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Periodic distortions near the Fredericks threshold in nematic liquid crystals
15 June 1998
PHYSICS LETTERS A
Physics Letters A 243 (1998) 71-74
ELSEVIER
Periodic distortions near the Fredericks threshold in nematic liquid crystals P.A. Santoro a, A.J. Palanganaa, M. Sim6es b a Departamento de Fisica, Universidade Estadual de Maring& Av. Colombo 5790, 87020400 Maringd. PR. Brazil b Departamento de Fkica, Universidade Estadual de L.ondrina, 86051 Londrina, Parand, Brazil
Received 5 February 1998; acceptedfor publication 27 February 1998 Communicated by V.M. Agranovich
Abstract The present work studies the behavior of one-dimensional periodic magnetic walls in the neighborhood of the Fredericks threshold of a nematic liquid crystal sample. The results show that in this region there is a linear relation between the inverse of the length of these walls and the square of the magnetic field, whose coefficients depend only on the elastic constants. This relation is verified experimentally and it is used to determine the ratio between the bend elastic constant and the diamagnetic susceptibility anisotropy of a lyotropic nematic sample composed of potassium laurate, potassium chloride and water. @ 1998 Elsevier Science B.V. PAC.? 61.30.Gd; 61.30.Jf; 64.70.Md
1. Introduction When a magnetic or electric field is applied to a previously oriented sample of nematic liquid crystal (NLC) the appearance of a one-dimensional and periodic set of symmetrical distorted textures can be observed. The separation between each of them is called a wall [ 11. The walls are a well-known phenomenon that have been studied both from the theoretical and experimental point of view [ 1,2]. The first experimental study of this kind of magnetic instabilities, in a lyotropic sample, was performed by Charvolin and Hendrikx [ 31. They suggested that their results could be interpreted in terms of a roll structure associated with a backflow effect induced by the orientation of the micelles along the magnetic field direction. In a remarkable work [ 41, Lonberg et al. have shown how this mechanism works: the external magnetic field ro-
tates the director, which stimulates a fluid flow, generating a non-uniform rotation pattern of the director, which reinforces opposite rotations of neighboring regions of the sample. The core of their result is that by this mechanism the wall formation has an effective viscosity that is lower than the one resulting from the matter movement forming a homogeneous alignment. In fact, the whole geometry of the walls is determined by the motion with the least possible effective viscosity [ 4,5], which was defined by Lonberg at al. [ 41 as a compromise between the elastic energy, that opposes short wavelengths, and the viscosity, that opposes long wavelengths. In this Letter, we present a study of these periodic distortions in the neighborhood of the Fredericks threshold of a nematic calamitic sample. Our main result is that in the neighborhood of the Fredericks threshold there is a linear relation between Ae2
0375-9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PIISO375-9601(98)00231-X
72
PA. Santoro et al. /Physics
and H2, whose coefficients only depend on the elastic constants and the diamagnetic susceptibility, and the viscosity of the fluid is irrelevant. This should be an unexpected result because, as explained above, the wall length is determined by the geometry of the fluid flow occurring as soon as the magnetic field is turned on. So it could be expected that this relation would also be a function of the sample’s viscosity. The idea of an effective viscosity is the key to understand the present discovery. In this magnetic field region the wall’s wavelength has a size so large that the effective viscosity then becomes so large (the time spent in the walls’ cons~uction goes to inanity) that the actual values of the viscosity coefficients become negligible. Of course their absence in the linear dependence between Ae2 and H2 does not mean, in these magnetic field regions, that the motion of the nematic material is not important for the walls’ cons~uction. This is the m~h~isrn by which the walls are constructed and the actual values of the viscosity coefficients would appear in a second-order approximation. These facts will be demonstrated here and used to determine the ratio between the bend elastic constant (K33) and the aniso~opy of the diamagnetic susceptibility ( xa) of a nematic lyotropic mixture of potassium laurate (KL), potassium chloride ( KCl) and water with the respective concentrations in weight percent: 34.5, 3.0, 62.5. The nematic calamitic phase (NC) upon heating changes to the hexagonal phase at about 40°C [6]. Fur~ermore, it will also be possible to determine the ratio between the bend elastic (K33) constant and the splay elastic constant ( KI 1) by comparing the present experimental data with other data of the same nematic lyotropic phase [ 61.
