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Surface order transition in nematic liquid crystals
G. Barbero, Z. Gabbasova, M. Osipov
To cite this version:
G. Barbero, Z. Gabbasova, M. Osipov. Surface order transition in nematic liquid crystals. Journal de Physique II, EDP Sciences, 1991, 1 (6), pp.691-705. �10.1051/jp2:1991199�. �jpa-00247550�
Classification Physics Abstracts 61 30
Surface order transition in nematic liquid crystals (*)
G. Barbero I'), Z. Gabbasova (2, **) and M A Osipov (3)
(~) Dipartimento di Fisica del Pohtecnico, Corso Duca degli Abruzzi 24, 10129 Tonno, Itaha (~) Theoretical Physics Department, Bashkir State Un~versity Frunze Street 32, 450074 UFA,
US S R
(~) Institute of Crystallography, Leninsky pr 59, 117334 Moscow U-S S R
(Received 27 December 1990, accepted m final form 22 February 1991)
Abstract. The surface order transition in nematic liquid crystals is theoretically analysed Starting from a pseudo-molecular model it is shown that the so called anchonng energy has two contnbutions one coming from the nematic-nematic interaction and another one coming from the nematic substrate interaction The two contnbutions have, generally, different angular dependence, and always a different temperature dependence This fact can explain recently observed surface transitions By using a simple phenomenologlcal model the tilt angle versus the temperature is analysed The agreement with the expenmental data is fairly good
1. Introduction.
The behaviour of the anchonng energy strength coefficient near a structural phase transition at a nematic-substrate interface has been experimentally studied by different authors [1-4]
In a very recent paper Di Lisi et al [5] reanalyse the phenomenon, by observing that a
nearly symmetric alkoxyphenylbenzoate monomer exhibits a surface transition from homeo- tropic to tilted alignment several degrees below the nematic-isotropic phase transition. By
means of the Freedericksz transition technique they determ~ne the anchonng strength
coefficient versus the temperature, finding a non-monotonic trend Similar results are obtained by Flatischler et al. [6] These surface order transitions are usually interpreted m
terms of the Parson theory [7]. According to this theory the nematic surface onentation is due to dipolar and quadrupolar interactions, having a different temperature dependence. As the temperature changes the relative strengths of the two interactions vary, and a vanation of the surface onentation can be observed Parson's expression of the surface energy works well, m
the sense that it is possible to fit the expenmental data. However from our point of view it is
necessary to modify the phenomenological surface energy proposed by him
In order to explain this fact let us recall that nematic liquid crystals are uniaxial materials charactenzed by an average molecular onentation n = (a) and by a scalar order parameter (*) Partially supported by Italian National Council of Research (CNR)
(**) Present address Academy of Science of the USSR, Institute of General Physics, Vavilova st 38,
117942 Moscow, US-S R
b92 JOURNAL DE PHYSIQUE II M 6
S
= (3/2) ((n a)~ l/3), where ) means a thermodynamical average and a is the major molecular axis
In the bulk n and n are equivalent The tensor taking into account the symmetry of the nematic phase is
Q~~ = S n~
n~ 3
~~ ,
known as tensor order parameter All the bulk properties of a nematic matenal can be
descnbed m terms of Q~~, and consequently, they are even in n Near a surface the symmetry of the phase can be different from the bulk and some polar order can ex~st
In fact recent measurements of surface polanzation show the ex~stence of a surface
ferroelectnc order [8], but this order ex~sts only over a few molecular lengths. This implies
that if one wishes to take into account this effect, one has to use a molecular point of view [9].
However, if the surface energy is estimated in a continuum theory, which is a pseudo-
macroscopic theory, the polar contnbution cannot be taken into account directly, because in this limit the presence of terms linear in n in the surface energy does not seem to be justified.
In fact m a continuum description the element charactenzing the nematic molecule is the
director n. This direction is obtained by averaging the molecular direction a over a volume element containing several molecules In the surface layer having ferroelectric order, the thickness is of one or two molecules, and hence it is difficult to define the thermodynamical
average n m the usual way Consequently the Parson expression
Y = Yo- YD(".kl+~YQ(n.k)~,
where yo is an isotropic contribution, k the surface normal, and y~ and y~ the temperature dependent coefficients of the dipole and quadrupole surface free energy, respectively, has to be modified
In our paper an expression for the surface free energy is deduced.
