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U.P.B. Sci. Bull., Series A, Vol. 71, Iss. 2, 2009
ISSN 1223-7027
SURFACE INFLUENCE ON THE OPTICAL
FREEDERICKSZ TRANSITION IN NEMATIC LIQUID
CRYSTALS CELLS
Eleonora - Rodica BENA1 , Cristina CÎRTOAJE2
Câmpurile critice pentru tranziţiile Freedericksz electrice, optice şi
magnetice sunt puternic influenţate de condiţiile de ancorare ale moleculelor de
cristal lichid la pereţii celulei. Utilizând o metodă de investigare bazată pe ecuaţiile
Euler-Lagrange şi considerând că energia de ancorare la suprafaţă a moleculelor
la interfaţa cristal lichid – perete este descrisă de formula Rapini-Papoular, am
găsit o corelaţie între intensitatea fascicolului laser care produce destabilizarea
alinierii homeotrope, energia de ancorare, grosimea celulei şi parametri de
material. Am găsit de asemenea intensitatea de saturaţie a fascicolului laser care
produce alinierea planară pe întreaga celulă, funcţie de proprietăţile cristalului
lichid şi parametrii de suprafaţă şi grosimea celulei.
The critical fields for electrical, magnetical and optical Freedericksz
transitions are strongly influenced by the anchoring conditions of the molecules of
the liquid crystal to the solid walls of the cell. Using an analytical method based on
the Euler-Lagrange equations and considering that the surface anchoring energy of
the liquid crystal-wall interface is described by Rapini-Papoular formula, we found
a correlation between the value of the critical intensity of the laser beam producing
the destabilization of the homeotropic alignment, the anchoring strength, the cell
thickness and the material parameters. We also found the saturation intensity of the
laser beam producing the planar alignment all over the cell, in terms of nematic
liquid crystal properties, surface and cell parameters.
1. Introduction
In confined structures of liquid crystals, the magnitude of the free energy
terms associated with elasticity, surface anchoring and coupling to the applied
external fields are frequently comparable.
The critical fields for electrical, magnetical [1, 2, 3, 4] and optical [5]
Freedericksz transitions are strongly influenced by the anchoring conditions of the
molecules of the liquid crystals to the solid walls of the cell [6, 7]. It may be also
changed in guest–host mixtures, such as those containing azo-dyes, when
illuminated with U.V. light [8, 9].
1
Prof., Physics Department, University POLITEHNICA Of Bucharest, Romania
Assist., Physics Department, University POLITEHNICA Of Bucharest, Romania, e-mail:
[email protected]
2
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Eleonora - Rodica Bena, Cristina Cîrtoaje
Rapini and Papoular have proposed a simple phenomenological
expression for the anchoring free energy per unit area [6]
1
f s  A sin 2  0
2
(1)
where 0 is the angle between the easy direction e and the director n of the
nematic liquid crystal at the nematic-wall interface. A is the so-called anchoring
strength. It can be strictly proved that when the anchoring energy takes the form
of Eq. [1], only a second order transition will occur [7].
By considering a new expression for the anchoring energy:
fs 
1
A sin 2  0 1   sin 2  0 
2
(2)
Yang et. al [10], [11], have shown that in the case of   0 the magnetic and
respectively the optical-field induced Freedericksz transition in NLC cells are first
order.
Using Eq. 2 for the anchoring energy Bena and Petrescu [12] have studied
the surface effects on magnetic Freedericksz transition in feronematics.
In this paper we study the influence of the surface anchoring and the cell
thickness on the optical-field induced Freedericksz transition in a homeotropic
cell.
2. Theory
2.1 Basic Equation
We consider an infinite flat layer of NLC between two identical
substrates. The NLC molecules are initially homeotropically aligned to the
transparent walls, considered parallel to the xOy plane and located at z  d and
z  d . In the presence of a monochromatic laser beam, the NLC director can be
reoriented: for p - polarization, the molecular director moves in the incidence
plane xOz. We assume the laser beam having the vector k parallel to Oz and the
electric field of the electromagnetic wave E parallel to Ox (Fig.1).
Surface influence on the optical Freedericksz transition in nematic liquid crystal cells
87
Fig. 1 Orientation of the molecular director with respect to the laser field B and E are the
magnetic and electric fields of the laser wave
The free energy of the liquid crystal is:
F  S  fv  , z  dz  SA sin 2 0
d
(3)
d
where S is the cell plates surfaces and fv  , z  is the free energy density in the
bulk of the system in the presence of the laser field [4].
1/ 2
1
I  II   
(4)
f v  , z   K1 sin 2   K3 cos 2   z2 
1/ 2
2
    a cos 2 




