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THERMODYNAMICAL PROPERTIES OF DILUTE BINARY MIXTURE OF NEMATIC LIQUID
CRYSTALS
C. Ishii, H. Yamada
To cite this version:
C. Ishii, H. Yamada. THERMODYNAMICAL PROPERTIES OF DILUTE BINARY MIXTURE OF NEMATIC LIQUID CRYSTALS. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-374-C3-375.
�10.1051/jphyscol:1979373�. �jpa-00218769�
JOURNAL DE PHYSIQUE Colloque C3, suppliment au no 4, Tome 40, Avril 1979, page C3-374
THERMODYNAMICAL PROPERTIES OF DILUTE BINARY MIXTURE OF NEMATIC LIQUID CRYSTALS
C . ISHII and H. YAMADA
Department of Physics, Science University of Tokyo, Tokyo, Japan
RksumB. - Les proprietes thermodynamiques des melanges diluks de cristaux liquide nkmatique sont considkrkes, en se basant sur le modkle statistique de Humphries, James et Luckhurst. En par- ticulier, la dkpendance de la pente initiale des lignes de skparation de phases en fonction #interactions orientationnelles entre les molQules du solvant et du solutk est analysee en dktail.
Abstract. - Thermodynamical properties of the dilute mixtures of nematic liquid crystals are considered, based on the statistical model of Humphries, James and Luckhurst. In particular, the dependence of the initial slopes of phase boundary lines on the orientational interaction between the host and guest molecules is analyzed in detail.
1. Introduction. - The statistical model of mul- ticomponent system of the nematic liquid crystals has been developed in a series of works [l] by Humphries, James and Luckhurst (HJL). When applied to the binary nematic mixture, the model predicts that the nematic transition takes place in two steps ; the IC-transition from the completely isotropic (I) phase to the intermediate coexisting (C) phase at the temperature T,, and the CN-transition from the C phase to the entirely nematic (N) phase at T,. The phase diagram and the other thermo- dynamical properties of the model are governed by the following parameters ; the orientational inter- action constants (i, j = 1, 2) between the consti- tuent molecules and their Van der Waals volumes V, and V,.
Thorough investigation of the model has' been done by HJL and Martire et al. in the case where the second molecules are quasi-spherical, E,, = 0.
Especially, Martire et al. [2] derived the expressions of the initial slopes of the phase boundary lines
P , ,
=-
dT:,,/dx in terms of the ratio of the Vander Waals volumes VJV,
We try here to generalize the relations (1) to the case where the guest molecule is also a nematogen with non-zero interaction constant
2. Free energy differences. - We confine ourselves in the dilute limit (X
<
1) of the guest molecules, and expand the HJL-free energy expressions around the point (To, X = 0) in the phase diagram in powers of X and temperature deviationIn the first order, the free energy difference AF" (or AFm) between the C-phase and I-phase (or N-phase) is given by
Here N is the total number of molecules and the constants a and y are introduced by
cc = ZG' exp 0.418
[
-21 ,
y = (a - 1)/0.418.
(3)Here, T& = Ti,,/T0 are the reduced IC- and CN- transition temperatures, To being the nematic transi- tion temperature of the host nematogens, and X is the mole fraction of the guest molecules in the mixture.
These are the functions of s12/sll through the orienta- tional partition function Z,,
(4) The function Y(z) in the eq. (2) is of the form
l
W(z) = z - l n z - 1
-
(1 - z)'/2, z 1.1 . (5)Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979373
THERMODYNAMICAL PROPERTIES OF DILUTE BINARY MIXTURE OF NEMATIC LIQUID CRYSTALS C3-375
3. Phase boundary lines. - It is clear from Eq.
that the phase boundary lines are described by equations
with the initial slopes
pi,,
1 - Zzo exp
(
- 0.418-
2)]/0.418 ,p,,
=[
ZG exp(
0.418- :>
- 1I1
0.418.
(2) the
From this result, we can conclude that :
i) the effect of the host-guest orientational inter- action is expressed by the function Z,,, eq. (4), and
ii) the crossover of the signature of the slopes
pi,"
occurs for the set of parameters V 2 / V l and ~ ~ ~ l e ~ ~ , related by
as shown in figure 1. The relation (7) defines a curve in the V 2 / V 1 vs. e l z / e l l plane as shown in figure 2.
The phase diagram will look like the one, shown by insets, in each region on the plane.
References
[l] HUMPHRIES, R. L., JAMB, P. G . and LUCKHURST, G . R., Symp. HUMPHRIES, R. L. and LUCKHURST, G. R., Proc. R. SOC. Land.
Faraday Soc. 5 (1971) 107 and J. Chem. Soc. Faraday A 352 (1976) 41.
Trans. 11 68 (1971) 1031. [2] MARTIRE, D. E. et al., J. Chem. Phys. 64 (1976) 1456.
