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On a generalization of the Rapini-Papoular expression of the surface free energy for nematic liquid crystals
G. Barbero, Z. Gabbasova, Yu. Kosevich
To cite this version:
G. Barbero, Z. Gabbasova, Yu. Kosevich. On a generalization of the Rapini-Papoular expression of the surface free energy for nematic liquid crystals. Journal de Physique II, EDP Sciences, 1991, 1 (12), pp.1505-1513. �10.1051/jp2:1991103�. �jpa-00247607�
J. Phys. II France 1 (1991) 1505-1513 DtCEMBRE 1991, PAGE 1505
Classification Physics Abstracts
61.30
On a generalization of the Rapini-Papoular expression of the surface free energy for nematic liquid crystals (**)
G. Barbero (~), Z. Gabbasova (~, *) and Yu. A. Koseiich (3)
(1) Dipartimento di Fisica, Politecnico_ Corso Duca degli Abruzzi 24, 10129 Torino, Italy (~) Institute of General Physics, Academy of Sciences of U.S.S.R. Vavilova st. 38, 117333
Moscow, U.S.S.R.
(3) All Union Surface and Vacuum Research Centre Andreyevskaya nab. 2, 117334 Moscow,
U.S.S.R.
(Received 17 June 1991, accepted 28 August 1991)
Abstract. The effect of the Kj~ and K, elastic constants on the equilibrium orientation of nematic liquid crystals is considered. Our analysis shows that these terms can be ignored if the surface energy is modified in a particular way. In this manner the difficulties in the elastic theory
of nematic liquid crystals connected to the proper minimization of the free energy can be solved.
Several suggestions have been made in the past to overcome this lacuna in the elastic theory of nematics. In our paper we suggest a different approach leading to a new surface free energy. The surface variation of the nematic average orientation induced by K13 and Ki is also estimated.
1. Introduction.
The presence Of second Order derivatives in the free energy density Of nematic liquid crystals
was first discussed by Nehring and Saupe a long time ago [I]. The connected elastic term, which gives rise to a surface contribution depending On the first derivatives, is characterized
by an elastic constant usually denoted by Ki~. The influence OfKj~ On the static properties Of nematic liquid crystals has been analysed by Hinov and coworkers [2]. The mathematical
analysis proposed by Hinov et al. was recently criticized, since the variational problem is ill posed [3]. In order to have a well posed variational problem we proposed to take terms
quadratic in the second order derivatives into account in the free energy density [4, 5]. By operating in this way the number of elastic constants necessary for an elastic description of the
nematic is generally very large [6].
However in some particular situations the analysis is made possible by introducing only a
new elastic constant [7]. In these cases the effect of the new elastic term is localized near the (*) Permanent address
: Theoretical Physics Department Bashkir State UrJiversity, Uliza Frunze 32, 450074 Ufa, U.S.S.R.
(**) Partially supported by Italian National Council of Research (CNR).
JOURNAL DE PHYSIQUE II T i, M 12, DiCEMBRE iwi 64
1506 JOURNAL DE PHYSIQUE II bt 12 surface over a layer whose thickness is of the order of the molecular forces giving rise to the nematic phase [8]. The nematic average orientation _changes across the surface layer of a
quantity independent of the new elastic constant connected with the square of the second order derivatives.
These facts suggest that the energy connected with the new term of higher order is
equivalent to a surface energy. The aim of the present paper is to show that it is possible to solve the problem connected with the presence of the Kj~ term without modifying the bulk free energy density of the nematic. To do this it is necessary to introduce a surface energy density taking the spatial variation of the elastic constants into account. In section 2 the elastic
theory of nematic liquid crystals is recalled, by underlying in particular the presence of new terms when a surface discontinuity is considered. The spatial variation of the elastic constants is discussed in tills section. In section 3 the equivalent surface energy, in presence of
distortions, is determined, as well as the surface variation of the average nematic orientation.
The results obtained there agree with those determined in different ways, some years ago
[8, 9]. In section 4 the Freedericksz transition is analysed by considering as surface energy the
proposed equivalent surface energy. In section 5 the main conclusions of our paper are underlined.
