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Submitted on 1 Jan 1975

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THERMAL EXPANSION AND SPECIFIC HEAT OF A NEMATIC LIQUID CRYSTAL

H. Gruler, F. Jones

To cite this version:

H. Gruler, F. Jones. THERMAL EXPANSION AND SPECIFIC HEAT OF A NE- MATIC LIQUID CRYSTAL. Journal de Physique Colloques, 1975, 36 (C1), pp.C1-53-C1-54.

�10.1051/jphyscol:1975108�. �jpa-00215885�

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JOURNAL DE PHYSIQUE Colloque Cl, suppliment au no 3, Tome 36, Mars 1975, page C1-53

Classification Physics Abstracts 7.130 - 7.500 - 7.520

THERMAL EXPANSION AND SPECIFIC HEAT OF A NEMATIC LIQUID CRYSTAL

H. GRULER (*) and F. JONES

Gordon McKay Laboratory, Harvard University, Cambridge, Mass 02138, U. S. A.

RBsumB.

-

On donne l'expression de I'energie libre de Gibbs en fonction d'une temperature rkduite. Les coefficients d'expansion thermique a et les chaleurs sptcifiques C devraient &re iden- tiques pour tous les nkmatiques. La loi de Griineisen (invariance du rapport a/C en fonction de la tempkrature) est vkrifiee pour le PAA.

Abstract. - The Gibbs free energy of a nematic liquid crystal is expressed in a reduced tempera- ture scale. Therefore the derived thermal expansion coefficient a and specific heat C should be similar for all nematics. Furthermore the ratio of a / C should be temperature independent. This Griineisen law holds at least for PAA where a complete set of data is available.

The Gibbs free energy of a nematic liquid crystal can be expressed with the mean field approximation of Maier and Saupe [I]. Here we use another way of describing the nematic phase by making the good approximation that the degree of order versus reduced temperature is an universal curve. Actually this univer- sal curve was first calculated by Saupe [2] and in a more generalized form by Humpheries et al. [3]. For such corresponding states the Gibbs free energy can be expressed (as in solids [4]) as :

GN = GI + Bf(7) (1) where z = T/O.

Such a form of the Gibbs free energy does not allow us to calculate the degree of order of the nematic phase.

The characteristic temperature 19 is a temperature where the degree of order would be zero. This temperature can be calculated by the Maier-Saupe theory.

The free energy given by eq. (1) gives very accurate results in calculating the thermal expansion coefficient a and the specific heat C,. For the thermal expansion we find that

a = - (ve2)-I ~ ( a e / a ~ ) f ' ~ . (2)

Similarly we get an expression for the specific heat : Cp = - T8-l f " . (3) Comparing eq. (2) and (3) we obtain the following relation :

./cP = (ve)- ( a e l a ~ ) . (4) Thus within the limits of applicability of the law of corresponding states, the ratio of the coefficients of thermal expansion and the specific heat is independent

(*) Present address : University of Ulm, Experimental Physics 111, Oberer Eselsberg, 0-7900 Ulm, West-Germany.

of the temperature. In solid state physics this effect is known as Griineisen's law [4]. We have the possibility of comparing Griineisen's law with experiment. For 4,4'-di(methoxy) azoxybenzene (PAA) we have all the data we need in order to check eq. (4). We use the specific heat from Arnold (see Fig. 1) and the thermal

-0 Y C - O - ~ - ~ + N.Q- O.CH,

-NEMATIC PHASE -

I I 9

110 120 130 1L0

- TEMPERATURE b]

FIG. 1. - Arnold's [5] specific heat data for PAA. Solid lines are predictions from mean field theory. (In the isotropic phase the fit

was done by Imura and Okano [lo].)

expansion coefficient from Saupe [6]. In order to eliminate linear background effects we take the deri- vative with respect to the temperature. In a temperature region of about 15 OC we get, for the left, side of eq. (4),

The right side of eq. (4) can be estimated with the assumption TN, = 8. From McColl's data [7] we get the dependence of the phase transition temperature

5

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975108

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C1-54 H. GRULER AND F. JONES

upon the pressure ((aT,,/d~) = (dB/aP)). From this with P' = 0.5 + 0.05. This exponent does not express

we get the critical behaviour of the specific heat near the phase

(V0)-l (aO/aP) = 5.1 x 10-l3 ergb1. (6) transition temperature ; it only shows that the degree A comparison of eq. (4) and (5) shows that for PAA

the relation (4) is fairly well fulfilled.

The complete set of data for checking eq. (4) is only available for PAA. Therefore we shall take a closer look at eq. (3) because a lot of specific heat data are published. Since f in eq. (1) should be an universal function which holds for all nematic phases, the contri- bution to the specific heat from the nematic ordering should be similar in all cases. The specific heat data of rhe homologous series 4,4'-di(n-alkoxy) azoxybenzene [5] can be expressed as

C: K ((T, - T)/T,)-~' (7)

of order is a universal curve which can be quite well expressed by the square root of the reduced tempe- rature. The latter formula was shown a long time ago by Foex [8]. If we assume that the power law of eq. (5) for the specific heat is correct, then we expect the same power law for the thermal expansion (see eq.

(4)). Press and Arrot [9] found an exponent of 0.56 for methoxybutyl-benzylindene-aniline (MBBA).

This value is close to the value which we expected.

In summary we can say that Griineisen's law which connects thermal expansion with the specific heat is a good approximation for the nematic liquid crystal phase.

References

I11 MAIER, W. and SAUPE, A., Z. Naturforsch. 14a (1959) 882 ; [5] ARNOLD, H., 2. Phys. Chem. 226 (1964) 146.

15a (1968) 287. [6] MAIER, W. and SAUPE, A., Z. Naturforsch. 15a (1960) 287.

[2] SAUPE, A., Angew. Chem. internat. Edit. 7 (1968) 97. [7] MCCOLL, J. R., Phys. Lett. 38A (1972) 55.

[3] HUMPHERIES, R. L. and LUCKHURST, G. R., Chem. Phys.

Lett. 17 (1972) 514. [8] FOEX, G., Trans. Faraday Soc. 29 (1933) 958.

[4] LANDAU, L. D. and LIFSCHITZ, E. M., Statistical Physics [91 PRESS, M. J. and ARROTT, A. S., phys. ~ e v . ~8 (1973) 1459.

(Pergamon Press LTD) 1958, p. 192. [lo] IMURA, H. and OKANO, K., Chem. Phys. Lett. 17 (1 972) 11 1.

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