HAL Id: jpa-00208270
https://hal.archives-ouvertes.fr/jpa-00208270
Submitted on 1 Jan 1975
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Simple model for the smectic A-smectic C phase transition
R.G. Priest
To cite this version:
R.G. Priest. Simple model for the smectic A-smectic C phase transition. Journal de Physique, 1975, 36 (5), pp.437-440. �10.1051/jphys:01975003605043700�. �jpa-00208270�
SIMPLE MODEL FOR THE SMECTIC A-SMECTIC C PHASE TRANSITION
R. G. PRIEST (*)
Naval Research Laboratory, Washington, D. C. 20375, U.S.A.
(Reçu le 18 décembre 1974, accepté le 16 janvier 1975)
Résumé. 2014 Un modèle basé sur les propriétés d’orientation du tenseur de second rang des phases smectiques est développé. Le modèle prédit une transition du second ordre smectique A-smectique C
du type superfluide si certains paramètres du modèle possèdent les valeurs appropriées. Une carac- téristique du modèle est que la biaxialité de la phase smectique C ne joue qu’un rôle mineur. La libre rotation autour du grand axe moléculaire n’est pas interdite dans la phase smectique C.
Abstract. 2014 A model based on second rank tensor orientational properties of smectic phases is developed. The model predicts a second order smectic A-smectic C phase transition of the superfluid type if certain model parameters have appropriate values. A feature of the model is that the biaxiality
of the smectic C phase plays only a minor role. Free rotation about the long molecular axis is not forbidden in the smectic C phase.
Classification Physics Abstracts
7.130
Recently there has been a high level of interest in the problem of developing models for the various smectic phases of liquid crystals [1-7]. In particular,
de Gennes has pointed out that the smectic A-smectic C
phase transition is formally analogous to the super- fluid phase transition [5, 8]. A molecular statistical
theory which agrees with the basic structure dis- cussed by de Gennes has been developed by Meyer
and McMillan [7]. The mechanism in the Meyer-
McMillan model which produces the tilt of the
smectic C layers also causes a loss of free rotation about the long axis of the rod like molecules. This last feature is not observed in experiment [9]. We develop here a different model of the smectic A- smectic C phase transition which also has the general
features predicted by de Gennes but does not forbid free rotation about the long molecular axes.
The most successful models [10, 11] of liquid crystal systems are those based on second rank tensor orientational properties of the constitutive molecules. This fact is the motivation for the cons-
truction of the model discussed here. We assume that there is an effective molecular second rank tensor
qij’ which is responsible for the orientational pheno-
mena in the smectic phase. It is useful to introduce
the average of qij :
The average is taken over molecules in the vicinity
of the point r [12]. Both qij and Qij are second rank
traceless symmetric Cartesian tensors. Note that
(*) National Research Council-Naval Research Laboratory
Postdoctoral Research Associate.
Qij may be uniaxial even if qij is biaxial. In the usual
case (nematic and smectic A) Qij is uniaxial and we
identify the unique principle axis of Qij as the director,
the preferred direction of orientation for the long
molecular axes.
We assume that the two body distribution function for the centers of mass can be decomposed as [12] :
Here p(r) is the density of centers of mass of the
molecules. It is not independent of r for smectics.
If the orientational interaction between two molecules is V12, the energy density is given by :
Using the idea that only second rank orientational forms are necessary we expand the product f V12 >
in a series bilinear in Q :
These are the only possible terms. The objects R"’(r)
are mth rank tensors composed from the components of the unit vector r.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003605043700
438
The functions gm(’12) are expected to have maxima
near some value ro comparable with a molecular
dimension. We shall assume that /2 and 14 are both positive. This is the sign that arises from interaction
favoring the side by side parallel configurations over
the end to end parallel configuration for the long
molecular axis. In fact, I2 > 0 is required for mole-
cules to be oriented perpendicular to the planes [14].
If I4 > 0 the smectic A-smectic C transition is
possible.
If /4 0 the smectic A phase is always stable.
We assume the form :
for the periodic (to the Xl direction) density. Actually
this form is not necessary but is convenient for calculations. Substitutions of eq. (4), (6) and (2)
into eq. (3) leads to the orientational energy density.
Furthermore, it has been shown that the correspond- ing Landau-Ginsberg free energy density, F, for the case 12 = I4 = 0 is in the form [2, 11] :
The coefficients A, B, t and u depend on the tempera-
ture and on (llp)2. As shown by McMillan [2], the Ap dependence leads to the nematic-smectic A phase
transition. We assume that near the smectic A- smectic C phase transition these coefficients are
essentially constant. We assume that 12 and 14 are sufficiently small so that eq. (7) need be augmented only by the additional energy density from eq. (3).
This contribution is, neglecting gradient terms :
where C and D are constants proportional to (,àp)’
and
The easiest case to work out in detail is when g2 and g4 may be approximated by delta shells at some
distance ro. Then Lm = L.(qro) and are easily given
in terms of elementary functions. We find L2[L4]
is monotonic decreasing (increasing) for qro n.
