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Submitted on 1 Jan 1979
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Smectic A to smectic A phase transition
Jacques Prost
To cite this version:
Jacques Prost. Smectic A to smectic A phase transition. Journal de Physique, 1979, 40 (6), pp.581- 587. �10.1051/jphys:01979004006058100�. �jpa-00209142�
Smectic A to smectic A phase transition (*)
J. Prost
Centre Paul Pascal, Domaine Universitaire, 33405 Talence Cedex, France
(Reçu le 20 novembre 1978, accepté le 6 février 1979)
Résumé. 2014 Nous étudions certaines propriétés de l’énergie libre que Meyer et Lubensky ont introduite pour la
description de la transition nématique-smectique A. Nous montrons que pour certaines valeurs des paramètres
entrant dans la théorie, une transition d’un nouveau type d’une phase smectique A à une autre phase smectique A
est possible. Les récentes observations de G. Sigaud et al. peuvent être interprétées dans ce contexte.
Abstract. 2014 We investigate properties of the Meyer-Lubensky free energy introduced in the description of the
nematic to smectic A phase transition, and show that a new transition from a smectic A to another smectic A is possible for some values of the parameters used in the theory. The recent observations by G. Sigaud et al. may
be interpreted within this context.
Classification
Physics Abstracts
61.30 - 64.70E
1. Introduction. - In a recent experiment,
G. Sigaud, F. Hardouin, M. F. Achard and H. Gasparoux, investigating the isobaric phase dia-
gram of a mixture of 4-n pentylphenyl 4-[4-cyano- benzoyloxy] benzoate (subsequently called diben- zoate for short) and terephtal-bis-butyl anilin (TBBA),
showed the existence of a first order phase boundary separating two smectic A domains. The relevant part of the binary diagram is reproduced on figure 1 :
- in the region of low TBBA concentration, there is a direct first order transition from the nematic to the smectic A phase (Branch CD);
- at TBBA concentrations larger than C *, cooling
frôm the nematic phase one obtains the smectic A
phase through a continuous transition (branch @)
and at still lower temperatures a first order SA to SA
transition (branch (3)).
Branch (D and (D have the same slope near the pseudo-triple point Q. Branch Q) obeys well
McMillan’s theory of the N-SA transition [2] : the
ratio T NA/TNI spans a domain including the critical value (T NA/TNI),, - 0.87, and indeed for
the phase change is continuous whereas for Fig. 1. - Dibenzoate, TBBA isobaric phase diagram according
to Sigaud et al. [1] : ---second order transition line; first order transition line. Note the existence of a first order boundary, separating two smectic A regions (branch 3) (labe;ed A and A’).
The point Q is a pseudo-triple point, at which the nematic and smectic A phases are indistinguishable and in equilibrium with
the smectic A’ which constitutes a different phase.
(*) Presented at the 7th International Liquid Crystal Conference, Bordeaux, July (1978).
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004006058100
582
it is of first order. On the contrary, branch 0 does
not obey this theory since TNA/TNI is notably smaller
than the critical value, and yet the transition is clearly
discontinuous. It was suggested by Meyer and Lubensky (subsequently M. L.) [3] that the coupling of
the harmonics of the mass density modulation with the fundamental, was responsible for the first order
character of this branch (in fact only the case of the
pure dibenzoate was discussed ; the first two Bragg spots obtained by X-Rays are indeed found to be
of comparable intensity [4]).
On the other hand, Sigaud et al. suggested that the SA-SA transition could correspond to a change from
a monolayer to a bilayer smectic A (TBBA is known to
be a monolayer whereas the dibenzoate is a bilayer
smectic A [4]). In fact, using the Fourier language the monolayer appears as the first harmonic of the bilayer order, which suggests that the M.L. free energy could be well suited to describe this problem. One simply
has to allow, in certain circumstances, the first har- monic to condense before the fundamental.
In the following, we will consider the TBBA concen-
tration
as an external parameter acting on the coefficients of the Gibbs free energy (per unit mass). The phase
boundaries are in principle obtained from equating
the chemical potentials of each compound in the phases considered. It is, however, easy to show that
one obtains the curve of first appearance of the most
symmetric phase, by minimizing the Gibbs free energy at constant concentration x. The width of the two-
phase region is subsequently determined by equating
the chemical potential difference of the two compounds (ilTBBA -¡LDB)’ across the boundary. Although there
is no difficulty in calculating this width, we will not discuss it, since no phase separation was evident experimentally (within a 0.1 K accuracy). Our calcu-
lations are directly applicable if we consider the
hydrostatic pressure as the external parameter which indeed suggests that pressure investigations might be interesting to perform on the pure dibenzoate
compound.
