Parallel and cross-like domains due to d.c. and low frequency (< 2 Hz) electric fields in nematic liquid crystal layers with negative dielectric anisotropy

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Parallel and cross-like domains due to d.c. and low frequency ( ² Hz)

electric fields in nematic liquid crystal layers ^with negative

dielectric anisotropy (*)

H. P. Hinov,

Institute of Solid State Physics, Bulgarian Academy ^of Sciences, ^Sofia 1113, Bulgaria and L. K. V istin,

Institute of Crystallography, Academy of Sciences of the USSR, ^Moscow 117437, USSR (Reçu ^{le Il octobre} 1977, ^{révisé le Il} septembre 1978, accepté ^{le 20 novembre} 1978)

Résumé.

On étudie les domaines parallèles ^et croisés obtenus par application ^de champs électriques ^continus

et basse fréquence ( ² Hz)

des cristaux liquides ^{MBBA et 440 à} anisotropie ^de susceptibilité diélectrique négative.

Les principaux ^{résultats sont :}

2014

la reproductibilité ^{des domaines} parallèles particulièrement pour des couches inclinées d’épaisseur ^5-90 03BCm

avec un

ancrage faible assuré _par

agent tensio-actif (savon) ;

2014

la création de domaines

croix dans des couches minces homéotropes ;

^domaines apparaissent ^{dans la} région ^de la cathode et sont dus

fort champ inhomogène générateur ^de structures flexion-éventail séparées par des disclinaisons.

Parmi les mécanismes électrostatiques ^et électrodynamiques connus, seul l’effet flexoélectrique ^{dû à}

gradient

de champ électrique peut expliquer la formation de

domaines.

Abstract.

Parallel and cross-like domains due to d.c. and low frequency ( ² Hz) ^electric fields, ^{in nematic} liquid crystal layers ^with negative dielectric anisotropy

^{MBBA and 440}

^obtained experimentally ^{and investi-} gated. ^{The basic} experimental ^results

the easy reproducibility ^{of the} parallel ^domains particularly in tilted LC

layers 5-90 03BCm with weak anchoring, synthesized

Schiff bases and when the

glasses

treated with

surfactant

soap; the creation of cross-like domains in thin homeotropic layers; the demonstration that these domains arise in the cathode region ^{due to the} strong inhomogeneous electric field which brings ^about

a

periodic bend-splay usually ^divided by disclinations.

Out of the known electrostatic and electrohydrodynamic mechanisms only the flexoelectric effect, ^{due to the} gradient ^electric field,

explain the initial formation of these domains.

Classification Physics ^Abstracts 61.30

Introduction.

Parallel, electrically stimulated,

domains aligned along the initial orientation of a

nematic liquid crystal ^{with a} negative ^dielectric anisotropy

^{domains of the second} type (DST) ^were

observed in PAA for the first time by ^{L. K. Vistin} [1].

As distinguished ^{from the R.} Williams’ domains [2]

which are in the perpendicular ^direction

^domains

of the first type (DFT) they exist under a specified

cut-off thickness of the LC layer (d ¹⁰ ym) ^and

below a specified ^cut-off frequency ^{of the} applied

electric field ( ¹⁰ Hz). ^{The fact that their} period

decreases with 1/E up ^to values of the electric field where a breakdown in the LC cell under investigation

takes place ^{is of} practical significance. ^{W. Greubel}

and U. Wolff [3] have observed similar domains in Merk IV. These investigations ^{have also been conti-}

nued by ^{S. Barret} [4] ^{in Merk IV. It has} been demons- trated experimentally by ^{Barret that in some cases}

DFT are formed at lower voltages ^{and DST are}

formed at higher voltages. ^{The LC cells} investigated by ^{Barret were} up ^to 50 ym in thickness.

Systematic experimental investigations ^{on these}

domains have also been made by ^J. Pollack, ^J. Flannery

and P. Watson [5], [6].

In all the experiments performed ^{the LC’s have}

been carefully purified ^{and were of low} conductivity

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004003026900

definite molecular 0 tilt and a weak _ç surface _energy of the LC molecules with the walls.

Back in 1970 the Orsay Group [ 13] reported ^observ- ing hexagonal domains for d.c. electric fields and the

importance of the surface interactions of the LC molecules with the walls was noted. This is a problem

which has been and is being considered by many

investigators (see ^for example ^{G. Elliot} [14], ^{S. Kai}

and K. Hirakawa [15], etc.).

Low voltage ^domains (3 V) ^{were obtained} by

G. Elliot and J. Gibson [16], ^S. Arora, ^J. Fergason

and A. Saupe [17], G. Heilmeier [18], ^{etc. The orien-}

tation of the domains described in these

papers ^was

not determined with respect ^to the initial orientation of the LC molecules.

Reproducible ^{d.c. and low} frequency parallel ^sur-

face induced flexoelectric domains (PSIFED) ⁱⁿ conducting NLC’s with a negative dielectric aniso- tropy ^are reported ^{in this} paper and the reason for their creation is clarified.

