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Submitted on 1 Jan 1985
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The smectic A - smectic C phase transition : sense and nonsense
W.J.A. Goossens
To cite this version:
W.J.A. Goossens. The smectic A - smectic C phase transition : sense and nonsense. Journal de
Physique, 1985, 46 (8), pp.1411-1415. �10.1051/jphys:019850046080141100�. �jpa-00210085�
The smectic A - smectic C phase transition : sense and nonsense W. J. A. Goossens
Philips Research Laboratories, P.O. Box 80000, 5600 JA Eindhoven, The Netherlands
(Reçu le 17 janvier 1985, révisé le 27 février, accepte le 28
mars1985)
Résumé.
2014On déduit
uncritère simple pour distinguer la transition smectique A - smectique C de la pseudo- transition, c’est-à-dire la transition qui n’existe pas, smectique A - smectique A incliné. Ce critère est utilisé pour classer et examiner les théories existantes pour la transition smectique A - smectique C.
Abstract.
2014A simple criterion is presented to discriminate between the smectic A - smectic C phase transition
and the pseudo- (i.e.
nonexisting) smectic A - tilted smectic A phase transition. This criterion has been used to
classify and examine the existing theories for the smectic A - smectic C phase transition.
Classification Physics Abstracts
64.70M - 61.30E - 61.30C
1. Introduction.
To describe the second-order smectic A - smectic C
phase transition it has been postulated in 1972 [1] that
the free energy could be written as
with a(T ) oc (T - Tc)Y and b( T)
oc(T - Tc}y-2/J.
The complex number qf
=wX + i(oy
= a)e"’ is the
order parameter of the smectic C phase ; mx and OJy
are the components of the two-dimensional rotation vector w that brings the director from the layer normal
in the smectic A phase to a tilted position in the smectic C phase. The tilt angle w = I ql | is proportional to (Tc - T)O. We do not wish to participate in the
discussion on what the exponents y and ought to be ;
y
=2 p
=1 is satisfactory. Since then many theories, based on a molecular field approach, have been proposed to describe and explain the smectic A-smec- tic C phase transition [2-6]. These theories do not
apparently have much in common. In an attempt to
create some order and understanding,
wewill first
show in section 2 that there is
arather simple criterion
for the SA-SC (smectic A-smectic C) phase transition
to exist. In connection with that criterion
wenotice the possible existence of
atilted smectic A phase, A ;
from the thermodynamic point of view this phase is
the same as the smectic A phase.
In section 3 it will then be shown that the phase
transitions described in [2, 3 and 4] do not exist. The phases described are at best tilted A phases. The phase
transitions described in [5] and [6] do exist. In section 4
we
will show that only the phase transition described in [6] corresponds to the SA-SC phase transition albeit based upon a rather hypothetical potential.
We apologize for the fact that when quoting
formulae from the literature we had to use (in order
to avoid misunderstanding) the original notations, badly attuned to each other, which indeed may be
confusing. The tilt angle however will everywhere be
denoted by
was introduced in [1].
2. A criterion for the existence of the SA -Sc phase
transition.
In a molecular field approach the free energy can be written as :
with
and
the molecular field can quite generally be written
as [7],
/
where an
=3P 2 (COS 0), cos qz and combinations thereof.
The interaction coefficients in equation (5), which
determine the strength of the various interactions,
may or may not be
afunction of the tilt angle cv,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019850046080141100
1412
dependent on the model used. The bracketi ( )
denote
astatistical average, that is,
with
where r denotes the configurational coordinates.
The a,, > then constitute the usual order parameters P2(COS 0), ( cos qz > and P2(COS 0).cos qz >,
which in view of the mean field approximation, ( a2 >= a )2, are independent of the tilt, i.e.,
With the above definitions the entropy can be written
as :
Consequently one finds
A potential as defined in equation (5) produces an
entropy that is independent of w, whatever the order parameters are. Consequently a mean field approach
can never produce
asmectic A-smectic C phase
transition in terms of
w.A smectic phase, described by the order parameters ( P2 (cos 0) >1 cos qz >
and ( P2 (COS 0).cos qz >, for which
with w, determined by 0 V )lam
=0, non zero,
must be classified as
atilted A phase, A. This phase
constitutes the same thermodynamic phase as does
the A phase, which it replaces; both are described by
the same order parameters. Here we encounter the
same situation as with the cholesteric or twisted nematic phase N* and the nematic phase N; at
agiven temperature the one or the other exists.
It seems reasonable to identify the smectic C1 phase, also denoted as the ordinary smectic C phase,
which has an almost temperature independent tilt angle and no corresponding A phase [8], with the
tilted A phase described above; in view of the thermo-
dynamic relationship with the A phase the latter
name is to be preferred.