2. Fundamentds Nematic calamitic samples were encapsuled in flat glass microslides (length a = 20 mm, width b = 2.5 mm, thickness d = 0.2 mm) from Vitr~ynamics. During the experiment the temperature was controlled at (25 f 1) “C. The uniform initial orientation of the director n, in a planar geometry, is achieved by applying a magnetic field (N 10 kG) parallel to the e, direction. After achieving the initial homogeneous orientation of the director, the field along the e, direction is switched off and a fixed-strength-controlled mag-
L.etters A 243 (1998) 71-74
netic field (H) is applied along the eYdirection [7,8]. When this is done there is a competition between the magnetic susceptibility, which tends to align the director along the magnetic field direction, and the elastic energies, which tends to produce a director orientation consistent with its orientation at the surface of the sample. When H is greater than the Fredericks threshold Hc the magnetic susceptibility overcomes the elastic resistance of the medium and the director, similar to in a second-order phase transition, begins to bend to the external magnetic field in order to become oriented either parallel or antiparallel to it. Let n(x,y,z)
=e,cosB(x,y,z)
+e,sin8(x,y,t),
be the NCphase director for a deformation of the kind discussed above. From a theoretical point of view, strong anchoring conditions on the glass bounds surfaces of the microslide are assumed for the nematic samples. The expression of the elastic Frank free energy, taking into account the magnetic field coupling, is [I]
s
F+
[U&n;
+ K33n;)GW2
V
+ 2tK33 -
-
~l~)~~~~(~~~)(~~~)
xaH2n$] dV,
-f- $22(W2
(1)
where K11, K22, and Ks3 are the elastic constants of splay, twist and bend, 0(x, y, z ) is the angle between the director it and the e, direction and V is the volume of the sample. To describe the matter flow and director bending, the so-called Eriksen-Leslie-Parodi (ELP) approach will be used [ 1,9]. In this picture the NLC motion is given by the anisotropic version of the Navier-Stokes equation,
where p the density of the system, V, the LYcomponent of the velocity, p the pressure and up0 the associated anisotropic stress tensor [9], which depends on the velocity V of the fluid, on the bending of the director ~9and on its time variation rate e.
EA. Santoro et al. /Physics Letters A 243 (1998) 71-74
In the ELF approach the equation of motion of the director in the geometry fixed above assumes the form [ 5,101
- y&&.&r:
- 8;) + (AYY- Axx)&aY]
+ (K,,n? + Ks&@ + K&f@ +
(K33
+ (K*,nZ + K33+$B
+ xaH2nxnY -
&,)(~,W.Y~)~
-
(M)=
-
Q-$J~I
- (4 - a;, (&@I(it,@))*
(3)
where the inertial terms were not considered, yt and y2 are the shear torque coefficients, A,p = 3 (&Vp + JpV,), W,p = ~(c?~V,, - 8pV,) and, as usual, the quid is considered incompressible. In this equation the walls are observed as being one-dimensional structures(d~8~0,8~~~0,V,~0,and~,V,w0) [S], and the following is obtained,
+
K22afO
+ wJx,H=
-
(K33
-
h>(&S)21.
