HAL Id: jpa-00218749
https://hal.archives-ouvertes.fr/jpa-00218749
Submitted on 1 Jan 1979
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
THE FIELD-INDUCED SQUARE GRID
PERTURBATION IN THE PLANAR TEXTURE OF CHOLESTERIC LIQUID CRYSTALS
M. de Zwart, C. van Doorn
To cite this version:
M. de Zwart, C. van Doorn. THE FIELD-INDUCED SQUARE GRID PERTURBATION IN THE PLANAR TEXTURE OF CHOLESTERIC LIQUID CRYSTALS. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-278-C3-284. �10.1051/jphyscol:1979353�. �jpa-00218749�
THE FIELD-INDUCED SQUARE GRID PERTURBATION
IN THE' PLANAR TEXTURE OF CHOLESTERIC LIQUID CRYSTALS
M. DE ZWART and C. Z. VAN DOORN Philips Research Laboratories, Eindhoven, The Netherlands
R6sum6. - Nous avons ktudik l'influence de la contraction et de la dilatation du pas d'hklice sur la deformation statique en rkseau quadratique de la texture planaire cholestkrique, provoquke par des champs magnktiques et klectriques. Des experiences ont Btk effectukes aussi bien dans des cellules en forme de coin d'un cristal liquide anisotropie diklectrique positive ou negative que dans des cellules parallkles d'un cristal liquide a anisotropie negative importante (BE =
-
5). Dans la cellule en forme de coin la contraction ou la dilatation du pas d'hklice s'allie a un accroissement ou une diminution de la tension de seuil pour la deformation. Dans la cellule parallkle la formation de nouvelles rkgions planaires associkes a un nombre croissant d'helices s'allie a un accroissement important de la tension de seuil. La longueur d'onde de la deformation est indkpendante de l'kcart par rapport au pas naturel. Notre developpement de la theorie de Helfrich-Hurault concorde quan- titativement avec les rksultats expkrimentaux.Abstract. - We have investigated the influence of pitch contraction and dilatation on the static square grid deformation of the cholesteric planar texture induced by magnetic and electric fields.
Experiments were performed both in wedge-shaped cells using liquid crystals with a positive or negative dielectric anisotropy and in a parallel cell using a liquid crystal with a large (BE = - 5) negative dielectric anisotropy. In the wedge, a contraction or dilatation of the pitch is accompanied by an increase or decrease of the threshold of the deformation, respectively. In the planar cell the formation of new planar regions with a larger number of helical turns is accompanied by a large increase of the threshold voltage. The wavelength of the deformation does not depend on the deviation of the pitch from its natural value. Our extension of the Helfrich-Hurault theory is in good quantita- tive agreement with the experiments.
1. Introduction. - When magnetic and electric fields with a value above a certain threshold are applied across a planar layer of a cholesteric liquid crystal, a distortion of the texture is induced. This distortion starts with a perturbation which can be observed under a polarizing microscope as a square grid pattern [I]. In the magnetic case this distortion is induced by a magnetic torque as a result of the positive diamagnetic anisotropy of the liquid crystal ( A x = X I , - xL > 0, where x l l and x, are the magnetic susceptibilities parallel and perpendicular, respecti- vely, to the director). In liquid crystals with a pdsitive dielectric anisotropy AE the distortion mechanism is similar, whereas in liquid crystals with a negative AE the square grid perturbation is produced by electro- hydrodynamic torques. In that case the conductivity anisotropy A o must be positive.
According to the theory by Helfrich [2, 31 and Hurault [4] the threshold field should be propor- tional to ( p d ) - 1 1 2 , where p and d are the actual pitch and thickness of the unperturbed planar layer.
The spatial periodicity of the square grids at threshold should be proportional to (pd)'I2. These predictions
were obtained with the condition d % p, but it has been shown experimentally that when d and p are of comparable magnitude the relations are still valid [5].
Therefore, the square of the threshold voltage (V,) should be proportional to the number of helical turns across the planar layer. However, several authors [6-81 found experimentally that a boundary- induced contraction or dilatation of the pitch also has a strong influence on the threshold value. A contraction of the pitch from its natural value po shifts the threshold to a higher voltage, whereas a dilatation decreases the threshold voltage. On the other hand the spatial periodicity of the square grids is not affected by the contraction or dilatation of the pitch. These effects were observed in wedge- shaped sandwich cells. In such a sample cell regions exist between two Grandjean-Cano lines where the pitch has a smooth transition from contraction to dilatation.