2. Elasdc tlieory of nematics.
Nematic liquid crystals are uniaxal materials, whose optical axis coincides with the average molecular orientation n. This direction is known as the nematic director. In the case in which
n is position independent, the nematic is undistorted, and its elastic energy has the minimum value fo. On the other hand, if n is spatially varying, and hence n;
j = an;laJ~ are different
from zero, the nematic is distorted and its elastic energy is greatir than fo. Let
us limit ourselves to the case in which the director's spatial variations are slow over a distance of the order of the range of the molecular forces giving rise to the nematic phase. In this situation the free energy density of the nematic has the form generally known as the Frank free energy
density [10].
The bulk elastic theory of nematics has been developed by Vertogen and coworkers [11].
The staffing point is the assumption of a two-body interaction of the type f(n, n', r )d~R d~R'
,
(I) between the volume elements d~R, d~R', where n and n' are the directors at R and R', respectively, and r
=
R' R [I I]. In order to write the energy corresponding to a given director distortion in terms of elastic constants we put n'(R')
= n (R + r)
= n (R) + An (R, r )
and furthermore An (R, r )
= n,,j(R) rj + (1/2) n, ~~(R)rj r~ + 0(3). In this way from (I) we
obtain
f(", ~', ~
" f0 + f< ~<,
e ~e + lfi ~i,
em + fij n;,
e ~j,ml ~e~m + (2)
where
fo = f(n, n, r' )
,
'~ 3n, n'=n~
f__ ~2f
~ 3~) 3~
n~ n,,n~ nj
bt 12 SURFACE FREE ENERGY FOR NLC 1507
The energy density F(R) at the point R is obtained by integrating f/2 over r. For nematic liquid crystals, from (2) we deduce that in the bulk the energy density can be written as
F
= Fo + [Kj
i(div n)~ + K~~(n curl n )~ + K~~(nx curl n)~]
2
(K~~ + K~4) div (n div n + nx curl n ) + Ki~ div (n div n ) (4)
In (4), Kii, K22,K~~ and K~4 are the Frank elastic constants. New terms in the expression of F are expected at a point R near a boundary surface, owing to the lower symmetry at the problem. In the hypothesis in which the plane defined by the nematic director and the surface
normal is a symmetry plane, only a new term appears which has the structure
Kj div n
,
(5) giving rise to a splay distortion. Ki is identically zero in the bulk, whereas it is generally
different from zero in the boundary layers. At first sight term (5) seems odd in n, and hence forbidden for nematic liquid crystals characterized by the invariance n - n. However a
deep analysis, performed according to Vertogen's theory [11], shows that Ki depends on n,
and it is found to be an odd function of n. Consequently, (5) is even in n, as expected (see Appendix I).
It is important to underline that Fo, K~~ (I
p 1, 2, 3 ), K~4,Ki~ and Ki
=
0) are constant and independent of the nematic orientation only in the bulk. An explicit dependence on the distance from the boundary surface and on n is expected in a boundary layer. The thickness of the boundary layer is of the order of the range of the molecular forces giving rise to the
nematic phase.
Since near a boundary the elastic constants and Fo are position and n dependent, an elastic
description in the usual sense is no longer possible. However since the thickness of the
boundary layer is usually very small compared to the sample thickness, the bulk free energy connected with the distortion localized there can be considered as a surface free energy. In this way the problem connected with the Kj~ elastic constant can be solved without
generalizing the Frank energy density.
3. Equivalent surface energy.
Let us consider a flat surface separating a nematic liquid crystal from an isotropic substrate.
Within this framework the surface energy of a uniform nematic (I.e. scalar order parameter and average orientation n position independent) has two contributions : one coming from a
direct interaction substrate-nematic, and the other coming from the nematic-nematic
interaction. The latter comes from the Fo term, appearing in (4), integrated over the surface
layer.
In the event in which a spatial variation of n is present, the surface energy also contains a term coming from the elastic » deformation.
In order to evaluate the equivalent surface energy let us assume thlt : I) the nematic sample occupies a half space,
it) the Cartesian reference frame has the z-axis normal to the interface, and the nematic is in the z
~ 0 side,
ifi) every physical quantity only depends on the z-coordinate, iv) the nematic director is always parallel to the (n, z ) plane,
and hence it can be expressed in terms of the tilt angle it makes with the z-axis
n = (sin 8, 0, cos 8 ).