The problems related to the surface polarization are discussed m section 2. The basic charactenstics of the surface energy are obtained m section 3 by using a pseudomolecular
model. In section 4 by means of symmetry considerations a phenomenological expression is proposed In section 5 the surface order transitions are analysed, and the phase diagram of the surface angle versus the temperature is reported In this section we also discuss the temperature behaviour of the anchonng strength, which is m agreement with the experimental
data of reference [5] In section 6 the possibility of first order phase transitions is discussed The main conclusions of our paper are summanzed m section 7.
2. The role of the surface polarization.
As mentioned in the introduction, the first interpretation of the surface order transition m
nematic liquid crystals, g~ven by Parson, is essentially based on the assumption that tliere is a balance between the dipolar contribution to the surface energy (the first term m Parson's
equation) and another contnbution which has another physical ongin and another (« quadru- polar ») symmetry. In this section we consider the influence of the surface polanzation on the surface energy. It is important to note that the polanzation P and the tensor order parameter
are two independent thermodynam~c parameters Indeed, the polanzation can be fixed by
the extemal electric field E, while the onentation of the tensor can be fixed independently by the magnetic field, for example.
It is obvious that m the general case the direction of the macroscopic polarization does not coincide with the onentation of the director (even in the absence of extemal fields). The same
conclusion is valid with respect to the surface polarization P~ and the surface order parameter Q~. In the frame of a particular model, however, the polarization and the director can be
approximately parallel Indeed, when the molecules possess only long~tudmal dipoles
djj = djj a, where a is the unit vector m the direction of the long molecular axis, the
corresponding dipolar order parameter p = d(a) coincides with the orientation of the
nematic tensor order parameter Q~~. In the ordinary nematics w~thout extemal fields the
polarization is absent and the sign of the director has no physical meaning. On the other hand,
m the thin boundary layer near the substrate the long~tudmal molecular dipoles can be ordered and the sign of the director can be put (for convenience only) equal to that of the polarization In this special model the surface polanzation and the tensor order parameter are are not completely independent and the surface free energy can be wntten m the form of
Parson's equation where the first term corresponds to the interaction between the surface polanzation and the surface field
Note, however, that from the general point of view Parson's equation is approx~mate and it is valid only m the frame of a particular model since it does not take into account any other
contribution to the surface polanzation. For example, there ex~sts a polarization contnbution which is proportional to the gradient of the scalar order parameter S which is always nonzero
near the surface This polarization is normal to the surface and it is not parallel to the surface director when the latter is tilted. Thus Parson's equation is valid when such additional
contnbutions are negligibly small. This assumption seems to be reasonable only m the case of
cyanobyphenyls or sim~lar molecules which possess very large terminal dipoles
It is important to stress that the liquid crystals which undergo the surface order transition
are composed by molecules which do not possess long~tudmal dipoles [5, 6] but only small transverse ones Then the properties of these real substances can hardly be described by the Parson theory in which it is assumed implicitly that the surface polanzation is always parallel
to the director However, from the principle point of view it is reasonable to estimate the contnbution from the surface polarization to the surface energy m the most favorable case,
i-e for mole~ules with large long~tudmal dipoles
The interaction energy between the permanent molecular dipole and the solid surface can be estimated as an energy of electrostatic interaction between the dipole d and its mirror
image d+ m the dielectric hem~space U~~ ~dd+li~, where I
is the distance between the
dipole and the surface and d+ s I d, where s is the dielectric constant of the substrate.
Taking d
=
4 x 10~ '~ CGSE, s I
=
I and I
=
L
= 20 I (L
is the molecular length) we
arnve at the estimate U4~ 10~' erg Now the contnbution to the surface energy from the first
monolayer can be estimated as U~ p~ U4~, where p~ = pL is the surface density of molecules
and p is the bulk number density of the nematic Taking p ~10~'cm~~
we arrive at U~ 10~ erg/cm~. Note that this energy is rather large compared with the typical values of the anchoring strength (10~~erg/cm~) for the director at the solid substrate Taking into
account that the dipole-dipole interaction is strongly anisotropic, we arrive at the conclusion
that it causes practically strong anchoring of the first molecular layer The dipolar interaction
of the second layer with the surface is (2)~
= 8 times weaker but it is still rather strong Starting from the fourth layer the dipolar interaction becomes negligibly small.