Here K1 and K3 are the bend and splay elastic constant respectively,  z 
d  z 
,
dz
I is the mean volumic density of the electromagnetic energy of the light
(connected with the intensity J of the light by relation J  cI ),  II and   are the
dielectric constants parallel and perpendicular to n respectively;  a   II    is
the dielectric anisotropy.
In order to find the critical intensity of the laser field producing a damage
of the homeotropic arrangement (i.e. the optical Freedericksz transition) we have
to solve the Euler-Lagrange equation:
d  f v  f v
(5)
0


dz   z  
with the boundary condition:
 d 
 K1 sin 2   K3 cos2   
(6)
 A sin 0 cos 0

 dz  z  d
In the middle of the cell ( z  0 ),   0   m ,  z  0  0 ; we assume the
distortions to be symmetric:   z      z  .
As f v does not depend explicitly on z , a prime integral of Eq. (5) has the
form:
88
Eleonora - Rodica Bena, Cristina Cîrtoaje
fv   z
f v
 C1
 z
(7)
Using Eq. (4), Eq. (7) leads to:
I   II   
1
 d 
K1 sin 2   K3 cos 2   
 C1

 
1/2
2
 dz       a cos 2  
1/2
2
In z  0 ,
d
dz
(8)
 0 and   0   m , so
z 0
I   II   
1/2
C1 

(9)
2
   a cos  m 
1/2
Eq. (8) can be written:
d

dz
2 I  II   
K1 sin 2   K3 cos 2 
1/ 2


1

2
     a cos  m


1/ 2


(10)
1/ 2 
    a cos 2  

1

a
1 , expanding into series the expression of the last bracket in Eq.

(10), we obtain:
Assuming
I a  II

dz 

K3 1   sin 2 

sin
 3 a 3 a
2
2
1  2  4 sin   sin  m





where:

d
2
 m  sin  
(11)
2
K1  K3
K3
Integrating in Eq. (11) we have:
I  a  II


d

K 3 1   sin 2  
 3

3
2
2
2
2
0 1  2 a  4 a  sin   sin  m   sin  m  sin  




Introducing the change:

sin 0 
sin 
sin  
;  sin 0 

sin  m 
sin  m 
we obtain:
d
(12)
(13)
Surface influence on the optical Freedericksz transition in nematic liquid crystal cells
d 
sin m 1  sin 2 
1  sin 2 m sin 2 
89
d
and Eq. (12) becomes:

I  a  II

K3 1   sin 2  sin 2  
2
d


o
d
 3 a 3 a
 1  sin 2  m sin 2 
2
2
1


sin

sin


1


m
 2

4 



(14)
2.2. The threshold intensity for Freedericksz transition
In order to find the critical value of I that induces a slow transition from
to
  0 m  0 in the middle of the cell we have to consider m  0 in Eq. (14).
We obtain:

I c a  II

2
d


0
K3
d
3
1 a
2 
and, by integrating we get:
I c a  II
  K3
 3 a 

1 
d   0
2
 2  
(15)
From Eq. (6) and Eq. (10) and using the above mentioned expansion into
series we get:
I a  II  3 a 3 a


sin 2  0  sin 2  m  K1 sin 2  0  K 3 cos 2  0 
1 
   2  4 

A sin  0 cos  0



sin 2  m  sin 2  0
With the change (13), Eq. (16) becomes

(16)
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Eleonora - Rodica Bena, Cristina Cîrtoaje
I a  II  3 a 3 a


sin 2  m sin 2 0  1  K 3 1   sin 2 0 sin 2  m 
1 
   2  4 


 
A sin 0 1  sin 2  m sin 2 0

cos 0

(17)
Fig. 2. The threshold light intensity J c (W/cm 2 as a function of the anchoring strength A for
optical Freedericksz transition.
At the limit of the transition (  m  0 ) Eq. (17) becomes:
I c a  II  3 a 
1 
  tan 0
   2  
Eliminating 0 from Eq. (15) and (18) we obtain:
K3
E I c  cot Ed I c
A
K3
A