Furthermore de Zwart [9] found that an increasing electric field across the cholesteric planar texture of a liquid crystal with a relatively large negative
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979353
FIELD-INDUCED SQUARE GRID PERTURBATION IN PLANAR TEXTURE C3-279
dielectric anisotropy (A& 5 - 1) induces sequentially new planar regions in which the threshold for square grid perturbation is increasingly higher. Each new planar region is formed after an intermediate stage of square grid perturbation and is accompanied by an increase of the number of helical turns by one, resulting in pitch contraction. These induced new planar textures remain stable until a new threshold for square grids is reached. In this case, too, Hurault's theory cannot explain the increase of V, in the induced planar regions.
The purpose of this paper is to discuss an extension of the Helfrich-Hurault model which includes the case of pitch contraction and dilatation. In section 2 a theoretical description of this extension is given.
The experimental conditions are reported in section 3.
In section 4 the new equations for threshold voltage and periodicity of the square grids are compared with experimental data. The conclusions are pre- sented in section 5.
2. Theoretical. - The model used consists of a cholesteric planar layer of thickness d with rigid boundary conditions (Fig. 1) and a number of m
FIG. 1 . -Geometry of the distortion in the cholesteric planar layer.
helical turns (defined as a 2 n rotation of the director) across the layer. The actual pitch p imposed by the cholesteric material and boundary conditions is given by p = dlm. The wave numbers t and q are defined by t = 2 nlp and q = nld. A magnetic field is applied perpendicular to the layer, i.e. parallel to the z axis. The induced perturbation in the cholesteric planes (to be defined below) is treated as a sinusoi- dal deformation with a period I in the x direction.
The corresponding wave number is k = 2 n/I.
We assume that
which means that
A cholesteric plane is defined as the plane to which the director is tangent everywhere and the twist angle cp is constant.
In an unperturbed layer we have
For the unperturbed layer the z position of a choleste- ric plane is a function of cp only :
but for a perturbed layer it is a function of cp and x, which we assume to be of the form
z(cp,x)
= f +
bsin - coskx, b -4 I . (2) (: m )
Here we have introduced a small perturbation that is periodic in the x direction and has zero value at the boundaries. We also need the twist angle cp as a function of x and z. This is obtained approximately from eq. (2) by substituting the unperturbed expres- sion eq. (1) into the small term of eq. (2) and solving for cp :
cp = tz - bt sin qz cos kx
.
(3) If n is the local director, the Frank free energy density for a cholesteric texture in a magnetic fieId is given by :1 1
F =
5
Kl ,(div n)'+
- 2 KZ2(n. rot n+
to)2+
1 1
+
K,,(n x rot n)2 - - 2 Ax(n. ,
(4)where K1 ,, K2, and K,, are the Frank elastic constants of splay, twist and bend, respectively, and H is the magnetic field. The natural twist ,2 is related to the natural pitch po through to = 2 nip,. The components of the director n are
n, = cos cp n, = sin cp n, = ti? cos cp ,
II/
is the angle between the cholesteric plane and the x axis (Fig. 1). Substitution of eq. ( 5 ) into eq. (4) gives1
F = - K ~ ~ 2
{-
s i n y l ( % + + % ) + C O S C ~ ~ ] ~ 8.2 +;K,,{%- to -- acp
ax+
sin cp cos cp - ax+
1
a* l2
+ ~ K ~ ~ { ~ O S ' ~ @ + J , ~ ) ~ + C O S ~ ~ ~ ( ~ ) ~ \ - Z A ~ H : ~ ~ s 2 C t , h 2 . ( 6 )
The various quantities of this equation are obtained from eq. (2) and (3). After substitution of these quantities in eq. (6) the mean free energy density F can be easily obtained (see the Appendix). Mini- mization of with respect to the distortion ampli- tude gives for the field H, at which the deformation starts :
Because of the assumption
A
$ d the Kll-term is left out of account in this expression.Putting dp/ak = 0 results in
When eq. (8) is substituted in eq. (7) and q, t and to are expressed in d, we obtain :
with
The term (dlp - d/po) in eq. (9) describes the influence of the deviation of the pitch from its natural value.