Similar domains were discovered in NLC’s with

a positive dielectric anisotropy. ^{It is} suggested ^{that in}

some cases the origin of such domains is the same

regardless ^{of the Ae} sign.

Sakagami et ^al. [19] have observed the creation of cross-like domains (CLD) ^{in thin} homeotropic ^MBBA layers (achieved by coating ^{a dilute} aqueous solution of sodium dodecyl ^{sulfate on} the electrode surfaces).

Such domains were also observed by G. Heimeier [18]

in butyl-p-anisylidene-p’-aminocinamate (BAAC) ^with

a positive dielectric anisotropy.

During ^the experiment the authors also observed cross-like domains in thin homeotropic ^MBBA layers

when the surface glasses ^were coated with a surfactant

-

common soap.

The authors postulate ^{that PSIFED are the one-}

dimensional analogue ^{of CLD.}

The experimental procedure and results are pre- sented in the first part ^{of the} paper. A simple ^theore-

tical model for explaining these domains is proposed

in the second part ^{of the} paper. The differences and the similarities between PSIFED and DFT, DST, chevrons, ^the volume flexoelectric domains of Yu.

and a periodic splay-bend in the cathode region always ^bounded by disclinations

CLD.

d) Above the threshold PSIFED are identical to PSIED.

e) As in the DST PSIFED and CLD exist below

a specified cut-off thickness of the LC layer ^investi- gated ^{and below a} specified ^cut-off frequency ^{of the} applied electric field « ² Hz).

The authors are of the opinion ^{that out of the}

known electrostatic and electrohydrodynamic ^mecha-

nisms the flexoelectric effect alone ^can explain ^the

initial formation of these domains.

An orienting (with respect ^to the dielectric aniso-

tropy) high frequency (20-200 kHz) electric field was

applied simultaneously with the d.c. or low frequency

2 Hz) voltage. ^An independent existence of these domains as well as Williams’ domains were observed at about 8 V. In some cases lattices are formed. At a

lower frequency ^{for the} additionally applied ^electric

field (20-100 Hz) ^{these domains were} replaced by ^a

Chevron mode [22].

A simple experiment ^{was used to determine the} dependence ^{of these domains on the} frequency ^{of the} applied electric field. Transformations PSIFED- Williams’ domains were observed.

1. Expérimental setup and results.

PREPA-

RATION OF THE CELLS.

The confining glasses ^{of the}

LC cells used in the investigation (0.1 ^{cm x 1.5 cm x 6} cm)

coated with transparent conductive layers ^of Sn02,

were cleaned by the standard method, rubbed with diamond paste (grain ^{size 0.5 and 1} gm) - ^deter- mining the easy direction of the LC molecules in the XO Y plane ^{and followed with a treatment by a}

surfactant

Na salt of the fatty ^acids (common soap). ^{The cell gap is} determined by teflon spacers ^{5 to} 120 jum thick.

The prepared ^{cells were} placed between metallic

plates ^{with central} openings

^a part ^{of a} spectroscopic

holder. The degree ^of compression between the glasses

and the teflon _spacers was regulated ^{with four screws.}

The NLC’s with a negative dielectric anisotropy

MBBA and 440 (a ^{mixture of} 2/3 p-n-butyl-p’- methyloxyazoxybenzene ^and 1/3 p-n-butyl-p’-hepta- noyloxybenzene) ^under investigation ^were initially quite pure with 10 -11 (n. cm - ’).

The orientation of the LC molecules was deter- mined by ^the simultaneous action of the rubbing,

soap deposition ^{and the direction} of the LC filling

flow coincident with that of the rubbing.

The cells with a 5 gm gap ^were carefully prepared

with two large ^teflon spacers by covering ^{the entire}

area of the cell and then moving ^{them in} opposite

directions with resistivity ^{control to} prevent ^{a short} circuit.

The thickness of the cells was measured as a dif- ference between the top ^and bottom, focusing ^{on the}

teflon edges ^for empty ^{cells and on the LC front} edge

for filled LC cells (see ^table II). ^The measurements

were performed with the aid of the UF-2 type microscope ^{with an} accuracy of + ^0.5 gm for thin cells and ± 2 gm for thick cells. Values 5.5-6.5 _gm

were obtained for filled cells. The value in the given

range depends ^{on the} degree ^of compression ^between glasses ^{and teflon} spacers.

1. 2 REMARKS. - 1) The thickness deviation in the

plane ^{of the} electrodes is of no great importance ^for

the creation of the domains investigated. 2) ^The soap

deposit quantity ^was controlled visually. During ^the experiment ^{we were not able to measure the thickness}

of the _soap deposition (this problem ^is going ^{to be}

solved with the aid of interferometric investigations).

But we can give ^some qualitative explanation ^{on this} point.

D. Berreman [23] has shown that the amplitude

of the surface _grooves is

200 A. When the thickness of _{the soap} deposition ^was smaller than this value the LC molecules are slightly ^inclined (an almost homeo-

tropic ^MBBA layer) ^and the PSIFED are formed under the d.c. exitation. When the thickness of _{the soap}

deposition ^is greater ^{than 200} A we observed CLD.