Thus we arrive at the following picture If the surface polarization is concentrated m the
first two molecular layers, as indicated by the expenments [8] the onentation of the long
molecular axes is fixed m these layers by the polar interaction w~th the substrate and these
layers do not take part m the reorientation of the director near the surface This is not m contradiction with the expenments [5, 6] since the expenmental observation of reorientation of the director at the surface corresponds m fact to a relatively broad boundary reg~on (of the
order of the light wavelength)
694 JOURNAL DE P@YSIQUE II M 6
It should be also noted that the surface free energy, which enters m the equations of the
phenomenolog~cal theory, corresponds to a certain boundary layer of finite thickness On the macroscopic scale the thickness of the boundary reg~on is « zero », but on the molecular scale it can be rather large and to include ten or more molecular layers. Then the surface free energy is determined as an extra energy of this boundary layer [10a]. Taking this into account,
one can conclude that the surface polarization can have a substantial effect on the director reorientation m such boundary region only if the whole region is strongly polarized. At the present, however, we do not know any experimental data which can confirm this fact.
Therefore it would be interesting to determine expenmentally the thickness of the polar layer
and to estimate the surface polarization m the same nematic cell where the temperature-
induced reonentation of the director has been observed It is also worth noting that the first term m Parson's equation can play an important role only if the surface electnc field is strong enough Indeed, this term can be estimated as rP~E~, where r is the thickness of the boundary layer, P~ the average polarization m the surface layer and E~ the surface electnc field. Now,
even if one assumes that all the longitudinal dipoles are completely ordered m the boundary
region, I e P~ pd, a relatively large surface electric fields is required to make the polar contribution of the same order of magnitude as the typical anchonng strength m nematics. In fact, taking r 300 I, P~
~ pd, d
= 4 x 10~ ~~ CGSE and rP~E~ ~ 10~ ~erg/cm~, we obtain E~ l CGSE. In real systems the surface polarization cannot be so large, of course, and the
very high polanzation P
~ pd 4 x 10~ CGSE can be achieved only in the first one or two layers Then the average surface field m the boundary region should be much stronger
3. Pseudo-molecular approach.
The pseudo-molecular approach has been w~dely used to determine the bulk elastic constants
of nematic hqu~d crystals In this section we extend it m order to take into account the
presence of a surface discontinuity, separating a nematic from another phase, that w~ll be
supposed isotropic.
Let us summarize the essential lines of the molecular approach to the elastic constants g~ven
m reference [10] The starting point is the assumption of a two-body interaction of the type
g(@, 4,
r d~R d~R'
,
(li
between two volume elements d~R, d~R', where 0 and Q' are the tensonal order parameters
r
il
n
R
o
Fig I -Physical quantities defining the two-body interaction in the nematic phase- d~R and
d~R' are two volume elements containing nematic molecules charactenzed by the average onentation n and n' and by scalar order parameters S and S', respectively
at R and R', respectively and r= R'-R (see Fig. I). The interaction energy g(l~,
Q', r) can be written m general as
g(Q, 0', ~
" z c~np(~) (Q Q')~ (" UT (U 0'
U Y, (2)
where u
= r/r, : Q'
= Q~~ Q[, and u : : u
= u Q~~ u~ By expanding g(Q, Q',
r given by equation (2) m power series of
Q~~ ~ = io~~lix~, one obtains
= = =
~(Q, Q', ~
" ~0(Q, Q, ~) + ~
tj Qy,e~e + (1/~)[~ij Qij,em + ~
tjkn Qtj,e Qkn,m ~e~m + (~)
where the uniform part, go, is given by
g0(0, Q,~)
~ z Cmnp (~)(0 0)~ (" 0
")~ ~~
,
(4) fpL~0
and the tensors appearing m the non-uniform part are
~ ig
~~ %Qlj Q'"Q
~"~~ ~2~
°Qlj join
Q'"Q
~~~
and can be evaluated by means of expansion (2).
The « elastic » free energy density is obtained by integrating expression (3) over the volume
v~ defined by the molecular interaction range of the forces g~ving nse to the nematic phase By operating m tbJs way from expansion (3) we deduce
f
" f0 + Bye Qij,
e
+ (1/2)lBtjem Qtj,
em
+ Bijkemn Qij,k Qem,n1 + (6)
where
fo = jjj go(Q, Q, r dU (71
and
B~~~ = A~~ r~ du
~v~
Bijem " ljj Au ~e~m d~ (81
V~
l~ljkemn "
l~
Ijke~m G dU
VN
As well known m the bulk, expression (6) can be written m the form known as Frank free energy density if the scalar order parameter S is assumed to be position independent [I I] In the opposite case m which S is position dependent from (8), in the bulk, one denves the
generalized Frank free energy density [12].