This term results from ;aking the cross product of
a4p
-
and@
- to) in eq. (6).ax
For th6 wavel&ngth of the perturbation we find via eq. (8) :
3 K,, 'I2
2 ( ) 2 ( + - - ) p d . (10) 2 2 K22
We notice that when p = p,, and without the term 112 in the square root, eq. (9) and (10) agree with the equations derived by Hurault [4].
Assuming that the square grid perturbation can be treated as two independent perpendicular distor- tion modes, eq. (9) and (10) are also valid in the two- dimensional case.
Furthermore, it will be clear that analogous results can be expected for a purely electrically distorted layer. The equation for the threshold voltage Vc is then :
For a liquid crystal with AE < 0 in an electric field the role of the magnetic torque is taken over by the combined dielectric and viscous torques. Taking
for the latter the expressions derived by Hurault [4]
we obtain for the electric case from eq. (9) by the substitutions
1 4 n ~ ~ + ~ l l 1 + c o 2 r 2
Hc + Ec and
-
+ - -AX zl El - Ell
5
- 1 - m2 r2I(;
3K3.)"' (1 ' ) I d x KZ2- + -
- + 4 - - - d - (12)2 K22 P Po, P
with
where o,, and a, are the electrical conductivities parallel and perpendicular to the director and Vc the r.m.s. value of the threshold voltage.
3. Experimental conditions. - The experiments were performed in sample cells which consisted of two glass plates coated with transparent electrodes and separated by spacers. The spacing of the empty cell is measured interferometrically with an accuracy of 0.5
%.
By rubbing the electrodes unidirectionally prior to assembling the cell a uniform planar layer is obtained, in which the number of helical turns across the inserted cholesteric layer equals an integer or half-integer value.When the sample cell is constructed in such a way that the spacing between the glass plates has a small Liniform variation in one direction, the cholesteric liquid crystal exhibits many single equally-spaced Grandjean-Cano disclination lines (Fig. 2). These
Grandjean-Cuno l lnes
FIG. 2. - Unidirectionally rubbed wedge-shaped sample cell.
The cholesteric planar layer possesses equally spaced single Grand- jean-Cano disclination lines. The open circles, shaded circles and triangles denote the positions of the corresponding threshold
voltages and grids in figures 3,4, 5 and 6.
lines are formed as a consequence of a discontinuity of a half in the number of helical turns. The pitch has the natural valuep, in the middle of a strip between two successive disclination lines but increases towards the wide side of the cell and decreases towards the narrow side. Then at the wide side of a strip d/p = d/po - 114 and at the narrow side of a strip dlp = dlpo
+
114.FIELD-INDUCED SQUARE GRID PERTURBATION IN PLANAR TEXTURE C3-28 1
For the experiments we used a liquid crystal with a positive dielectric anisotropy : 4-cyano-4'-n-heptyl biphenyl, available as K21 from the BDH company.
The static dielectric anisotropy and the elastic constants for the twist and bend mode were respec- tively : AE =
+
11.0, K Z 2 = 8.2 x dyne and K 3 , = 23.2 x dyne at 25 OC. The elastic constants were taken from reference [lo]. The cholesteric phase was obtained by dissolving 2.9 wt%
cholesteryl nona- noate (CN) in the nematic liquid crystal. The natural pitchp,, determined in a Cano wedge, was 5.5 f 0.2 pm.For the liquid crystal with a negative dielectric aniso- tropy we used a mixture of 68 wt
%
of 4-butoxy-4'-n- heptyl-a-cyano-trans-stilbene and 32 wt%
4-ethoxy-4'- n-hexoxy-a-cyano-trans-stilbene-
-
[9]. The static dielec-tric anisotropy was- 5.0, K2, = 5.2 x lop7 dyne and K,, = 9.6 x dyne at 25 OC. To this liquid crystal mixture 2.3 wt
%
cholesteryl nonanoate was added, resulting in a natural pitch p, of 3.3 f 0.2 pm.A polarizing microscope was used to observe the square grid patterns. The measurements were performed at a temperature of 25 oC.
FIG. 3. -Threshold dependence on the number of helical turns (clip) across the cholesteric planar layer of the K21/CN mixture in a wedge-shaped sample cell. The open circles and triangles are the values on the wide and narrow side, respectively, of the strips bet- ween two successive single Grandjean-Cano lines. The shaded circles are the data in the middle of these strips. The solid curves are the corresponding values calculated with eq. (11). The dashed curve represents the theoretical calculations made with Hurault's theory.