(4) Furthermore, as the external magnetic field is just a little above the Fredericks threshold it affords a very small bending (0 +C 1, such that rzYz 8, $ M 0 and RXM 1) . So this equation becomes y1&0= ;(y, -y2)a,v,+K3:,3a,2e+K22a2e+x,H2e. (5)
To determine d, V, the Navier-Stokes equation must be used, Eq. (2). When this is done it is found that [ 591 dxQ M a&8, where LYis a parameter that depends on the viscosities of the sample and the length A of the walls. From Eq. (5) it is easy to see that when the magnetic field approaches the Fredericks threshold the time spent in wall formation is of the order, r N ( Ia I)-‘, where h = H/He. Note that in the limit h2 -3 1, T goes to infinite and, for any practical application, a,e x 0. The interesting point about Eq. (5) is that even in this limit (7 + cc) it predicts a very large, but finite, length A to the walls. To see this it is enough to assume, as usual [ 4, I 1,121,
13
and r3& M 0, where $0 is a very small constant, to obtain that in this limit A would be given by A-2 = ct H= - c2,
(7)
where ct = ~~/472~Kss is the slope of the linear dependence of X2 as a function of H2. The constant c2 is connected to the boundary conditions. For a study of the anchoring strength of a nematic lyotropic liquid crystal in the NC phase, see Ref. [ 131. Eq. (7) will be used in the sequence to determine the ratio K33/xa of a lyotropic mixture of potassium laurate, potassium chloride and water in the nematic calamitic phase. 3. Results and discussion A typical texture observed in a poi~izing microscope shows a periodic distortion of n with walls formed in the eY direction (see Fig. 1 of Ref. [ 141). As presented in the last section, due to the fact that the magnetic field is close to the Fredericks threshold, the time spent in wall construction is too long. In the present experiment, and in the range chosen for the magnetic field, this time varies from 8 hours (H N 3 kG) up to about 14 hours (H N 2.5 kG). Once formed, these walls are sufficiently stable to perform several measurements, and the relaxation rate is so slow that after about 8 hours, in the presence of the magnetic field (H N 2.5 kG), and in about 4 hours for H w 3 kG, the periodic distortion of the texture remains stable. In Fig. 1, the experimental values of hB2 are plotted as a function of H2. Many independent measurements of A have been made at a controlled magnetic field and a mean value was obtained. As expected, for a magnetic field close to Hc (H < 2.5H,), the experimental points can be fitted by a straight line (see Eq. 7), where the approximation of small distortions is satisfied. From the slope of the linear dependence of 1\-2 as a function of H2 one obtains the ratio Ks1(33/xa N 316 dyn and the value of the critical magnetic field obtained from an extrapolation of the fitting is Hc N 1.22 kG. The experimental error in Ks3/xa (about 6%) was evaluated taking into account the possible inhomogeneities in the textures with the creation of the periodic distortions. To our knowledge there are no independent measurement of this ratio regarding this lyotropic mixture. A recent measurement [6] of
I?A. Santoro et al. /Physics Letters A 243 f1998) 71-14
dealing with a metastable system. Furthermore, as a consequence of these experimental results the ratio between the bend elastic constant and the splay elastic constant was also determined.
6
1
/
t
Acknowledgement
Many thanks are due to AK. Zvezdin (Moscou) and G. Barbero (Turim) for useful discussions, and the Brazilian Agency (CNPq f for partial financial support. 0
2
’
4
’
’ 8
1
IO
II@)” :2) Fig. 1. Dependence of the wavelength of the periodic distortion (A) on the applied magnetic field (H). The linear behavior is described by Eq. (7).