Considering the above one has to conclude that the smectic A-smectic C phase transition must be due to the appearance of
anew order parameter ( X ),
characteristic for the smectic C phase, which drives the transition; this is in agreement with the approach
of [5] and [6]. In addition the tilt must be coupled to
that new order parameter such that 0 V )law
=0 yields a) oc X )
oc(Tc - T)1/2 [6]. Then indeed
the free energy can afterwards be written as an expan- sion in 0)2, similar to equation (1), which eventually
may elucidate the meaning of the complex order parameter 1/1. The
«unusual » smectic C phase or C2 phase [8], which has
astrongly temperature dependent
tilt angle and a second order phase transition to the smectic A phase, can then be identified with the real C
phase discussed above, which indeed is
adifferent
phase than the A phase.
3. Smectic C or tilted A ?
3.1.
-In reference [2]
arather unusual potential
of the mean field type has been proposed, which after
some calculations, leads to the following coupled equations for the smectic order parameter p >>
and the tilt angle 0) :
Here S
=S(p) is
amean field energy due to the smec-
tic order and Kp is connected with the energy due to
fluctuations of that order. The functions G and F, defined as
determine the energy due to the dipole-dipole inter-
action which depends on the tilt.
The above coupled equations, which indeed cons-
titute the base of the theory, can be solved imme-
diately to yield the exact solutions,
that is
and
There is no phase transition. Even the existence of
a tilted A phase seems unlikely in view of equation (17) ;
the smectic order parameter p increases with increasing temperature.
The phase transition obtained in [2], however,
is the outcome of an unjustified approximation used
to solve the coupled equations (12) and (13). The spirit of the approximation can be illustrated as
follows. Multiply both sides of equation (16) with
kT
=kTc + k(T - Tc)’ where kTc is defined by G(0) + Kpc = - gkTc/2 f
= -KkT j2(iJS/iJp)c; this
also defines Pc and implies fK
=g(iJS/iJp)c. Then
substitute the equations (14) and (15) and neglect
«
higher order » terms, such as g’ (J)2(T - Tc). Then
one finds with fKTc
=g(iJS/iJp)c Tc, the following equation for (J)2,
leading to ro2(T)
=ro2.(T - Tr
The longer course actually pursued in [2] was to
obtain from equation (12) an approximate solution for
« p - p, >> and to use that solution in the same approxi-
mate way as sketched above to recalculate
(Kp + G) (OF/Oco) in equation (13). The result is
essentially equation (18), which violates the equality
in equation (16).
3.2.
-In reference [3] the free energy per molecule is given as :
Here E is put forward as the mean field energy v
=vnem. + Vm., corrected for the packing entropy of rotationally symmetric, non spherical molecules.
This packing entropy for the nematic phase, as derived
in [9] is given by,
where y and a2 are numbers, dependent on the packing
fraction and the form anisotropy respectively;
P2(cos 0) > with P2 the second-order Legendre polynomial, is the order parameter for the nematic
phase. Then with y.em. > = _ Bo p2(COS 0) >2
one may write [3, 9], Fp
= -TSP,
where Tp
=Bo/Z kya2.
Now the averaged mean field ( V... > for the
smectic phase, derived in [3], is, written in shorthand
notation, given by :
where a = (2 A2 - B2 P2 (cos 0) ») P2(cos 0) >, b
=2 C2 ( P2 (cos 0) ) and
c =B4 ( P2(COS 8) ) Z are inter-
action coefficients ; ( cos qz ) is the order parameter for the smectic phase and the P.(cos ro), n
=2, 4,
are Legendre polynomials of order n, describing the tilt-, that is the (u, dependence.
Minimizing V,.. > yields ro2
=(4/7) (1 + (3(b- a)/
10 c)), which for ro2 > 0 may give rise to a tilted A phase but not to a phase transition.
The appearance of a phase transition in [3] however
is the result of the daring assumption, prompted by
the form of equation (21), that the mean energy
density E,.. > defined by E,.. > = ( V... > +
2 Fp,,..IN, can be obtained from the averaged mean
field ( V... > by multiplying a part of it by (1 + T/Tp),
that is ( E... > is assumed to be :
Minimizing this ( E... > leads to (02(T) = co2.((Te - 1)IT c).(T p/(T p + T)), where T,
ocTp oc
Bo, the strength of the interaction giving rise to nema-
tic order.
Equation (23) however implies the introduction of a packing entropy for the smectic phase of the very
special form, Sp
= -Fp/T
which indeed is the essential part of ( E... > for producing a phase transition. Such packing entropy, however, if it exists, ought to be derived in a proper way, with due regard to the tilt dependence, and not
to be invented on an unjustified analogy of equa- tion (21).
3. 3.