4. Results. - 4.1 WEDGE-SHAPED SAMPLE CELL. -
Upon application of an increasing electric field across the cholesteric planar layer in a wedge-shaped cell, the occurrence of the square grid pattern always starts at the wide side of the strips between the Grand- jean-Cano lines. A further increase of the field induces a uniform expansion of this pattern to the narrow side. In figure 3 the square of the threshold voltages in the K21/CN mixture is plotted as a function of the number of helical turns. The values near the Grandjean-Cano lines at the wide and narrow side of the strips are represented by the open circles and triangles, respectively. The threshold voltage in the middle of the strips, where p = p,, is given by the shaded circles. The corresponding theoretical values calculated with eq. (1 1) are given-by the solid curves.
The same experiments were performed in the a-cyanostilbenelCN mixture. The results for the threshold voltage, reported earlier in reference [ll], are shown in figure 4. The calculated values using eq. (12) are represented by the solid curves. These
FIG. 4. - Dependence of the threshold voltage at low frequency on the number of helical turns (dip) across the cholesteric planar layer of the a-cyanostilbenelCN mixture. The open circles and triangles are the values on the wide and narrow side of the strips, respectively. The shaded circles are the values in the middle of these strips. The values calculated with eq. (12) are given by the solid curves. The dashed curve represents the values found with the aid of Hurault's theory. The squares, obtained from a parallel sample cell with d = 20 pm, correspond to the original (labelled A) and induced (B'.
. .
C ) planar regions [9]. The dotted curve gives thecorresponding values found with eq. (12).
(rll
results were obtained with - = 1.25, e, = 11.0 and
01
the condition oZ '2 3 1.
In the same wedge-shaped cells we determined the period of the deformation wavelength by measuring the size of the square grids. The square of these values vs. p.d is shown in figure 5 for the K2l/CN mixture
FIG. 5. - The square of the wavelength of the square grid pertur- bations at threshold versus the product of thickness and actual pitch of the K21/CN mixture in the wedge-shaped cell. The circles and triangles are the values on the wide and narrow side respectively of the strips between the Grandjean-Cano lines. The solid line gives the values calculated with eq. (10). The dashed curve represents
the theoretical calculations made using Hurault's theory.
and in figure 6 for the a-cyano-stilbenelCN mixture.
The circles are the values on the wide side, the triangles are the values on the narrow side of the strips. The solid curves represent the calculated values using eq. (10). The dashed curves in the figures are the values predicted by Hurault's theory. The difference between the predictions of our theory (solid curves withp = po in Figs. 3 and 4 and solid curves in Figs. 5 and 6) and those of Hurault is due to the presence of an extra constant 112 in the square root term of eqs. (1
O),
(1 1 ) and (12).In view of the accuracy of f 10
%
in the measure- ments and the elastic constants of the liquid crystals used, there is good agreement between the experi- mental data and the calculated values for the liquid crystal with the positive dielectric anisotropy. The agreement for the liquid crystal with the negative dielectric anisotropy is somewhat less, which is pro- bably due to an attendant inaccuracy of 10%
in the conductivity.FIG. 6. - Dependence of the wavelength of the square grid pertur- bations at threshold on the product of the layer thickness and the actual pitch of the u-cyanostilbene/CN mixture in the wedge- shaped cell. The circles and triangles are the data on the wide and narrow side of the strips. The solid line gives the values calculated with eq. (10). The dashed curve represents the values found with
the aid of Hurault's theory.
4 . 2 PARALLEL SAMPLE CELL. - AS mentioned in section 1 and in detail in reference [I 11, an increasing electric field across the cholesteric planar texture of a liquid crystal with a relatively large negative A&
induces a new planar texture after the occurrence of the square grid pattern. This new planar texture has subsequently a new threshold for square grid perturbation which is relatively much higher. Further, after the occurrence of the square grid perturbation in this new planar texture a transition occurs to another new planar texture. This effect of the alternate occurrence of square grid pattern and planar textures can be observed several times in succession, but usually the planar textures become increasingly less stable. Finally the electrohydrodynamical turbulence is observed. We found that the occurrence of the new planar textures is accompanied by pitch contraction.
In a unidirectionally rubbed parallel sample cell with 6 helical turns across the constant layer thickness (called texture A) we found 7 and 8 helical turns in the successive planar textures, denoted B and C respectively.
Furthermore it is possible that new planar regions will again occur in the textures B and C. These planar regions, denoted B' and C' dislodge B and C in course of time. The number of helical turns in B' and C' appears to be 6 112 and 7 112 respectively.