2. Formulation of the problem. - Using the Lifschitz approach to phase transitions [5], de Gennes describes
the nematic to smectic A phase transition as the onset of a periodic sinusoidal one dimensional modulation of the mass density [6] ; namely, it is always possible to expand the density in a Fourier
series :
In the nematic phase Pq > = 0 ; below the tempe-
rature TNA one of the components takes a non zero
average value p.. * 0(qoen the nematic axis) thus leading to the smectic A symmetry. The Landau free energy describing this transition can be written [6] :
In the absence of any coupling to other variables of the system, B is assumed to be positive and inde- pendent of T. The sign change in A oc a(T-TNA)
drives the (continuous) transition.
In fact, as pointed out by M.L., the smectic sym- metry does not correspond necessarily to the onset
of just one Fourier component ; the development of
the mass density has to be periodic but may contain
higher harmonics : p2qo, p3qo and so on.
As it is clear from (1), the Pnqo’S are complex num-
bers, a change in the coordinate system resulting in
a change of the phase of the pnqo’s. Only those combi-
nations which do not depend on the origin choice
can enter into the free energy calculations. Keeping only the fundamental and the first harmonic one
gets [3] :
Again, as always in Landau theories, the sign change in A1 or A2 will drive the transitions (we will
see that three different types of transition are possible depending on whether A 1 changes sign before A2 or
the reverse).
B > 0 is a constant ; the expression of the fourth
order term is not the most general but renders the
algebra simple without greatly changing the problem.
D > 0 is also a constant. In fact the most general coupling between pqo, and P2qo has the form :
with
However, it is straightforward matter to show that
the free energy is minimized by locking the phase I/J 2
of p2qo on the phase gli of PqO (or reverse) ; indeed (4)
is equivalently written :
obviously minimized by choosing :
Thus 0 describes the intrinsic property of the system of locking the two Fourier components at a
constant phase and (3) is justified. Relation (5) will
be useful in the understanding of the SA-SA’ transition.
Further calculations will be simplified by the
variable change :
(3) then reads :
The equilibrium phases are obtained by minimi- zing F with respect to xi and X2. We find it convenient to first minimize f(x1, X2) with respect to xi defining
thus f(x2) (which can be multivalued) and then
minimize f(x2) with respect to X2. This procedure
is at variance with [3] in which Y2 is supposed always
to be strictly positive which allows the Rodbell Bean [7] analysis to be made. As previously stated
we consider yi and Y2 as independently varying parameters and investigate the (Yl, Y2) phase dia-
gram (Fig. 2).
8fl8xl = 0 leads to :
either
Fig. 2. - (Y1’ Y2) phase diagram obtained from the Meyer Lubensky free energy. N stands for nematic, SA for a smectic A phase in which only one Fourier component of the density modula-
tion is condensed (namely the first harmonic p2qo), SA, a smectic A phase in which both the fundamental P qO and the first harmonic p2Qo of the density modulation are condensed : --- second order transition line ; - first order transition lines. Note the existence of two tricritical point P and R, and one pseudo-triple point Q.
The NSA and NSA, transitions are helium like, while the SA SA’
transition is Ising like.
Note that for (10) to have a meaning, X2 has to lie within the domain D of figure 3(X2 1 > 0) :
One futhermore shows easily that :
(xf given by (10».
Hence f2(x2) is always smaller than fl(x2) except for
(that is on the boundary of D) where the two curves
are tangential. In general, the curve
exhibits a minimum f2(xi) and a maximum f2(x2)
with :
Thus, when the minimum of f2 will fall within D,
f2 will drive the transition. In the opposite case Il will
Fig. 3. - D : Domain in which equations (10) have a meaning.
The lines CD, @ , @ correspond to the value x2(yl) at the tran-
sition temperature (NSA,, for 0) SI SA, for (3) and il’ The (D
phase boundary corresponds to the point (0, 0) of this diagram.
Note that the second order transition lines CI) and D fall on the
boundaries of D.
584
provide the minimum. As a consequence, there are
only three different ways of minimizing the free
energy :
(Note the absence of a x2 = 0, xl 0 SA phase.)
2.1 THE NEMATIC TO SMECTIC A’ TRANSITION. - This
problem is basically that discussed by Meyer and Lubensky, and the results we will get are identical to those of reference [3] in the region where Y2 is not
small (compared to 1). Our results are original in the
small y2 region.