K1 g/,ya for the same lyotropic mixture, at 25”C, gives the value Kr 1/,ya II 126 dyn. Combination of these ex~rimen~ data gives KssfKt 1 N 2.51. These results are in the same range as similar ones reported in the liquid crystals literature [ 93, and as expected K33 > Ki 1. Another point to be remarked upon is that the known order of magnitudes of xa [ 151, combined with our experimental results, leads to the conclusion that lyotropic elastic constants are of the same order of magnitude as those of thermotropics liquid crystals, ( 10m6dyn). To sum up, we have carried out a detailed study concerning the behavior of one-dimensional periodic magnetic walIs in the Fredericks ~reshold neighborhoods. As in this range of magnetic field (H < 2SH,) the growth of the director bending is very slow, we have used Eq. (7) to describe the periodic structure that appears near the Fredericks threshold. In this case, it was possible to determine the elastic parameter of a nematic calamitic phase, despite the fact we are
References [ 1I F?G.de Gennes, The Physics of Liquid Crystals, 2nd Ed. (Clarendon, Oxford, 1993). [2] E.B. Priestley, P.J. Wojtowickz, P. Sheng, eds., Introduction to Liquid Crystals (Plenum, New York, 1979). [31 J. Charvolin, Y. Hendrikx, J. Phys. (Paris) Len. 41 ( 1980) 597. 141 F. Lo&erg, S. Fraden, A.J. Hurd, R.B. Meyer, Phys. Rev. Lea. 52 (1984) 1903. [?I] M. Sim&s, Phys. Rev. E 56 (1997) 3061. 161A.V.A. Pinto, L.Q. Amaral, J. Phys. Chem. 94 (1990) 3186. [71 T. Kroin, A.M. Figueiredo Neto, Phys. Rev. A 36 (1987) 2987. [81 A.J. Palangana, M. Simoes, L.R. Evangelista, A.A. Arroteia, Phys. Rev. E 56 (1997) 4282. [91 G. Vertogen, W.H. De Jeu, Thermotropic Liquid Crystals, Fundamentals (Springer, Berlin, 1988). [lOI M. Sirnoes, A.J. Palangana, L.R. Evangelista, Phys. Rev. E 56 (1997) 7326. 1111 T. Kroin, Ph.D. thesis, Institute de Fisica, Unive~idade de Sgo Paul0 ( 1990). [121 E. Oliveira, G. Barbero, A.M. Figueiredo Neto, Phys. Rev. 54 (19%) 5830. 1131 A.M. Ribas, L.R. Evangelism, A.J. Palangana, E.A. Oliveira, Phys. Rev. E 51 (1995) 5204. [I41 A.J. Palangana, L.R. Evangelista. A.K. Zvezdin, Phys. Len A 200 (1995) 56. 1151 M. Stefanov, A. Saupe, Mol. Cryst. Liq. Cryst. 108 (1984) 309.
PHYSICS LETTERS A
Physics Letters A 243 (1998) 71-74
ELSEVIER
Periodic distortions near the Fredericks threshold in nematic liquid crystals P.A. Santoro a, A.J. Palanganaa, M. Sim6es b a Departamento de Fisica, Universidade Estadual de Maring& Av. Colombo 5790, 87020400 Maringd. PR. Brazil b Departamento de Fkica, Universidade Estadual de L.ondrina, 86051 Londrina, Parand, Brazil
Received 5 February 1998; acceptedfor publication 27 February 1998 Communicated by V.M. Agranovich
Abstract The present work studies the behavior of one-dimensional periodic magnetic walls in the neighborhood of the Fredericks threshold of a nematic liquid crystal sample. The results show that in this region there is a linear relation between the inverse of the length of these walls and the square of the magnetic field, whose coefficients depend only on the elastic constants. This relation is verified experimentally and it is used to determine the ratio between the bend elastic constant and the diamagnetic susceptibility anisotropy of a lyotropic nematic sample composed of potassium laurate, potassium chloride and water. @ 1998 Elsevier Science B.V. PAC.? 61.30.Gd; 61.30.Jf; 64.70.Md
1. Introduction When a magnetic or electric field is applied to a previously oriented sample of nematic liquid crystal (NLC) the appearance of a one-dimensional and periodic set of symmetrical distorted textures can be observed. The separation between each of them is called a wall [ 11. The walls are a well-known phenomenon that have been studied both from the theoretical and experimental point of view [ 1,2]. The first experimental study of this kind of magnetic instabilities, in a lyotropic sample, was performed by Charvolin and Hendrikx [ 31. They suggested that their results could be interpreted in terms of a roll structure associated with a backflow effect induced by the orientation of the micelles along the magnetic field direction. In a remarkable work [ 41, Lonberg et al. have shown how this mechanism works: the external magnetic field ro-
