On the elastic constants of nematics ; a comparison between experiment and molecular theory

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On the elastic constants of nematics ; a comparison between experiment and molecular theory

S.D.P. Flapper, G. Vertogen, F. Leenhouts

To cite this version:

S.D.P. Flapper, G. Vertogen, F. Leenhouts. On the elastic constants of nematics ; a compari- son between experiment and molecular theory. Journal de Physique, 1981, 42 (12), pp.1647-1650.

�10.1051/jphys:0198100420120164700�. �jpa-00209362�

On the elastic constants of nematics ; ^a comparison ^between experiment

and molecular theory

S. D. P. Flapper, ^G. Vertogen

Institute for Theoretical Physics, ^Catholic University, Toernooiveld 1, ^{6525 ED} Nijmegen, ^The Netherlands

and F. Leenhouts (*)

Siemens AG, Optoelectronics Development, ^{RS FKA} TE, St. Martin-strasse 76, D-8000 Munich 80, Germany.

(Reçu ^{le 11 mai} 1981, accepté ^{le 31 août} 1981)

Résumé.

Il est montré _{que le} comportement expérimental ^des constantes élastiques ^de substances nématiques

ne peut pas être interprété à l’aide des théories moléculaires statistiques ^actuelles.

Abstract.

It is shown that the experimental ^behaviour of the elastic constants of nematics cannot be interpreted

in terms of present molecular statistical theories.

Classification Physics ^Abstracts

64. 70E

Purpose of this article is ^to investigate ^the adequacy

of present ^molecular statistical theories [1, 2] ^{for an} understanding ^{of the} experimental behaviour of the elastic constants of nematics. Summarizing ^{it can be}

said that these theories, ^{which are based} upon the

assumption ^{that a truncated} expansion ^{of the inter-} molecular repulsivé and attractive interactions in terms of spherical ^{harmonics is} relevant, give ^{rise to}

the following expressions for the elastic constants

where the constants K1,, K22 ^and K33 ^{are associated}

with splay, twist and bend respectively,

and Pn > denotes the expectation value of the nth order Legendre polynomial. ^The quantitiesa ^{and L1}

can be approximately ^{written as}

(*) ^{Present address :} Videlec, Hardstrasse 5, ^CH-5600 Lenzburg, Switzerland.

LE JOURNAL DE PHYSIQUE

T. 42, N° 12, DÉCEMBRE 1981

where T denotes the temperature. ^The constants a,

c and e describe the attractive intermolecular interac-

tions, whereas the constants b, ^{d and} f represent ^the repulsive intermolecular interactions. The expression

for K reads

where _p denotes the particle ^number density N /V.

For a systeni ^{of hard} spherocylinders, ^i.e. a = c = e = 0,

Priest showed

with R

L/B, where B and L + B denote the breadth and the length ^{of a} spherocylinder respectively.

In order to analyse the relevant theoretical expres- sions the temperature dependence of the six quantities K11, K229 K33, p, P2 / and P4 > ^{has to be known} experimentally. ^At present these six quantities ^have only been measured for MBBA and 5CB [3-6] ^{as far}

as we know. In both substances ( P4 > changes sign

near the clearing point, ^whereas K33 ^remains larger

than K1,. According ^to equations (1)-(3), however, ^the

107

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420120164700

1648

bend and splay ^{elastic constants are} equal ^{at the tem-} perature, ^where P4 ) changes sign. Clearly experi-

ment and theory ^{are in flat} contradiction for these materials.

In a number of nematics, e.g. HNBP [7] ^and

BBOA [3, 8], ^{the order} parameter P4 ) ^{does not} change sign ^{as a} function of temperature. ^Unfortuna- tely these nematics cannot be used for our analysis

because of lack of experimental data of their elastic constants and densities. In other nematogens ^like MBCA and the homologues ^of APAPA, ^{which will}

be denoted by APAPAn, ^{where n} denotes the number of carbon atoms of the alkyl chain, ( P4 ) ^{has not}

been measured. Obviously experiment ^and theory

contradict each other if ( P4 > changes sign ^{as a}

function of temperature ^{in the nematic} region ^of

MBCA and APAPAn, because either K33 ^is larger

than K11 ^or K11 ^is larger ^than K33 ^{for these mate-}

rials [5]. Therefore let ^{us assume} that P4 ) ^{does not} change sign in MBCA and APAPAN and then fmd

out whether the expressions (1)-(9) ^{have any} signi-

ficance.

In figure ^{1 the observed} dependence ^{on the} reduced

temperature ^{of the} quantity 2 p - 2 ( P2 ) 2 K ^is

shown _up to TITNI

^{0.995 for} MBBA, OHMBBA, MBCA and the homologues ^APAPAn (n

1-5,9).

Generally ^{the errors in the} experimental ^{data are about}

5 %, ^but they ^increase quite ^{near the} nematic-isotropic

transition temperature TNI.

Fig,1. - The observed ^reduced temperature dependence ^of

2 p-2 ( P2 >-2 K ^{for the} homologues ^APAPAn (n

1-5, 9) [An],

MBBA [BA], ^MBCA [CA] and OHMBBA [OH]. ^{The data are}

taken from Leenhouts [5].

The data have been taken from Leenhouts [5]. It is

clear that a trend for, ^the quantity 2 p-2 p 2 >-2 K

within the homologous series of APAPA is indicated in figure ^{1. The results of a least} square fit for the constants a and b appearing ⁱⁿ expression (6) ^are given in table I. The observed and fitted reduced tem-

perature dependences agree within 2 % except ^{in the} region ^0.99 T /T NI ^0.995, where the deviation between the observed and fitted data is slightly larger.