In this case the LC molecules were not sensitive to

microgrooves (homeotropic layer). 3) ^The conductivity

increased gradually ^{with the} operation ^{of the cells} (e.g. ^{10- 11 Q 10 - 9} (ÇI.cm)-’).

1.3 EXPERIMENT PROCEDURE.

d.c. or low fre- quency voltage ^was applied ^{to the LC cell thus}

constructed. A high frequency orienting electric field

was applied ⁱⁿ some cases as well. Both voltages ^were separated ^{with an RC} circuit. The observations were

performed with the aid of a polarization microscope (type ^UF 2) with transmitted polarized ^monochro-

matic or white light. ^{The nicols were} normally ^crossed

with a polarizer ^oriented along the direction of

rubbing ^{or rotated at an} angle 9 - ^when lp-defor-

mations are being investigated. ^When investigating

deformations along ^{0 the} easy axis was placed ^at

an angle ⁴⁵⁰ between the crossed polarizer ^{and the} analyser - ^{see for} example ^R. Chang [24] ^and

R. Soreff and M. Rafuse [25].

The greater part ^{of the} investigations ^were per- formed at room temperature. Observations were

performed ⁱⁿ some cases throughout ^the temperature range (see ^for example ^the experimental results of L. Vistin et al. [26]).

Due to the nature of the confining glasses ^used (weak anchoring, ^tilted layers) ^{PSIFED were observed}

up ^to 30 _gm without using soap ^{as a} surfactant as well.

Domains _up to 90 _gm for MBBA and _up to 60 _ym for the LC 440 were formed in cells treated with _soap.

In cells constructed with glasses displaying ^strong anchoring ^{these domains exist} only when the surfaces

are treated with _soap.

Soap deposition ^{was realized} using ^a soap ^water solution with the glasses dried afterwards. It has been _proven experimentally ^{that the} orientation of the nematic layer is tilted with two states of tilt (inclined

at an angle ç) existing (Fig. la, b, c). ^{The first} (the

darker on the figure 1 a) ^{was more} stable and appeared

for larger quantities ^{of the} deposited soap molecules.

It was close to a homeotropic ^nematic layer - 50; ;3 for some cases the angle ^{reached 0°} (a homeotropic layer). ^The second tilt was smaller. It was obtained for

a smaller quantity ^of deposited soap and was consi-

derably ^more homogeneous. Usually ^{after a certain} period ^{of time} (minutes ^or hours) ^{the LC} layer

becomes homeotropically ^{nematic or a more stable}

tilt is established. One could _go from the more stable tilt to the less stable one by ^a short time application

of high voltage creating dynamic scattering ^{in the LC.}

Regions ^{with a} various tilt divided by ^{0 and} ç discli- nations were formed as well (see ^{G. Porte} [27], [28],

W. Crossland [29], etc.).

The application of weak d.c. voltages (3-6 V) bringing ^{about a domain structure} also stabilizes the less stable tilted orientation of the LC molecules. In

rare cases an almost planar layer ^was obtained with the aid of _soap treated glasses. ^{A check} of the Chevron

regime demonstrated that the soap coating ^{does not}

contaminate the LC and has no influence on its

properties. ^It became clear that the quantity ^of soap molecules introduced is of great importance ^{for the}

behaviour of the LC layer. ^This problem ^remains

unresolved for the moment and further investigation

is _necessary.

The thickness of the disclinations obtained and the threshold of the Freedericks transition demonstrated the weak anchoring ^{of the LC} molecules with the walls.

The 0 anchoring energy ^was determined from the Freedericks transition. The results :

2 x 10-4-5 ^x 10-4 erg/cm2

are in agreement ^{with the} latest results of

G. Ryschenkow ^{and M. Kléman} [30], [31]. ^The

anchoring energy ^was determined from the thickness

of the disclinations observed in this experiment :

Fig. ^1.

a) ^{Two states} of tilt inclined at

angle (p in

MBBA

layer ⁴⁵ pm thick ; initially ^crossed nicols ; ^the polarizer is rotated

along ^the projection ^{of the} LC molecules with the smaller tilt in the XOY plane. Magnification ^12.5

^10. b) ^With applied voltage

U

10 V; ^{10 kHz.} c) ^With applied voltage ^U

¹⁰⁰ V ; 10 kHz.

3

10-4 erg/CM2 in accordance with R. Meyer [32].

It is clear that soap deposition determines a conical weak alignment of the LC molecules [33] ^and rubbing

alone determines the difference between the 0 and ç anchoring. ^The strong ^thermal fluctuations, usually ignored ^{for a fixed surface} angle (rigid coupling)

Fig. ^2.

^A threshold of parallel domains in

MBBA layer 45 pm

thick ; initially ^crossed nicols ; ^the polarizer is rotated at

angle jp ;

magnification ^12.5

^25.

Fig. ^3.

a) Cross-like domains in

thin homeotropic ^MBBA layer 15 pm thick ; crossed nicols (the polarizer ^is along ^the long

side of the photo) ; applied voltage

^{6 V (the} sign indicates the top position ^{of the} cathode). b) Cross-like and parallel domains in

MBBA layer ^{30 pm} thick ; ^crossed nicols ; applied voltage ^{+ 4.5} V ;

magnification ^12.5

^12.5.