tates the director, which stimulates a fluid flow, generating a non-uniform rotation pattern of the director, which reinforces opposite rotations of neighboring regions of the sample. The core of their result is that by this mechanism the wall formation has an effective viscosity that is lower than the one resulting from the matter movement forming a homogeneous alignment. In fact, the whole geometry of the walls is determined by the motion with the least possible effective viscosity [ 4,5], which was defined by Lonberg at al. [ 41 as a compromise between the elastic energy, that opposes short wavelengths, and the viscosity, that opposes long wavelengths. In this Letter, we present a study of these periodic distortions in the neighborhood of the Fredericks threshold of a nematic calamitic sample. Our main result is that in the neighborhood of the Fredericks threshold there is a linear relation between Ae2
0375-9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PIISO375-9601(98)00231-X
72
PA. Santoro et al. /Physics
and H2, whose coefficients only depend on the elastic constants and the diamagnetic susceptibility, and the viscosity of the fluid is irrelevant. This should be an unexpected result because, as explained above, the wall length is determined by the geometry of the fluid flow occurring as soon as the magnetic field is turned on. So it could be expected that this relation would also be a function of the sample’s viscosity. The idea of an effective viscosity is the key to understand the present discovery. In this magnetic field region the wall’s wavelength has a size so large that the effective viscosity then becomes so large (the time spent in the walls’ cons~uction goes to inanity) that the actual values of the viscosity coefficients become negligible. Of course their absence in the linear dependence between Ae2 and H2 does not mean, in these magnetic field regions, that the motion of the nematic material is not important for the walls’ cons~uction. This is the m~h~isrn by which the walls are constructed and the actual values of the viscosity coefficients would appear in a second-order approximation. These facts will be demonstrated here and used to determine the ratio between the bend elastic constant (K33) and the aniso~opy of the diamagnetic susceptibility ( xa) of a nematic lyotropic mixture of potassium laurate (KL), potassium chloride ( KCl) and water with the respective concentrations in weight percent: 34.5, 3.0, 62.5. The nematic calamitic phase (NC) upon heating changes to the hexagonal phase at about 40°C [6]. Fur~ermore, it will also be possible to determine the ratio between the bend elastic (K33) constant and the splay elastic constant ( KI 1) by comparing the present experimental data with other data of the same nematic lyotropic phase [ 61.
2. Fundamentds Nematic calamitic samples were encapsuled in flat glass microslides (length a = 20 mm, width b = 2.5 mm, thickness d = 0.2 mm) from Vitr~ynamics. During the experiment the temperature was controlled at (25 f 1) “C. The uniform initial orientation of the director n, in a planar geometry, is achieved by applying a magnetic field (N 10 kG) parallel to the e, direction. After achieving the initial homogeneous orientation of the director, the field along the e, direction is switched off and a fixed-strength-controlled mag-
L.etters A 243 (1998) 71-74
netic field (H) is applied along the eYdirection [7,8]. When this is done there is a competition between the magnetic susceptibility, which tends to align the director along the magnetic field direction, and the elastic energies, which tends to produce a director orientation consistent with its orientation at the surface of the sample. When H is greater than the Fredericks threshold Hc the magnetic susceptibility overcomes the elastic resistance of the medium and the director, similar to in a second-order phase transition, begins to bend to the external magnetic field in order to become oriented either parallel or antiparallel to it. Let n(x,y,z)
=e,cosB(x,y,z)
+e,sin8(x,y,t),
be the NCphase director for a deformation of the kind discussed above. From a theoretical point of view, strong anchoring conditions on the glass bounds surfaces of the microslide are assumed for the nematic samples. The expression of the elastic Frank free energy, taking into account the magnetic field coupling, is [I]
s
F+
[U&n;
+ K33n;)GW2
V
+ 2tK33 -
-
~l~)~~~~(~~~)(~~~)
xaH2n$] dV,
-f- $22(W2
(1)
where K11, K22, and Ks3 are the elastic constants of splay, twist and bend, 0(x, y, z ) is the angle between the director it and the e, direction and V is the volume of the sample. To describe the matter flow and director bending, the so-called Eriksen-Leslie-Parodi (ELP) approach will be used [ 1,9]. In this picture the NLC motion is given by the anisotropic version of the Navier-Stokes equation,
where p the density of the system, V, the LYcomponent of the velocity, p the pressure and up0 the associated anisotropic stress tensor [9], which depends on the velocity V of the fluid, on the bending of the director ~9and on its time variation rate e.