The largest ^{deviation concerns APAPA3} being ⁵ %

at T /T NI

^0.995. Taking ^{into account the} experi-

mental ^errors it may be stated that the agreement between expression (6) ^and experiment ^is quite ^satis- factory. Although the values for b ^are small, ^the

influence of b is considerable, ^because b TNI ^{and a are}

of the same magnitude. Table 1 shows clearly ^the importance of attractive interactions in order to des- cribe these nematics. According ^to this table an even-

odd effect for bla exists within the homologues ^of

APAPA. An insurmountable problem, however, ^con-

cerns the negative sign ^{of b. Hard core} repulsions give ^{rise to a} positive b, ^see e.g. expression (9). ^Thus

the molecular statistical expression (6) looses its phy-

sical meaning.

Table I.

Values of ^the parameters ^{a and b} giving

rise to the best fit of ^{the curves shown in} figure ^1.

This conclusion is further sustained by calculating

the quantities _q ^{A and} A’ P4 > for MBCA and the

P2 > ^{o C and the} homologues ^APAPAn (n=1-5,9) using ^the experi-

mental data of the elastic constants and the _expres- sions (1)-(3). ^The resulting dependences ^{of L1 and}

P4>

A’ Pz> on ^{the reduced} temperature ^{are shown in}

B P2 >

figure ^{2 and} figure ³ respectively. ^{For APAPA9 the}

/ P B

quantity A’ P4 > ^is _négative, because Ki i ^is larger

q Y

P2 > ^{g m g}

than K33 in this material. Clearly ^{both L1 and}

4 ’ / P4) P4 > ^are temperature dependent. ^{For a further} P2 >

comparison ^with theory ^{we have to} investigate ^whether

these curves can be fitted with the expressions (4)

and (5). Because P4 > is unknown we can only ^inves-

tigate expression (4). The results of a least _square fit

Fig. ^2.

^{The reduced} temperature dependence ^{of the} quantity d, for the homologues ^APAPAn (n

1-5,9) [An] ^{and for MBCA} [CA].

Fig. ^3.

^{The reduced} temperature dependence ^{of the} quantity

A 1 P2 >for ^the homologues ^APAPAn (n

1-5, 9) [An] ^{and for} CP2i

MBCA [CA].

for the constants c and d appearing ⁱⁿ expression (4)

are given ^{in table} II, ^{where we} used the values of a and b as given ⁱⁿ table I. The observed and fitted reduced temperature dependences agree within 4 %

if T /T NI ^{0.99. In the} region ^0.99 T /T NI ^0.995

the deviations between the observed and fitted reduced temperature dependences ^{of L1} increase till 6 % except in the cases of APAPA and APAPA4, ^{where the} deviations increase till 10 % ^and 15 % respectively.

Table II.

Values of ^the parameters ^{c and d} giving

rise to the best fit of the observed temperature depen-

dence of ^{the quantity A, using the} parameters ^{a and b} given ^{in table 1.}

The experimental ^{errors for} K11 ^and K22 ^{are less}

than 4 %, whereas those for K33 ^{do not exceed 6} %.

Therefore the fit procedure ^{is not} entirely satisfactory

for some of the nematics in question. A second fit

procedure ^{is to treat a and b as free} parameters ^as well. In this _way we get ^an idea about the internal

consistency ^{of the} theory. The results of a least square fit for bla, cla ^and dla ^are given in table III. With the

exception ^{of APAPA9 the observed and} fitted reduced

Table III.

Values of ^{the relative} parameters bla, cla ^and dla giving ^{rise to the best} fit of the observed temperature dependence of ^the quantity ^Li.

temperature dependences ^{agree now within 1} % up to T/TNI

0.995. In the case of APAPA9 the first fit procedure appears ^to be superior. ^A comparison

between the values of b/a, cla ^and dla given ^{in the}

tables 1 and II and those given ^{in table} III indicates

clearly ^{that the} expressions (4) ^and (6) ^are incompa-

tible.

The conclusiôn of this analysis ^{is rather} disappoint- ing. ^An adequate interpretation ^{of the} experimental

behaviour of the elastic constants of nematics is not

possible using ^{the available truncated} theoretical

expressions.

This conclusion is sustained by related work of Karat et al. [9]. It should be remarked, however, ^that

our point of view differs from that taken by ^Karat

et al..

1650

References

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[2] ^{VAN DER} MEER, B. W., Thesis, Groningen (1979) ;

VAN DER MEER, B. W. and VERTOGEN, G., Phys. Lett. 71A

(1979) 486.

[3] JEN, S., CLARK, ^N. A., PERSHAN, ^{P. S. and} PRIESTLEY, ^E. B.,

J. Chem. Phys. ⁶⁶ (1977) ^4635.

[4] MIYANO, K., ^{J. Chem.} Phys. ⁶⁹ (1978) ^4807.

[5] LEENHOUTS, F., Thesis, Groningen (1979);

LEENHOUTS, F. and DEKKER, ^A. J., ^{J. Chem.} Phys. ⁷⁴ (1981) ^1956.

[6] KARAT, P. P. and MADHUSUDANA, N. V., Mol. Cryst. Liq.

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[7] ^{HEGER, J. P., J.} Physique ^{Lett. 36} (1975) ^L-209.

[8] LEADBETTER, A. J. and WRIGHTON, ^P. G., J. Physique Colloq.

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