When a d.c. voltage ^{of the order of 2-3 V was} applied alternating ^{dark and} light ^{bands were formed} (see ^also Hinov, Vistin and Magakova [34]), ^oriented

in the _easy direction of the molecules OX (Fig. 2)

when the layer is tilted (the polarizer ^is partially

rotated at an angle ⁺ ç ; if the polarizer is rotated at

an angle - ç the bands change places), ^cross-like

domains when the layer is almost homeotropic (Fig. 3a, b) and intermediate domains with different

periodicity along X ^{and Y} (Fig. 4) (crossed nicols).

These bands are due to the alternating splay-bend ^of

the director n in the Y direction out of the plane ^XZ

determined by ^the easy axis OX and the direction of the

applied ^{electric field OZ. A} further increase of the

voltage brings about the establishment of parallel

domains of long ^term stability, ^oriented everywhere ⁱⁿ

the _easy direction (Figs. 4 ; 5a, b, c ; 6a, b). ^{It is evident}

from the figures ^{that these domains are} homogeneous

for a homogeneous ^{tilt of the LC molecules at the}

surfaces. When the thickness of the LC cells investi-

gated ^{was small} (below ¹⁰ gm) the domains formed

were observed after irradiation with white polarizing light in the field of a microscope ^as alternating ^red

and _green bands almost equal in width. These obser- vations as well as those of thicker cells demonstrated

Fig. ^4.

Intermediate domains with different periodicity along X

and Y in

MBBA layer 45 gm thick ; crossed nicols ; applied voltage ^{+ 6 V ;} magnification ^12.5

^25.

that these domains have a double period. ^The periodi- city of CLD is almost equal ^to that of PSIFED. This fact demonstrates that these domains follow the same

objective physical ^{laws. The} magnitude of the initial tilt along ^{0 is of no} great importance for the creation of the parallel domains and for this ^reason for thin

Fig. ^5.

Parallel domains

oriented everywhere ^{in the} easy direction : a) ^MBBA layer ³⁰ gm thick ; ^{two states} of tilt ; initially ^crossed nicols ; ^the polarizer ^{is rotated at}

angle cp; magnification ^12.5

^10. b) ^With applied voltage

^{3 V} (top focusing). c) ^With applied

voltage

^{4 V} (top focusing). d) ^With applied voltage

^{6 V} (top focusing).

Fig. ^6.

a) Cross-like and parallel domains in

MBBA layer

30 _gm thick ; ^crossed nicols ; applied voltage - 4 V ; magnification

12.5

10. b) ^With applied voltage - ^{6 V.}

cells with different tilt at the threshold (around ⁴ V)

the splay-bend deformation along ^{O Y is} accompanied by ^a multicoloured picture ^which disappears ^{with the}

increase in voltage. ^The observation of the below threshold regime in crossed nicols with _{the easy} axis at 450 demonstrated a threshold deformation along ⁰

which includes first of all dielectric moments. A constant 0 along ^Y (in many cases) ^{was observed when}

the domain picture ^{was formed as well as for a} higher voltage ^{4-5 V} except around the disclinations and the focal lines (see figure ^{7a, b}

^the discrepancy ^{of the}

band is due _{to 9} deformations). The simultaneous

application ^{of a} deforming d.c. electric field and an

orienting high frequency ^electric field with a frequency

above the _space charge relaxation frequency clearly

confirmed the bulk nature of the threshold and

distinguished ^the deformations in the XOZ plane (the angle 0) and in the XOY plane (the angle cp) (Fig. 8a-d, Fig. la-c). ^{For a} large orienting high frequency voltage (up ^to 170 V) ^the deformation almost disappears ^{due to elastic} anchoring ^{of the}

director at the walls. For CLD and PSIFED the

Fig. ^7.

a) 6 deformation remaining ^{constant in}

^MBBA layer

90 _J.1m thick ; crossed nicols (the polarizer ^is along ^the long ^{side of}

the photo) ; applied voltage - 6 V ; top focusing ; magnification

12.5

12.5. b) Focusing ^down.

deformation is entirely eliminated by ^a relatively

weak high frequency electric field (10-20 V) ^{for a} large

tilt 00 which demonstrated that the deviation of the molecules from the XOZ plane ^{is small} (the angle 9 is small) and when the molecules take to the XO Y

plane they go easily ^{into the} position ^{of minimum}

elastic _energy in the _easy direction which is in _agree- ment with the theoretical and experimental ^results

of D. Berreman [23], [35] ^{and W.} Greubel et al. [36].

The larger deviation from the XOZ plane requires

a larger high frequency voltage ^{due to the} simultaneous action of the elastic surface _energy and the considerable dielectric orienting torque ^{of the} applied voltage.