Let us limit ourselves to consider an ideal situation m which Q, and hence n and S, is
constant across the nematic sample. In this case
Q~~,~ = Q~~,~~ = 0 Consequently f reduces to the uniform part fo
696 JOURNAL DE PHYSIQUE II M 6
Let ri be the range of the molecular interaction forces giving nse to the nematic phase, and
z the distance of the considered molecules defining Q(R) from the surface. In the event in which z
» r,, the integration over u, appearing m fo defined by equation (7), is performed
w~thln a sphere of radius r,, by assuming the contribution of the outside reg~on to the energy
density at point R to be neglig~ble (see Fig 2a) In this case a simple calculation g~ves
fo= z L~~~A~~~S~~+~+P=a~S~+ a~S~+a46l'+... (9)
m=I n,p=0 where
Lmnp = (2/3)m ~'
cmnp(r) r2du (lo)
and
,"
~ i n+p
A~ ~~ = cos ~b sin ~ d~ (I I)
o 3
To obtain expressions (9), (10) and (11) we have considered n as polar ax~s and wntten du
=
r~drdJ2 where the solid angle is defined m the usual way da
= sin ~b d~b dfP, in
which ~ is the angle of u with respect to n Note that (9) reduces to the well known Landau expansion near the N
~ I phase transition temperature. We underline that m the considered
case z » r,, fo is independent of the nematic director n, as expected.
n
NLC
NLC p
z>r,
a) SUBSTRATE b) SUBSTRATE
Fig 2a. Nematic interaction m the bulk (± > ri) The free energy density of the uniform nematic
fo is obtained by integrating the molecular interaction over a sphere of radius rj, defining the range of the molecular forces giving nse to the nematic phase fo is found to be undependent of n.
Fig 2b -Nematic interaction near a surface (z<r,) fo is obtained integrating the molecular interaction over a part of the interaction sphere V(z)
=
V~ AV(z) Now fo depends on z and on the
average onentation n too The equivalent surface energy can be defined as the integral of
fo over AV(z), integrated from z
=
0 to z = ri
On the contrary if z < rj the integration over u is performed only on a part V of the sphere
of interaction, as shown in figure 2b. In this situation in the boundary layer of thickness r,, fo depends explicitly on z and on the director n, since
fo(z < r,
= jjj go(Q, 0, r du
= jjj go(Q, Q, r dU iii go(Q, Q, r dU
,
(12)
m which Av(z) is the excluded part of the interaction sphere, dashed m figure 2b By putting
gl'~(4, k, z)
= jjj go(4, 4, r) du, (13)
where k is the surface normal, parallel to the z-ax~s, the quantity
fj'i(§, k
=
~' gj')(4, k,
z dz, (14)
o
plays the role of surface energy due to the presence of the surface, com~ng from the nematic- nematic interaction.
In the event m which the molecular forces responsible for the nematic phase are known,
expansion (2) is known too. In this case fl'~ given by equation (14) can be estimated Until now we have considered only the part of the surface energy com~ng from the nematic itself.
Of course a direct interaction, like Van der Waals [13], between the substrate and the
nematic can also ex~st. This interaction tends to induce on the nematic a preferred
onentation, according to the molegular properties of the nematic, and furthermore it is
responsible for the isotropic part of the surface tension. Since the substrate is supposed to be
isotropic, the «surface field » taking into account the direct coupling substrate-nematic is normal to the surface E(z)
= E(z) k The relevant free energy density can be wntten m
general as
gj2)(4, k,
z =
( D~ (z) (k : #
:k)m (15)
In the Van der Waals case D~(z)
= 3~, c/z~, where c is a constant, as discussed m [13]
E(z) and hence D~ (z) are different from zero only if z
< r~, defining the interaction range of the surface interaction. By integrating (15) over z from z
=
0 to z
= r~ one obtains
f)~~(0, k
= g)~~(Q, k, z dz
= z p
~
(k : : k)~
,
(16)
where
.r2
p~ = 0 D~(Z) dZ.
The quantity f)~~(Q, k is the surface energy due to the direct interaction substrate-nematic.
In the hypothesis that r, r~, m the « surface layer » is localized the surplus of energy The effective surface energy is, as follows from (14) and (16),
fs(Q, k
=
fl'~(Q, k + fl~~(0, k (17)
We stress that m f)'~ the first term in S is of order two, whereas m f)~~ the expansion starts with a term linear m S.
In the following we consider equation (17), and by means of symmetry considerations we expand f m power series of S. By operating in this way we obtain an expression for f showing
the possibility of surface order transitions