EA. Santoro et al. /Physics Letters A 243 (1998) 71-74
In the ELF approach the equation of motion of the director in the geometry fixed above assumes the form [ 5,101
- y&&.&r:
- 8;) + (AYY- Axx)&aY]
+ (K,,n? + Ks&@ + K&f@ +
(K33
+ (K*,nZ + K33+$B
+ xaH2nxnY -
&,)(~,W.Y~)~
-
(M)=
-
Q-$J~I
- (4 - a;, (&@I(it,@))*
(3)
where the inertial terms were not considered, yt and y2 are the shear torque coefficients, A,p = 3 (&Vp + JpV,), W,p = ~(c?~V,, - 8pV,) and, as usual, the quid is considered incompressible. In this equation the walls are observed as being one-dimensional structures(d~8~0,8~~~0,V,~0,and~,V,w0) [S], and the following is obtained,
+
K22afO
+ wJx,H=
-
(K33
-
h>(&S)21.
(4) Furthermore, as the external magnetic field is just a little above the Fredericks threshold it affords a very small bending (0 +C 1, such that rzYz 8, $ M 0 and RXM 1) . So this equation becomes y1&0= ;(y, -y2)a,v,+K3:,3a,2e+K22a2e+x,H2e. (5)
To determine d, V, the Navier-Stokes equation must be used, Eq. (2). When this is done it is found that [ 591 dxQ M a&8, where LYis a parameter that depends on the viscosities of the sample and the length A of the walls. From Eq. (5) it is easy to see that when the magnetic field approaches the Fredericks threshold the time spent in wall formation is of the order, r N ( Ia I)-‘, where h = H/He. Note that in the limit h2 -3 1, T goes to infinite and, for any practical application, a,e x 0. The interesting point about Eq. (5) is that even in this limit (7 + cc) it predicts a very large, but finite, length A to the walls. To see this it is enough to assume, as usual [ 4, I 1,121,
13
and r3& M 0, where $0 is a very small constant, to obtain that in this limit A would be given by A-2 = ct H= - c2,
(7)
where ct = ~~/472~Kss is the slope of the linear dependence of X2 as a function of H2. The constant c2 is connected to the boundary conditions. For a study of the anchoring strength of a nematic lyotropic liquid crystal in the NC phase, see Ref. [ 131. Eq. (7) will be used in the sequence to determine the ratio K33/xa of a lyotropic mixture of potassium laurate, potassium chloride and water in the nematic calamitic phase. 3. Results and discussion A typical texture observed in a poi~izing microscope shows a periodic distortion of n with walls formed in the eY direction (see Fig. 1 of Ref. [ 141). As presented in the last section, due to the fact that the magnetic field is close to the Fredericks threshold, the time spent in wall construction is too long. In the present experiment, and in the range chosen for the magnetic field, this time varies from 8 hours (H N 3 kG) up to about 14 hours (H N 2.5 kG). Once formed, these walls are sufficiently stable to perform several measurements, and the relaxation rate is so slow that after about 8 hours, in the presence of the magnetic field (H N 2.5 kG), and in about 4 hours for H w 3 kG, the periodic distortion of the texture remains stable. In Fig. 1, the experimental values of hB2 are plotted as a function of H2. Many independent measurements of A have been made at a controlled magnetic field and a mean value was obtained. As expected, for a magnetic field close to Hc (H < 2.5H,), the experimental points can be fitted by a straight line (see Eq. 7), where the approximation of small distortions is satisfied. From the slope of the linear dependence of 1\-2 as a function of H2 one obtains the ratio Ks1(33/xa N 316 dyn and the value of the critical magnetic field obtained from an extrapolation of the fitting is Hc N 1.22 kG. The experimental error in Ks3/xa (about 6%) was evaluated taking into account the possible inhomogeneities in the textures with the creation of the periodic distortions. To our knowledge there are no independent measurement of this ratio regarding this lyotropic mixture. A recent measurement [6] of
I?A. Santoro et al. /Physics Letters A 243 f1998) 71-14
dealing with a metastable system. Furthermore, as a consequence of these experimental results the ratio between the bend elastic constant and the splay elastic constant was also determined.