On the other hand the CLD cannot be related to the walls combined with the disclinations ^since they ^are

obtained only ^{when a} high frequency electric field is applied

^{the well} known Schlieren texture (see

the works of J. Nehring ^{and A.} Saupe [37],

A. Saupe [38] ^{and A.} Rapini [39], [40]). The observa- tions made with a polarizer alone showed that the _ç deformation ^is much smaller and only focal lines

were seen in the view field of the microscope, dividing

each half of the domain. Hexagonal ^{domains were}

observed _very often depending ^{on the state of the} boundary ^surface.

An increase of the d.c. voltage ^{to about 8 V} brings

about the independent existence of Williams’ domains in the normal direction OY as well (Figs. ^{8 ;} 9a-f ; 10a, b). The increase of this threshold by ^{2-3 V is} explained by ^the difficult conditions under which

they ^{are formed} (the existence of another domain

mode). Williams’ domains were not observed in thin cells « ¹⁰ ym) by the authors. A careful observation demonstrated that the focal lines of Williams’ domains

are at a planar angle lp. The application ^{of a} high frequency electric field as well, ^raises the Williams’

domains threshold and widens the voltage range where

parallel ^domains (PSIFED) ^{exist. In} some cases this

gives way ^to the herring ^bone texture, arising ^{from the}

simultaneous action of the orienting dielectric torques in XOZ plane ^{and the} deforming bulk flexoelectric torques (Fig. 11). (Similar optical ^{texture was obtained} by ^{I. Rault and P. E.} Cladis in cholesterics [41].) ^A

new hydrodynamics predominantly ^{in the XO Y} plane appears for large values of both voltages (Figs. 12, 13).

Figures ^{12 and 8} demonstrate that the periodicity

of PSIFED is determined by regularly ^{oriented discli-}

nations along ^the easy direction which as all evidence demonstrates are formed around the threshold where the _ç deformation is formed as well (see Fig. 2).

Similar results are demonstrated by figure ^{14 where}

domains are seen under relatively unsteady ^and inhomogeneous boundary conditions. These disclina- tions are clearly visible for the LC 440 (Fig. ^15a, b)

and as will be demonstrated they play ^a considerable part ^{in the} dynamic scattering

^{an idea} developed by ^{De Gennes} [42] ^and experimentally ^observed by

J. Nehring ^{and M.} Petty [43] ^and by ^M. Bertolotti’s

Group [44]. ^It is difficult to observe the disclinations without the high frequency electric field due to their

overlapping with the focal lines brought ^about by ^{the Z} deformation, ^{also due to the} hydrodynamics (compare Figs. 8a-d). Establishment of this fact is more difficult for thicker cells (&#x3E; ⁵⁰ ym) (Figs. ^{7 and} 9). ^{On the}

other hand a comparison ^between figures ^{11 and 8}

shows that the disclinations _appear where there is an

interrupted change ^of (p - the replacement ^of ç with

-

(p. The stable division of the domains with discli- nations demonstrates the hydrodynamic ^{behaviour of}

Fig. ^8.

^The applied simultaneous high frequency electric field distinguished ^the deformation in the XO Y plane (the cp angle) ^{and in the}

XOZ plane (the ⁰ angle). a) ^MBBA layer 45 gm thick ; initially ^crossed nicols ; ^the polarizer is rotated at

angle cp ; applied voltage ^{+ 5} V ; magnification ^12.5

^10. b) Applied voltages ^{+ 5 V and 20 V 10} kHz ; top focusing. c) Applied voltages ^{+ 5 V and 20 V 10 kHz ;} focusing

down. d) Applied voltages ^{+ 5 V and 100 V 10 kHz.}

Fig. ^9. - a) Parallel domains in

MBBA layer 45 um thick ; ^crossed nicols ; applied voltage ^{+ 5 V ;} top focusing ; magnification ^12.5

^10.

b) Focusing ^down. c) Parallel domains and Williams’ domains ; applied voltage ^{+ 8 V.} d) Parallel domains in

MBBA layer ⁹⁰ nm thick ; applied voltage - 6 V ; focusing down ; magnification ^12.5

^12.5. e) Parallel domains and Williams’ domains ; applied voltage - ^{10 V.}

f ) Parallel domains and Williams’ domains ; applied voltage - ^{10 V - after}

few seconds.

Fig. 10.

a) Everywhere parallel ^domains

^{in the} easy direction ; ^MBBA layer ³⁰ gm thick ; crossed nicols ; applied voltage - 5 V ; focusing down ; magnification ¹²

^10. b) With the Williams’ domains ; applied voltage - ^{8 V.}

Fig. ^11.

^A herring bone texture, arising from the simultaneous action of the orienting dielectric torques ^{in the XOZ} plane ^{and the}

bulk flexoelectric torques ; ^MBBA layer 45 um thick ; ^crossed nicols ; applied voltages ^{+ 14 V and 17 V 10} kHz ; magnification ^12.5

^10.

Fig. ^12.

^New hydrodynamics, predominantly ^{in the XO 1’} pLnc ,

MBBA layer 45 J.1m thick ; crossed nicols ; applied voltages + 15 V and 90 V 10 kHz ; magnification ^12.5

^25.

Fig. ^13.