-In reference [4] the free energy is written as
where F. is in fact the mean potential energy density
E = V(co) >. Based upon assumptions and trun-
cated calculations, the biaxiality y in fact being put
equal to zero, F. is presented as,
1414
leading to
The arguments concerning the different dependences
of D’ and C’ on qro and therefore on T, q
=q(T) =
2 n/d can only imply that it is possible to have
anon vanishing Q)2, i.e. a tilted A phase. The statement that
the above leads to a
Q)oc (Tc - 7)1/2 is without any foundation as is the statement that it is possible for
asecond order SA -Sc transition to occur if the ratio of the interaction coefficients 14/,2 is large enough.
4. The smectic A - smectic C phase transition.
4. l.
-The phase transition described in [5] is based
upon a two particle potential of the form,
where
n =1, 2, 2-bis denotes the three perpendicular
molecular dipoles, Jln
=(Pn,x, 0, 0), i.e. Jln,x =’ Jln.
Here the X-axis is a molecular axis perpendicular to
the long molecular 3-axis, which is taken along the layer normal, i.e. the Z-axis. The X-axis is then in the
smectic plane. Then pn,x
=Jln,X.XX
=It.. cos cp, where Xx
=X. is the direction cosine of the X-axis with respect to the X-axis. In equation (28) only the inter-
action between dipoles on the same level, defined with respect to the Z-axis, is taken into account. The
neglect of the interaction between dipoles on different
levels seems rather unrealistic and hardly to justify.
However, according to the model, the averaged mean
field can be written as
where vo
=(nn)312/g . and
The distribution functions fl(2) are defined by
where
The order parameters
aand P are defined selfconsis-
tently by above equations. There are three ordered
phases, i.e., C with
a =0, P -# 0, C, with ot :0 0, fl
=0 and C2 with
a0 0, p :0 0. We do not wish to
discuss the physical relevance of these dipole-ordered
smectic phases, but only to comment on what happens
when the outboard dipoles P2 have a component Jl2,3
=6-92 along the molecular 3-axis. In [5] it is
stated that with these dipoles ordered antiparallel in
the smectic plane there
aretwo competing torques,
namely 3 fl 6kT,,, and KúJ, where
mis the tilt angle,
cos m = 3z and K an elastic constant. Equating torques yields
m =3 fl 6kTr IK oc (T,, - T) . 112 The
existence and special form of these torques have not been argued nor have they been derived; we are afraid
it can not be done. Doing things properly, i.e. repeating
the above derivation with IZ2,X
=112X’XX + Jl2,3.3x,
there are two possible solutions. If one considers cp
as the azimuthal angle with respect to the layer normal,
i.e. the Z-axis one has xx
=cos cp cos
mand 3x
=cos 9 sin
co.The relevant part of the averaged mole-
cular field is then given by - vo lt2 #2(COS
ro+
6 sin ro)2 leading to
(o =6
=Jl2;J Jl2.
The tilt angle is a constant and the corresponding phase is a tilted, ordered-dipole phase but not the
smectic C phase. One also may consider 9 as the azi- muthal angle with respect to the long molecular axis,
i.e. the 3-axis ; then Xx
=cos 9 cos
mand 3jf
=sin
m.For the averaged molecular field
wenow find
-
vo 2(p cos
Co+ 5 sin ro)2, leading to tg 2
cv =2 61(p’ _ 6 2) with fl
oc((T,, - T)/Tc)1/2.
Whatever the meaning of this exotic temperature dependence may be, it seems rather unphysical and is
not in agreement with the law predicted in [1]. There-
fore we conclude that these dipole-ordered smectic phases cannot be classified as the smectic C phase.
4.2.
-The phase transition described in reference [6]
is based on a steric model giving rise to a two-particle
«
zig-zag interaction » which is assumed to be of the form
Here R., = 31. 1:J1, where 3 and 1:J refer to a molecular fixed coordinate system and a, fl
=X’, Y’, Z’ refer
to a coordinate system fixed to the smectic layer; the
Z’-axis coincides with the layer normal; 5z,, etc., is
the direction cosine of the 3-axis with respect to the Z’-axis;
ris the absolute value of the intermolecular vector
r =(rx,, ry,, rz,). Taking a statistical average with a properly defined one-particle distribution func- tion, the averaged molecular field is of the form
With the transformation Z’ - Z cos
OJ -Y sin
cowhere the Z-axis defines the tilted director one finds
( v(O)(OJ) > = V(O) > + Cl.(o’ and Rz,
z, =Rzz -
(Rzy + Ryz).(o. Minimizing V yields
The tilt angle
mis proportional to the new order parameter ( RzY.cos qZ’ ), which drives the smectic A-smectic C phase transition; then
m oc(T c - 7-)1/2.
From the thermodynamic point of view the description
and derivation of the phase transition and the tempe-
rature dependence of a) are satisfactory and up to now the only consistent derivation.
Nevertheless there is a serious objection to the physical picture. There is no connection between the
physical model proposed and the potential used.
According to the model the tilting term is due to the zig-zag interaction between zig-zag molecules in the smectic plane, favouring optimum packing in that plane. This interaction, however, is assumed to be
-