(18)
(19)
where
E
I c a  II
  K3
 3 a 
1 

 2  
(20)
Eq. (19) gives a correlation between the value of critical intensity of the
laser beam producing the damage of the homeotropic alignment, the anchoring
strength A , the cell thickness 2d and the material parameters ( K3 ,  a ,  II ,   ).
Surface influence on the optical Freedericksz transition in nematic liquid crystal cells
91
In the rigid anchoring limit A   and considering  a    we obtain
  K3
the formula for optical Freedericksz transition in a
4 d  a  II
homeotropic cell [13], [14].
I C 
2
2
2.3 The saturation intensity
If the laser beam intensity is high, it can produce the transition from
homeotropic to planar alignment all over the cell, i.e. the saturation of the
Freedericksz transition.
In order to find the saturation intensity I s producing the distortion

m 
in the middle of the cell, we reconsider Eqs. (12) and (16) with the
2
change of variables:
cos m 
cos m 
(21)
cos  
;  cos 0 

cos 
cos

0 

Eq. (12) becomes:
I  a  II

0
d
0

 cos 2  m  
K 3 1    1 

cos 2   


 3 a 3 a  cos 2  m
 

2
1  cos 2    

1 

2
cos 
 2  4  
 
d
cos   cos 2  m
2
(22)
When  m 

2
we have:
I s a  II

d
0

0
K1
d
cos 
(23)
By integrating:
I s a  II

From Eq. (16) we get:
d
K1  1  sin 0 

ln 
2
 1  sin 0 
(24)
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Eleonora - Rodica Bena, Cristina Cîrtoaje
I s a  II

K1 
A
sin 0
(25)
Eliminating 0 from Eqs. (24) and (25) we obtain the relationship:
 I 

I 
(26)
A coth  s a II K1   s a II K1






which enables us to calculate the saturation intensity of the laser beam as a
function of the material parameters and the anchoring strength A .
In the rigid anchoring limit A   there does not exist a laser intensity
that produces the saturation transition.
For material parameters we can use the following values:
K 3  7.4 1012 N, K1  5.3 1012 N,  II  2.89 ,    2.25 , (  a  0.64 ). The cell
thickness is 2d  2 104 m and the anchoring strength A is in the range
107  103 J/m 2 , [15], [16].
The threshold and the saturation intensities of the light obtained from Eqs.
(19) and (26) respectively, as a function of the anchoring strength A are shown in
Fig. 2 and Fig. 3 respectively.
Fig. 3 The saturation light intensity for optical Freedericksz transition as a function of the
anchoring strength.
Surface influence on the optical Freedericksz transition in nematic liquid crystal cells
93
3. Discussion
As can be seen in Fig .2, the critical intensity J C for optical Freedericksz
transition is slightly influenced by the anchoring strength: when A increases
hundred times J C increases only three times; for values of A greater than 5 106
J/m 2 the threshold intensity is practically constant and equal to the value obtained
in the case of the rigid anchoring.
Regarding the saturation intensity, it increases rapidly with A ; for A of
the order of 3 107 J/m 2 the laser intensity exceeds the acceptable value for a
liquid crystal cell experiment.
4. Conclusion
We have studied analytically the laser -field induced Freedericksz
transition in an homeotropic aligned NLC, taking into account the surface
anchoring effects. We have calculated the threshold intensity of the laser beam
producing the transition and the saturation intensity as a functions of the
anchoring strength.
For obtaining the Fredeericksz transition using convenient laser intensities
it is possible to take a large interval for the anchoring strength ( 107  104 J/m 2 )
while for obtaining the saturation, the anchoring strength must be small
( 107 J/m 2 ).
Our results can be useful in designing of practical devices used in
optoelectronics. Our results can be useful in designing of practical devices such as
light shutters or laser writing devices.
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Eleonora - Rodica Bena, Cristina Cîrtoaje
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