Hydrodynamics ^with

chaotic fluid and disclination motion; ^MBBA layer ³⁰ gm thick; applied voltages ^{+ 18 V}

and 40 V 200 kHz; crossed nicols; magnification ^12.5

^25.

Fig. ^14.

Domains under unsteady ^and nonhomogeneous ^boun- dary conditions; ^MBBA layer 60 pm thick; applied voltages ^{12 V}

and 20 V 10 kHz; ^crossed nicols; magnification ^12.5

^12.5.

polarizer is rotated at

angle ç ; top focusing ; magnification ^12.5

^12.5. b) Parallel domains in LC 440 50 _gm thick ; applied voltage

-

6 V ; crossed nicols ; top focusing; magnification ^12.5

^12.5.

the LC as well, particularly for slower laminar flow.

The question of domains bounded by disclinations’

was considered experimentally by ^{J. Nemec and}

B. Cook [45] (a ^resume only ^is given) ^{and theo-} retically by ^R. Meyer ^{and P. Pershan} [46] ^and

P. G. De Gennes [47], [48].

Depending ^on the initial state of the LC and the thickness of the LC cell investigated ^{a more} complex

deformation is possible ^as evident from figures ^{7 and 9}

where determination of the disclinations is _very difficult. All evidence demonstrates that besides the

boundary conditions the formation dynamics ^{of these}

domains is of importance ^as well in this case. In agreement ^{with the} theory ^{of A. Penz} [49] ^{for a fast} voltage ^{rate the} faster Williams’ mode is established and PSIFED are formed after a longer ^time.

In the CLD the deformations are divided by disclinations as well and these domains exist in a

homeotropic ^matrix.

The application ^{of a} high frequency electric field increases the threshold where the periodic parallel

domains appear without changing ^their period ^which

shows that it is determined by purely surface aniso-

tropic (in ^{this case} electric) properties. ^{The existence}

of the system ^of disclinations also supports ^this thesis.

For larger voltages ^and dynamic scattering ^the

system of disclinations is bent along X ^{and there}

appears ^a zig-zag shape ^{similar to} that observed by

A. Petrov and B. Markovski [7] ^{and later} by ^{H. Hinov}

and A. Derzhanski [8]. ^The dynamic scattering ^{can be}

followed with particular ^{success in the LC 440 where}

d.c. and low frequency voltages ^are applied ^simulta- neously. ^{Due to} the presence of stable disclinations for dynamic scattering ^{the LC} is divided into regular

cells carrying ^{out a} fast oriented rocking ^motion.

This is not so obvious for MBBA, ^{the fluid motion} is chaotic (Fig. 13) ^{and the} disclinations move about

in a chaotic manner. There is no doubt that the different hydrodynamic behaviour of these LC’s is determined by ^{their nature}

^{the various} viscosity, elasticity, ^{etc. values.}

It has been unambiguously demonstrated in the process of the experiments ^that these domains arise in the cathode region and that their threshold is determined by ^a certain threshold value of the electric field in the cathode region. ^The first fact was established

by ^the different behaviour of the LC cells with a

clean cathode and with a cathode treated with _soap

(introduced ^cell asymmetry) ^{with the} simultaneous

application of d.c. and high frequency voltage ^and

with the optical picture apparently ^{the same} (the

anode was treated with soap). Domain modes much stronger in deformation (cancelled ^{with a conside-} rably larger high frequency voltage) ^were established in cells where the cathode was clean (see ^table I)

Table I (MBBA)

The _soap deposited ^on the cathode caused a tilted surface orientation. The domains were then formed at a large ^{tilt and were} erased with a weak a.c. electric field. On the other hand the cathode was clean the molecules in the surface layer ^{were tilted} slightly ^and

the domains were erased with a strong ^{a.c. electric} field. At the same time for _very thin cells ( ⁵ ym)

in a strong d.c. electric field the authors have observed

the formation of domains in the cathode region,

while in the anode region only ^the dynamic scattering

mode exists. This fact can also be established from

figures ^{8a-d. It} is clear that the disclinations and the _(p deformations (typical ^{for these} domains) ^{are seen}

better by focusing down in the LC layer ^{with the} microscope (the top plate being ^the anode).

The second fact was established at the time of appearance of these domains. It is strongly dependent

on the thickness of the LC cell under investigation,

also on the electric field strength. ^{For thin} cells, they

appear for parts ^{of a} second and for thicker cells

(90 gm) - for minutes. This time is well in agreement with the formula derived for the appearance of Felici’s instabilities by ^J. Lacroix and R. Tobazéon [50]

and by ^D. Meyerhofer ^and A. Sussman [51] :

wher Vc is the threshold voltage, k ^{is the ion} mobility

and E is a mean value of the applied electric field.

On the other hand it is strongly dependent ^{on the}

current prehistory of the LC cell investigated. ^Such

domains do not appear immediately ^{when the cell}

is freshly ^{filled with} pure LC (conductivity

lO-11 (ÇI. cm) - ’). ^A long waiting ^{time is} required ^for

their appearance. These domains are obtained in the easiest manner after a switching ^the LC cell from

dynamic scattering ^{to a} domain mode which leads ^to the presence of _many ions and therefore to a strong electric field gradient.