1508 JOURNAL DE PHYSIQUE II M 12 In this framework in the boundary layer the bulk free energy density (4) after taking into
account the spontaneous splay term (5) becomes
F
=
Fo(z, o) + Ki(z, o ) ( (cos o)
+
+ [Kij (z, 8) sin~ 8 + K~~(z, 8 ) cos~ 8 8~ + Ki~(z, 8) ~ (- cos 8 sin 8 )
,
(6)
2 dz
where
= do /dz. For 8 «1, expression. (6) up to the second order in 8 becomes F
= A (z, 0) + y (z, 0) 8 + ~ (z, 0) 8~ Ki (z, 0) 8 ~) +
+ K~~(z,0) #~ Ki~(z, 0) ~ (8 )
,
(7)
where
A (z, o
= Fo(z, 0), Y (z, 0)
= (afola 8)o, ~ (z, 0
=
(a~fola8~)o (8)
The fact that y(z, 0) # 0 only means that Fo(z, 8) can have a minimum for 8 # 0. This depends on the molecular forces giving rise to the nematic phase.
The surface energy coming from the nematic-nematic interaction fl'~~~ is obtained by integrating (7) over the boundary layer, whose thickness will be denoted by b, I-e-
b
fl'~''~~
= F dz (9)
o
The surface energy coming from the substrate.nematic interaction f)~'~> is obtained
integrating the interaction energy [12]
u(z, e )
= ~ (z) e2 (io)
over the range of the surface forces p (f.I. Van der Waals forces). Since the substrate is assumed to be isotropic, no linear tennis are present in u(z, 8 ). In the hypothesis that
p b, one obtains
f)~''~~ b
= u(z, 8) dz (I I
o
The total surface free energy is then
~~ =
~jN,N> ~ ~js,N) ~j~~
By substituting (6) and (10) into (9) and (I I) we obtain, by applying the average theorem,
f~=B+~ W(80- 8)~- (Ki)(8)-8()+
+
~/~~ (8s 80)~ (Ki~) (8s ds 80
°~
~
°° ('3)
In (13) ( means the value of the function evaluated in some point of the range
(0,b), f.I.
i~ Kj(z)
~ (8~) dz
=
Kj(z*) j~ (8~) dz = (Kj) (8~(b) 8~(0)), (14)
o 2 dz 2
o
bt 12 SURFACE FREE ENERGY FOR NLC 1509
I.e. (Kj
= Kj (z*), z * e (0, b). Furthermore we have put 8(b
= 8
~,
8 (0)
= 8
o, and
supposed that in the boundary layer #
= (8~ 80)/b and that 8~ 80 is small, that seems reasonable. Finally in (13)
f~
= <~~ll ~'5)
is the easy axis of the uniform part of the surface energy, B
= ((~ + p )) b8~, (16)
is an unimportant constant, independent of the nematic orientation, and w
=
j(~ + a )j b (17)
is the usual anchoring energy strength. By taking into account that in the interesting case
#~ « (8~ 80)/b, expression (13) for the equivalent energy can be rewritten as
f~
=
~'(80 8)~ + fl80(8~ 80) +
y(8~ 80)~. (18)
where
~ j~ j ~ lK13) j~j lK33)
~ b ~ " b
In this way the surface energy is characterized by two angles : one on the geometrical surface
80
=
8(0) and the other at the end of the «diffuse» layer 8~
= 8(b), and from three
anchoring parameters. The actual values of 80 and 8~, considered as free parameters, are
determined by minimizing f~ given by (18). By taking into account that
af~ =p80+y(8~-80)=0 3°s
(19)
~~~
= W(80 8) + P(8~ 2 80) + y (80 8~) = 0
3°o
we deduce
8o =
~ 8
,
(20) YW
which can be considered as the surface easy axis, and
8s =
~ j ~ 80, (2')
which is the macroscopic easy axis (at the end of the diffuse » layer).
From (20) and (21) one easily obtains for the surface variation of the average orientation the value
ho
= 8~ 80 = ~
80. (22)
Y
1510 JOURNAL DE PHYSIQUE II bt 12
In the evdnt in which W
- co (strong anchoring hypothesis), equation (20) #ves 80 - 8 and from (22) it follows that
ho
=
~ 8
,
(Strong anchoring (22')
Y
This result (22') has been obtained some years ago by neglecting the spontaneous splay (I.e.
(Kj) w 0), in the framework of a non-linear elastic theory [8].