This transition occurs in the y2 > 0 domain, where the minimum of fi is zero. One has thus to compare
f2(X2) to zero (X2 E D).
a) y2 > 1 : the observation of the figure 4a clearly
shows that the :
line defines the second order Q) phase boundary
between the nematic and the smectic A’ phase (Fig. 2).
b) y2 = 1 : figure 4b displays the corresponding
variation of the free energy for différent y, values.
Again, the transition is continuous but for Y, = 0, f2 exhibits an inflexion point at X2 = 0 : this defines the point P of figure 2 y1 = 0, Y2 = 1 [ as a tricri-
tical point.
c) 0 Y2 1 : as shown on figure 4c, there is a
domain for which with y, > 0, x2 and xM lie in the
domain defined (this is also apparent on Fig. 3). As
yi is lowered one reaches a value y’ for which f(x’) = 0. For y, > yb, the free energy is minimized
by fl(X2) (X2 = 0), while for y, y’ 1 it is minimized
by f2(x’). This defines a first order nematic to smectic A’ transition, since
is non zero for yi = y i . ’
yl is easily computed as a function of y2 by remark- ing that for this particular value the equation f2(X2) = 0 has a double root. This is a standard algebraic problem. One gets :
(14) may have three positive roots for some values
of Y2, but y’ is anyway found as the largest y, value
satisfying (14).
YB(Y2) defines branch Q) of the phase diagram :
the curve displayed on figure 2 has been computer
Fig. 4. - Plot of the (reduced) free energy as a function of X2 :
--- fi(x2) ; - f2(x2) in the physically relevant
domain (X2 E D) ; --- f2(x2) in the physically irrelevant
domain (X2 rt D) ; = locus of the free energy minimum.
The structure of the different cases can be easily infered from the relations (9), (10), (11) and (12), together with the knowledge of f2(0) and df/dx2(0). a) Y2 - 1 > 0 ; cases (D, (2), (3) correspond respectively to y, > 0, y, = 0, y, 0. b) y2 = 1 ; the same convention as in 4a apply. c) 0 Y2 1 ; cases (D, (D, (2)
correspond respectively to y, > yb, Yi = y’, Yi y’.y’ is the
value for which f2(Xf,) = 0. d) - i Y2 0; the same conven-
tion as in 4c apply, except that yt1’ is the value for which
/2M) = - y22/4. e) Y2 = - ?; same as in 4d. f ) Y2 - ? ;
same as in 4d.
calculated. 0 and l’ meet at the tricritical point P
where they have the same slope (e.g. :
On the other hand CD meets the line Y2 = 0 at the point Q (yz = 0, yi = 1/4) (in the vicinity of Q :
2.2 THE NEMATIC-SMECTIC A LINE. - When Y2
is negative, and for yi > -1, since the transition toward the SA- phase has not yet taken place, f is
still minimized by fl, but with non zero values of x2.
The sign change of Y2 drives a second order phase
transition from the nematic to the smectic A phase
(X2 * 0, xi = 0). This boundary corresponds to
branch @ of the phase diagram (Fig. 2). This is in fact the line which corresponds most closely to the
initial description of [2] and [6]. In particular if one
adds a coupling with the nematic order parameter, it should obey McMillan’s criterion for the location of a tricritical point.
2.3 THE SA-SA, -PHASE BOUNDARY. - This transition is novel in the literature of liquid crystàls. It corres- ponds to the onset of a density modulation pqo, in a matrix which has already the 2 qo periodicity.
In other words, this would correspond to a transition
from a monolayer to a bilayer smectic.
The minimization procedure is still the same, except that the free energy minimum in the SA phase
is obtained for
value which then has to be compared to f2(xm2 ).
It is furthermore clear (Fig. 4d, 4e, 4fl that the transi-
tion will be obtained for that value of yi = g(Y2)
for which f2(X2) = _ y2/4 has a double root.
Again, this is a standard algebraic exercise. One obtains :
a) - 1/8 y2 0. The variation of the free energy is drawn on figure 4d ; a first order transition is reached for
which is the largest root of (18) x2 t = 1/4 lies within D
(Fig. 3).
For yi > yt1 the free energy is minimized by
e.g. the SA phase is stable. For y, y’ the free energy is minimized by x2 (given by equation (13) and
x 1 # 0, given by (10)). This defines a first order transition with the discontinuities
b) For y2 = - i, YI 3 : the discontinuity
vanishes. This is a second tricritical point (called R
on the phase diagram Fig. 2). As for P, x2 = x2 at the
transition point, and f2 has no extrema for YI > 3
and classical expressions at a tricritical point hold.
c) Y2 - 1/8 : when Y2 gets smaller then - 1/8, the
solution (19) leads to x2t values outside D. The relevant solution to (18) is then
which corresponds to the case of figure 4f.
falls exactly on the boundary of D. For yi > y’, x2 is
outside and the free energy is minimized by
For y, yt1, x2 enters the defined domain, and the free energy is minimized by f2.