As noted the cut-off thickness of the LC cell _up to which these domains are observed was increased to 90 _gm with the aid of _soap as surfactant. A test

experiment ^was performed ^{with a 100} gm cell where

one side was ion doped. ^{No domains at all were}

formed in the part without ion doping.

An experimental measurement of the dependence

of the period of these domains on the thickness of the LC cell investigated ^{for the same} boundary ^conditions (the ^same glasses) ^{was carried out} in three cells

2 with MBBA and one with the liquid crystal ^{440. The}

results obtained are given ^{in table II} (see ^also Fig. 16) :

Fig. ^16.

^The determination of Ulh ^{and the} extrapolation length K 11/ WSfP for MBBA from the extrapolation ^{of the} experimental

curves

Ulà(d) and E ^{d (d ).}

It was experimentally verified that for frequencies higher ^{than 0.06 Hz} (for ^LC 440, d

¹⁰ )Lim) disap-

pearance of the domains starts with a rocking ^motion

,in the XOY plane ^{and for} frequencies ^{above 2 Hz} only ^{Williams’ domains are observed. A} change ^of polarity ^{in the} CLD leads to the formation of Williams’ domains and again ^{to CLD} (in ^{the formation}

of the gradient ^electric field).

A special ^{circuit was} designed ^{for the} experimental investigation ^{of the} dependence of these domains in 440

on the value of the applied ^d.c. voltage. ^{Here the}

Table ^Il

that rubbing ^{with diamond} paste ^{does not} change ^the

surface properties ^of glass except ^for giving ^a preferred

direction. Parallel domains, ^not oriented in one

direction in the XOY plane ^{however, were formed for} polished glasses. Their orientation in the given region (of the order of hundreds of microns) ^is probably

determined by ^{local causes. A} covering ^{of the} glass

with SiO as a blocking cover removes the appearance of these domains (it ^{has to be taken into account that}

SiO changes considerably ^{the surface} properties ^and

therefore the surface _energy of the LC molecules). ^The

inclusion of _soap, as already ^noted,, decreases consi-

derably ^{the 0} energy of the molecules and what is of great importance ^{in this case}

^the 9 energy ^as well, creating ^a possibility ^{for the} development ^{of a} ç deformation. The increase of the quantity ^of soap molecules (soap layer thickness 1 000 À) ^decreases

this _energy even more and eliminates the periodicity

of the domains and the arising deformations _express themselves as wide disclinations and walls or defor- mations freely moving ^{in the fluid} [30], [31]. ^Thermal

fluctuations are increasing ^{and the LC flows} freely

between the limiting glasses. Naturally under such

boundary conditions the whole electrohydrodynamic

behaviour of the liquid crystal changes. ^{A further}

increase of the _soap (soap layer ^thickness &#x3E; 5 000 Á)

increases the anchoring ^{of the} liquid crystal molecules, eliminates the conditions under which these domains exist and also changes ^{the whole} electrohydrodynamic

behaviour of the liquid crystal.

Breakdown occurs easily ^for higher voltages ^{in LC}

cells whose glasses ^are soap treated. This fact demonstrates that _soap is directly linked with cathode carrier injection (see ^{for example P. Atten [52]).}

Small size impurities ^{in the} liquid crystal ^{do not}

exhibit _any noticeable and orienting motion around the threshold (2-3 V). (One should be reminded at this

point that the Felici’s instabilities [21], [50] ^{as well}

as the hydrodynamic ^{domains of the Williams’ type}

start with a finite fluid velocity of motion (see

W. Helfrich [53]). ^{Around 4 V the} motion of dust

particles ^is regular, ^{similar to} Williams’ domains however, ⁱⁿ planes ^{normal to the XZ} plane. ^As

demonstrates their dependence ^{of the} degree ^of

elastic anchoring ^{of the LC} with the cell walls. The

following experimental ^{fact was obtained} indirectly :

after switching ^{over from} dynamic scattering ^{to a}

domain regime (more ions, ^a larger ^field gradient)

a 20 % decrease of the period of the domains was

observed.

In thin cells, 5 gm where the thickness error is

larger, ^a considerable change ^{in the} domain width

was observed in the XO Y plane.

A short _summary of the results presented ^above

follows : cross-like (CLD) ^and parallel (PSIFED) domains, ^divided by disclinations demonstrating ^bulk

deformation static around the threshold, hydrodyna-

mic of the Pikin’s type [11], [12] ^{at 4-8 V and} existing simultaneously with Williams’ domains above 8 V

were observed in d.c. and low frequency electric fields for weakly anchored tilted layers along ^{0 and} ç.

These properties demonstrate that the domain existence is determined by ^the anisotropy ^{of the} boundary ^{surface at the cathode} (in ^{an electric} field).