6
1
/
t
Acknowledgement
Many thanks are due to AK. Zvezdin (Moscou) and G. Barbero (Turim) for useful discussions, and the Brazilian Agency (CNPq f for partial financial support. 0
2
’
4
’
’ 8
1
IO
II@)” :2) Fig. 1. Dependence of the wavelength of the periodic distortion (A) on the applied magnetic field (H). The linear behavior is described by Eq. (7).
K1 g/,ya for the same lyotropic mixture, at 25”C, gives the value Kr 1/,ya II 126 dyn. Combination of these ex~rimen~ data gives KssfKt 1 N 2.51. These results are in the same range as similar ones reported in the liquid crystals literature [ 93, and as expected K33 > Ki 1. Another point to be remarked upon is that the known order of magnitudes of xa [ 151, combined with our experimental results, leads to the conclusion that lyotropic elastic constants are of the same order of magnitude as those of thermotropics liquid crystals, ( 10m6dyn). To sum up, we have carried out a detailed study concerning the behavior of one-dimensional periodic magnetic walIs in the Fredericks ~reshold neighborhoods. As in this range of magnetic field (H < 2SH,) the growth of the director bending is very slow, we have used Eq. (7) to describe the periodic structure that appears near the Fredericks threshold. In this case, it was possible to determine the elastic parameter of a nematic calamitic phase, despite the fact we are
References [ 1I F?G.de Gennes, The Physics of Liquid Crystals, 2nd Ed. (Clarendon, Oxford, 1993). [2] E.B. Priestley, P.J. Wojtowickz, P. Sheng, eds., Introduction to Liquid Crystals (Plenum, New York, 1979). [31 J. Charvolin, Y. Hendrikx, J. Phys. (Paris) Len. 41 ( 1980) 597. 141 F. Lo&erg, S. Fraden, A.J. Hurd, R.B. Meyer, Phys. Rev. Lea. 52 (1984) 1903. [?I] M. Sim&s, Phys. Rev. E 56 (1997) 3061. 161A.V.A. Pinto, L.Q. Amaral, J. Phys. Chem. 94 (1990) 3186. [71 T. Kroin, A.M. Figueiredo Neto, Phys. Rev. A 36 (1987) 2987. [81 A.J. Palangana, M. Simoes, L.R. Evangelista, A.A. Arroteia, Phys. Rev. E 56 (1997) 4282. [91 G. Vertogen, W.H. De Jeu, Thermotropic Liquid Crystals, Fundamentals (Springer, Berlin, 1988). [lOI M. Sirnoes, A.J. Palangana, L.R. Evangelista, Phys. Rev. E 56 (1997) 7326. 1111 T. Kroin, Ph.D. thesis, Institute de Fisica, Unive~idade de Sgo Paul0 ( 1990). [121 E. Oliveira, G. Barbero, A.M. Figueiredo Neto, Phys. Rev. 54 (19%) 5830. 1131 A.M. Ribas, L.R. Evangelism, A.J. Palangana, E.A. Oliveira, Phys. Rev. E 51 (1995) 5204. [I41 A.J. Palangana, L.R. Evangelista. A.K. Zvezdin, Phys. Len A 200 (1995) 56. 1151 M. Stefanov, A. Saupe, Mol. Cryst. Liq. Cryst. 108 (1984) 309.