Note that the transition is continuous since
(20) thus describes the @ second order SA-SA,
phase boundary.
Note that at variance with the cases displayed on figure 4d, e we show a decrease of x2 with the onset
of xi on figure 4f. x2 still given by (13) exhibits this behaviour only for Y2 - !; for - ! Y2 - j,
it does increase with the onset of xi and at exactly
Y2 = - j, the fluctuations in ÔX2 and xi are decou-
pled. In other words, the onset of a bilayer order may be favorable or unfavorable to the preexisting mono- layer order in the second order transition région ;
on the contrary, it is always favorable in the first order domain.
3. Conclusion. - The study of the M.L. free energy shows that one should indeed expect three différent
phases in the [yi, Y2] phase diagram Branch l’
and part of (l) have been already discussed by Meyer
and Lubensky, branch (2) by de Gennes and McMillan. All these transitions are isomorphous to
the normal fluid-superfluid case [6]. More novel is the
finding of two different smectic A phases, seperated by first or second order transition lines. The conden- sation of the fundamental of the layer density modu-
lation always drives a non zero harmonic, whereas,
on the contrary one may get a smectic phase with the
harmonic modulation only. The subsequent conden-
sation of the fundamental then defines the SA-SA- phase transition. Although pi is a complex order
parameter, this transition is not helium like : indeed the phase t/J 2 of P 2qo is well defined (if one excepts its hydrodynamic fluctuations) and t/J 1, is not an inde- pendent variable ; (5) can be rewritten :
Thus there are two possible choices for t/1l and the
transition is isomorphous to an Ising transition
lu - 1 . J -
586
The observation of the corresponding critical
behaviour will depend on the width of the critical domain, that is on the zero temperature correlation
length. Another consequence of the actual dimen-
sionality of the order parameter is that topologically
stable defects are surfaces [8]. An example of such
defects is given figure 5. The Landau theory we use,
is not expected to be fully correct, but gives a qualitati- vely satisfactory picture of the phase diagram. It predicts the existence of two tricritical points (one
helium like P, the other Ising like R), and one pseudo- triple point Q. At the point Q, the nematic and smectic A phases are undistinguishable, and in equilibrium
with the distinct smectic A’ phase. The branches Q)
and Qj) have the same slope, and the second order branch (2) seems to end on a unique first order line.
Fig. 5. - The two possible phase lockings of PqO on P2qO in the SA, phase : on the left hand si de of the drawing !/i 2013 § = + n/2,
on the right hand side t/J 1 - t/J 2 = - x/2. (The x/2 value has been
arbitrarily chosen.) The defect thus generated in the middle of the
picture, which is a surface perpendicular to the plane of the figure,
is topologically stable in agreement with the theory of Toulouse and Kléman [8]. Note that P2., does not undergo any discontinuity,
when crossing this surface. The molecular structure of the double
layer and of the defect core schematized on the figure is a simple
guess. The heavy segments are intended to represent the strong dipole C - N of the dibenzoate molecule, the rectangle its aromatic
rings, and the wavy line the alkyl chain. The hatched area indicate the mean location of the dipolar part. In the SA phase PqO loses its long range correlations, probably via a large number of such defects.
Let us also point out that the SA SA’ part of the phase diagram exhibits a reentrant behaviour, around the point (yi = 0.5, y2 = - 0.5), which might in certain
circumstances be experimentally observable.
The connection with the experiment is easily made by assuming :
(ai, a2 and a are supposed positive). Figure 2 is then easily transposed into figure 6, which is in good qualitative agreement with the experimental diagram
of figure 1. Even the curvature of the branches l’
and 0 are observed which suggests that the tricri- tical Ising like point R might in fact lie in the experi- mentally accessible domain. The comparison can
Fig. 6. - Temperature, concentration (or pressure) phase diagram obtained from the Meyer Lubensky free energy after the transfor- mation defined in the text. Note the close similarity with the expe- rimental diagram displayed on figure 1.
be brought further by calculating, the entropy discon-
tinuity across Ol and Q) (assuming ai = a2) :
across @ :
across
The plot of the corresponding curves is displayed figure 7 together with the experimental result. Here
again the qualitative agreement is good. It is striking
to observe in both experimental and theoretical results a maximum in the entropy discontinuity
on branch Q).
One may wonder if one could imagine alternative
interpretations to the experimental phase diagram