Note that if 8
=
0 (I.e. the easy direction is the homeotropic one), ho
= 0, I.e. no surface distortion appears.
From equation (20) we see that the quantity
W'
=
~~, (23)
Y
which is always negative, since y is a positive quantity, has the dimensions of a surface energy. It is connected only with the elastic properties of the nematic. The influence of this term will be reconsidered in the next section, where the Freedericksz transition is analysed.
4. Freedericksz transition.
Freedericksz' transition has been widely used to determine the nematic elastic constants. The
theory of this phase transition in different geometries has been developed by H. J. Deuling
and coworkers [13-15] in the strong anchoring case. In this section we are interested in
evaluating the Freedericksz transition threshold of a homeotropic sample submitted to a
magnetic field by taking into account : I) the finite surface anchoring energy, and it) the Ki~ and Kj elastic terms. Up to now this problem has been solved in a correct way only within
the frame of a non-linearized elastic theory [4].
Let us consider a nematic slab of thickness d, limited by two identical surfaces at
z = ± d/2. In this situation 8 (z) is an even function of z, due to the symmetry of the problem :
e (z)
=
e( z) (24)
The total energy of the sample near the threshold is, as is well known [13],
where f~is given by (18). By minimizing 3 with respect to 8 (z) we obtain for the bulk the well known first integral of the Euler-Lagrange equation [10]
j~ Xa H~
~ ~ ~ ~ ~~~~
3~33
~~~~ ~ ~~~
~
where 8M
=
8 (0), and the boundary conditions af~
K~~ 8 + = 0, at z
= (d/2) + b
~~~ ,
(27) af~
= 0
,
at z = d/2
3°o
bt 12 SURFACE FREE ENERGY FOR NLC lsll
Equations (27) can be rewritten as
-K~~8 +P80+ y(8~- 80),
W80+P(8~-280)+ y(80- 8~) =0, ~~~~
if (18) is taken into account.
From the second equation of (28) one obtains
~°
W y ~2 p ~~ ~~~~
By putting (29) into the first equation of (28) we have
-K~~#+W~8~=0, (30)
where the equivalent anchoring energy strength is w- (p~/y) Kq (31)
= i + (wjy) 2(p/y)
Equation (31) shots again that the W'introduced before (see Eq. (23)) renormalizes the anchoring energy strength, Equation (31) generalizes that proposed some years ago [4].
Experimentally only W, can be deduced. From this parameter it is impossible to determine the « true » anchoring strength W, since (Ki), (kj~), (K~~) and b
are unknown. However the only physical observable, near the threshold, remains W~. The other quantities do not
enter directly in the analysis, and hence can be ignored. Of course, far from the threshold field the tilt angle is no longer small, and our linearized analysis has to be extended to the case of large 8. In this situation equation (31) could be no more valid. However since all
experimental data show that W~ ~
10~~erg/cm~ [15, 16] our linearized analysis is practically always good.
S. Conclusions.
In our paper the effects of Kj~ and Ki elastic constants on the equilibrium orientation of nematic liquid crystals has been donsidered. By taking the spatial variation of the elastic
constants into account, we have introduced a generalized surface anchoring energy. This
energy contains a bulk uniform part coming from the nematic-substrate interaction and from the nematic-nematic interaction. It also contains a non-uniform bulk part connected with the
spatial variation of the nematic director, in which the so-called «elastic constants» are
position and orientation dependent. The equivalent surface energy is obtained by integrating previous contributions over a boundary layer having the thickness of the range of the intermolecular forces giving rise to the nematic phase. Our analysis shows that the presence of
the Ki~ and Ki tennis modifies the easy direction and the anchoring strength. Consequently
these terms can be completely ignored if we accept to modify the surface free energy, without
modifying the bulk free energy density. Furthermore the analysis pertaining to the Freedericksz transition shows in particular that the effect of the Ki and Ki~ elastic constants is
equivalent to a destabilizing surface energy. This fact is easily understood since Ki~, Ki # 0 imply that the homeotropic or planar orientation are not stable. Finally we have
evaluated the surface variation of the nematic average oRentation. Our calculations show that this variation, appearing over a layer of thickness b, is proportional to (ply) 80 = [( (Ki~) b (Kj) )/(K~~) 80. Recent measurements of Shen et al. [17] show that this sharp
variation of average orientation is detected in some cases.
