III (continued).

Table III (continued).

with

where

and

If i’deduced from _eq. (13) ^{is more} nearly independent

of temperature ^{than oc} obtained from _eq. (11) ^then

one shall have some method for descriminating

between the model of de Jeu and Bordewijk ^{and that}

of Vuks. However both quantities vary with tempe-

rature to about the same extent though ⁱⁿ opposite

senses, the variation being ^small enough ^{to be} negli- gible ^{for most} purposes.

Eqs. (8) ^and (14) ^{are used to} calculate ^{values for} oc{ ^and o. ^The quantities ^{of 3} Da’/(n2 ⁺ 2) ^which play ^the role, according ^{to de Jeu and} Bordewijk,

of Aoc in the Vuks model, ^{are tabulated in table III.}

In table III, ^{the entries in rows} (i) ^and (ii), ^and (iii)

should be unity if Vuks and de Jeu-Bordewijk ^models

are right, respectively. ^{The entries in row} (ii) ^{are not} unity. ^The explanation may be that there are errors

in the calculated polarizabilities, implying ^{errors in}

Aoc. Unfortunately the literature data for the type ^of

compounds ^discussed in this paper ^are rare, and even

less is known about the validity ^of the additive bond

polarizability ^{scheme. The extent to which the} figures

in the last four rows of table III vary is presumably

a measure of the uncertainty ⁱⁿ the final results for S.

Using ^{the mean value of the} figures in the last four

rows of table III and _eqs. (5) ^and (7), ^the order para-

meters are deduced. These S are shown in figures ^2-4

as functions of a reduced temperature (r) ; ^{the broken}

line is obtained from the MS theory. ^The temperature dependence ^{of S} agrees reasonably ^{well with that}

predicted ^from the MS theory, ⁱⁿ general ^{the curves}

are shifted, but differs at temperatures ^{close to} TNI.

For the mixture (nematic phase ^ZLI 1052) ^{and its} components 4-methoxy-benzoic ^acid- [4’-n-pentyl- phenylester] ^and 4-n-hexyloxy-benzoic ^acid- [4’-n- pentyl-phenylester], ^{the order} parameters ^{as a function}

Fig. ^2.

^Order parameter ^{vs. reduced} temperature ^for

and the nematic phase ^{ZLI 1052} ( x ). The broken line is from

the MS theory.

Fig. ^3.

^Order parameter ^{vs.. reduced} temperature ^for

198

Fig. ^4.

^Order parameter ^{vs. reduced} temperature ^for

of T (Fig. 2) ^{are found to} justify ^the additivity ^rule

with

where _xi and Si ^are the mole fraction and the order parameter ^of component i, respectively.

Figures ^{3 and 4} show that the temperature depen-

dence of S for the isomeric compounds ^are nearly

identical. The discrepancy in the absolute values of S for the two isomers shown in figure ^{3 is not} meaningful

because it corresponds ^{to the} uncertainty ^{in S.}

For the compounds ^{of the structure}

and

it is inferred that the order parameters ^{for structures} with R’

CN and R

C5 H11 ^are higher ^than

those with R’ = OC6H13 ^{and R}

OCSH11 (Fig. 4).

This is probably ^{due to} short-range interactions between the molecules resulting in better parallel

arrangements. Generally, ^{we can} conclude that the

alkyls ^{make the order} parameter higher than do the alkoxides. This is in agreement ^with literature results

[18]. Moreover, figure ⁴ shows that the central _groups esters and thioesters have nearly ^{the same contribu-}

tion to the order parameter.

Finally, ^a comparison between the order parameters obtained from the optical data and those from the

magnetic data [23] shows that the agreement ^is satisfactory.

Acknowledgments.

^{One of us} (I.H.I.) gratefully acknowledges ^{the financial} support by ^{the Mission} Department ^of Egypt. ^{We thank E.} Merck, ^Darm- stadt, ^for providing compounds. ^Part of this work

was supported by the Deutsche Forschungsgemein-

schaft.

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