The quantities J2 and J4 are quadratic uniaxial
invarients
To complete the mean field approximation analysis
of this model it is necessary to minimize eq. (7) as augmented by eq. (8) neglecting the r dependence of Q.
That is to minimize :
To this end we introduce the form [ 15] :
Here ml, m2 and m3 are a set of orthogonal unit
vectors giving the orientation of the principle axes
of Q. The angle y is a measure of the biaxiality of Q.
Without loss of generality we take :
Since Xl is the direction normal to the layers, 0=0
in a uniaxial smectic A. In smectic C, 0 is the tilt angle.
Substituting into eq. (11) we find
Eq. (14) must be minimized with respect to 0, y and M. In the case y = 0 :
Since D’ and C’ are different functions of qro and D’
is more rapidly increasing than 1 C’ for small values of the argument, it is possible for a second order smectic A-smectic C phase transition to occur if the ratio I4lI2 is large enough. The smectic C will be the low temperature phase because q increases
as T decreases. It is easy to see that it is sufficient that the coefficients of the higher harmonics of q
increase as the temperature decreases. This leads toao - (Tc - T) 1/2 law as predicted by de Gennes [5]
and in agreement with Meyer and McMillan [7].
As q increase the quantity on the R.H.S. of eq. (15)
does not increase without limit -thus giving saturation
of 0 an expanation in terms of this model.
For () =F 0, y * 0. However the cubic and higher
order terms in eq. (14) favor small y. It is not useful to calculate y and M from eq. (14) because the terms of order M5 and higher are important. However
from eq. (14) we can deduce :
The effect of non zero y on eq. (15) is to introduce a
factor of order 1 multiplying the right hand side. It should be noted that in contrast to the model of reference [7], the biaxiality is induced by the symmetry of the smectic C phase rather than the smectic C
tilt being the result of a tendency to form a biaxial phase. Biaxiality plays only a minor role in the model.
If the molecular tensor qij is uniaxial there will be free rotation about the long molecular axis in the smectic C phase.
The nature of the phase transitions can be seen by
an alternate analysis of eq. (11) in the smectic A region.
We decompose Qij as [ 15]
Substituting in eq. (11) gives :
We introduce M via the substitution a 1 = L + M and demand that terms linear in L vanish. This leads
to the equation :
Using this relation we find the coefficients of the
quadratic terms in F are :
So at the transition only the coefficients of a24 and a’
vanish. Thus only 2 independent components of Q
have large fluctuations. Although this phase transition
is not of the usual order-disorder type, the critical exponents will be the same as for the superfluid case
as discussed by de Gennes [5].
As discussed above the effective value of q is the
crucial variable in the vicinity of the phase transition.
This has two important consequences. The first is
440
that it is easy to induce the phase change by applying
pressure. This has been observed by Durand [16].
The second is that density fluctuations will be strongly coupled to orientational fluctuations near the tran- sition temperature. Consequently the nature of the phase transition will be affected and the analogy to
the superfluid broken. The magnitude and nature of
this effect will be discussed elsewhere.
The author would like to thank J. M. Schnur for useful suggestions. The National Academy of Sciences- National Research Council is thanked for providing
funds that made this work possible.
References [1] KOBAYASHI, K. K., J. Phys. Soc. Japan 29 (1970) 101.
[2] MCMILLAN, W. L., Phys. Rev. A 4 (1971) 1238.
[3] MCMILLAN, W. L., Phys. Rev. A 6 (1972) 936.
[4] DE GENNES, P. G., Solid State Commun. 10 (1972) 753.
[5] DE GENNES, P. G., C.R. Hebd. Séan. Acad. Sci. B 274 (1972) 758.
[6] LEE, F. T., TAN, H. T., YU MING SHIH and CHIA-WEI WOO,
Phys. Rev. Lett. 31 (1973) 1117.
[7] MEYER, R. J. and MCMILLAN, W. L., Phys. Rev. A 9 (1974)
899.
[8] For a discussion of smectic A and smetic C see : DE GENNES, P. G., Mol. Cryst. Liq. Cryst. 21 (1973) 49.
[9] Luz, Z. and MEIBOOM, S., J. Chem. Phys. 59 (1973) 275.
[10] MAIER, W. and SAUPE, A., Z. Naturforsch. 14a (1959) 882 and 15a (1960) 287.
[11] DE GENNES, P. G., Phys. Lett. 30A (1969) 454.
[12] This idea is discussed more fully in FREISER, M. J., Phys. Rev.
Lett. 24 (1970) 1041.
[13] This type of decomposition was first introduced by KIRKWOOD, J. and MONROE, E., J. Chem Phys. 8 (1940) 845.
[14] Tilt angles of 90° have never been observed.
[15] PRIEST, R. G. and LUBENSKY, T. C., Phys. Rev. A 9 (1974) 893.
[16] DURAND, G., report on work prior to publication.