In figure 4 the square of the threshold voltages in the original and induced planar textures of the a-cyanostilbenelCN mixture is plotted as a function of the number of helical turns (squares). The corres-
FIELD-INDUCED SQUARE GRID PERTURBATION IN PLANAR TEXTURE C3-283
ponding theoretical values calculated with eq. (12) are given by the dotted curve.
5. Conclusion. - Our study of the static distortions of cholesteric planar textures by external fields has resulted in a theory which predicts the threshold for square grid perturbation. This theory is able to describe the influence of pitch contraction and dila- tation on the threshold in contrast to the Helfrich- Hurault theory which appears to be reasonably valid in textures with the natural pitch only.
Taking account of the experimenta1'~a~uracy the measurements performed in the wedge-shaped sample cells show a good agreement with our extension of the Helfrich-Hurault theory.
The observed large increase of the threshold voltage in the newly induced planar textures of the liquid
crystal with the relatively large negative dielectric anisotropy can be predicted very well by our theory.
The size of the square grid pattern should be independent of contraction or dilatation of the pitch.
This result agrees with the experimental observations.
Appendix. - We assumed in section 2 that an external field applied across the cholesteric planar layer induces a sinusoidal deformation of the choleste- ric planes. For the twist angle cp in the perturbed layer we find :
cp = tz - bt sin qz cos kx
.
Now the following quantities can be expressed as a function of the coordinates x and z :
I/I
=- a~
= - bk sin qz sin kx = a sin qz sin kx ax(where a = - bk is the amplitude of
I/I)
- a* ax
= ak sin qz cos kx - -a'
- - a t sin qz sin kxax
sin cp = sin tz
+
at - cos tz sin qz cos kx+
a2.. .
k
c o s q = costz - -sintzsinqzcoskx + a 2 . . . at
.
k
Substitution of these quantities into eq. (6) results in :
F =
I
K2,{
t - to+
a($) cos qz cos kx+
a' t sin2 qz sin2 kx+
ak sin tz cos tz sin qz cos kx+
2
+
a2 t cos 2 tz sin2 qz cos2 kx K,, a 2 k2 cos4 tz sin2 qz cos2 kx - - 1 AXH2 a2 cos2 tz sin2 qz sin2 kx,
2
up to fourth order terms in a. The splay term has been disregarded on account of q
+
k. In order to obtain the equilibrium values of a and k we calculate the mean free energy density :- 1 1 27 1 1
F = - K 2 , 2 { ( t - to)2+Sa2($) + - a 4 t 2 + - a 2 k 2 + - a 2 t ( t - t o ) 128 32 2
3 1
+
- K 3 , a 2 k 2 64 - -AXa2 16 H'.Because of the symmetry of the problem this expression contains even powers of a and k only. If we want to calculate the equilibrium value of a we need all a4 terms, but, since we are preliminary interested in the question of whether a is different from - zero or not, we have retained only the a4 term in the KZ2 term, as being representa-
dl" -
tive of the others. Putting - = 0 gives : aa
Finally the condition a = 0 results in an expression for the threshold field H, (eq. (7)).
References
[I] GERRITSMA, C. J. and VAN ZANTEN, P., Phys. Lett. 37A (1971) [7] GERRITSMA, C. J. and VAN ZANTEN, P., Liquid Crystals and
47. Ordered Fluids 2, Edited b y J. F. Johnson and R. S. Por-
121 HELRICH. W.. ADD^. Phvs. Lett. 17 (1970) 531. , , ' z .. , ter (Plenum, New York) 1974, 437.
[3] HELFRICH, W., J. Chem. Phys. 55 (1971) 839.
[4] HURAULT, J. P., J. Chem. Phys. 59 (1973) 2068.
[8] RAULT, J., idem page 677.
[9] DE ZWART, M. and LATHOUWERS, Th. W., Phys. Lett. 55A (1975) 41.
[5] HERVET, H., HURAULT, J. P. and RONDELEZ, F., Phys. Rev. [lo] KAR~T, P P. and MADHUSUDANA, N. V., Mol. Cryst. Liq.
8A (1973) 3055. Cryst. 40 (1977) 239.
[6] SCHEFFER, T. J., Phys. Rev. Lett. 28 (1972) 593. [ l l ] DE ZWART, M., J. Physique 39 (1978) 423.