In accordance with the theory ^{of R.} Meyer ^and

P. Pershan [46] ^{and R.} Meyer [54], the rise of similar domains bounded by disclinations in nematics near

N-Sm A phase transition obtained by ^{T. Scheffer}

H. Grüler and G. Meier [55], ^C. Gebhardt et al. [56]

and by ^R. Meyer ^{and N. Clark} [57] ^{in Sm A, the}

above domains can be related to the _presence of flexoelectric properties ^{of the} liquid crystal, ^introduced by ^R. Meyer [58] ^and developed by ^{J. Prost and}

J. Marcerou [59].

2. Theory.

^2.1 PRELIMINARY REMARKS.

Do- mains divided by disclinations have also been observed in ChLC by many researchers. A. Saupe [38] ^remarked

that regular disclinations + 1/2 ^{and -} 1/2 ^{have been} generated ^{as a} transition between homeotropic boundary conditions and planar cholesteric texture.

J. Rault [60] ^also pointed ^out that several kinds of

periodic patterns ^divided by disclinations _appear

in ChLC’s submitted ^to an external field. P. Cladis

and M. Kléman [61] observed characteristic striped

domains in a large pitch cholesteric held between

parallel glass plates. They explained ^{these domains} theoretically with the aid of surface disclination, separating ^the regular deformations. These result

were also confirmed later by ^T. Akahane et al. [62]

and by ^{M. Kawachi et al.} [63].

In the observed bubble domains [64]-[71] ^{as a}

natural extension of Cladis and Kléman’s model

(ring ^{of a} single striped domain) there exist discli- nations as well.

Solving ^the problem ^{of a PSIFED we will use the}

ideas presented in these works as well the results from P. G. De Gennes [47], [48] ^{and R.} Meyer ^and

P. Pershan [46], [54].

In this section of the paper only ^the theoretical

explanation ^{for the} creation of PSIFED is treated.

This theory, however, ^{can be extended to the cross-} like domains as well.

The electric enthalpy given by ^R. Meyer [58] ^{is of}

the known form :

The following assumptions ^can be made in accor-

dance with the experiment performed :

a) ^{There is a} strong electric field gradient ^{in the}

cathode region ⁱⁿ agreement with the results of S. Lu and D. Jones [72], ^G. Assouline et al. [73] ^{and one}

of the authors (H. ^P. H.) [74].

b) ^The coupling ^{of the LC} layers ^{with the walls is}

very weak. The estimate of the surface _energy

(1/2) W. ^{sin2 0} sin2 from the thickness of the observed disclinations in MBBA layers (Fig. 8d)

(see ^R. Meyer [32], [75], ^{N. Gilra} [76], ^{V. Vitek and}

M. Kléman [77] gives roughly ²

^10-4 erg/cm2).

This _energy is really smaller in accordance with the results from J. Sicart [78]. ^An estimate of the surface energy (1 /2) Ws8 ^{sin’ 0} from the Fredeericks threshold of a homeotropic ^nematic layer gives

c) Around the threshold the 0 deformations along

Y are negligible ^{due to} the surface _energy type.

In thick cells where ôlêz :0 ⁰ there exists a coupling

between the bend and splay flexoelectric polarizations

and the applied electric field. The case is complicated by dielectric deformations as well. In thinner cells the

approximation ôfôz - ^{0 is} increasingly valid. When the thickness of the cell is in _{the range} of the extrapo- lation length ^b (introduced by ^De Gennes) the l{J ^or 0 deformations _may exist only in the X and Y directions

(planar deformations).

As pointed ^out by the authors the dielectric aniso- tropy ^{of the} investigated LC’s is of no importance ^for

the formation of PSIFED especially ^{in cases where the}

electrodes were treated with _soap. Therefore in the

expression ^{of the electric} enthalpy ^the dielectric terms were omitted. This makes a simple calculation possible

of the deformations separated by regular array of

disclinations ^{at the surfaces.}

The disclinations observed in the LC layers ^are usually regular ^without singular points ^{on the line.}

The authors observed junctions with other discli- nations only ^{in the} glass defect. The line disclination

can sometimes detached from the glass plate ^and following ^R. Meyer [75] they have been found at a

distance greater ^than dis

^{0.3. In this case the}

authors did not observe the parallel domains due to the larger energy of these disclinations. In DS mode the disclinations were usually repelled by the surfaces

getting ^a zig-zag ^form. The energy of the surface disclinations is _very low (see ^{C. Williams, V. Vitek}

and M. Kléman [79]) ^and they ^{are not related to}

localized defects from the rubbing. ^{In the authors} opinion ^these disclinations (at ^the threshold) ^{are of} strength ^r

Açf2 n ^{and attract} with their images

-

the complementary disclinations of strength Ic

defined by ¹ + Ic

S, where S is an integer ^or half-integer. ^The theory of these disclinations was

given by ^{V. Vitek and M. Kléman} [77]. Consequently

these disclinations have ^a continuous core with a sur-

face _energy ^gy E, = 7r 2 K 2 2

₂ ^{spread p on the surface.}

V. Vitek and M. Kléman found that the energy barrier for detachment of a disclination when 1 is neither an integer ^nor half-integer is infinite _e.g. that such a line cannot go into the bulk except by ^nucleation

of other singularities (e.g. singular points). ^These

authors calculated also an energy of some surface disclinations especially ^{when 1 +} Ic

n/2.

We prefer ^at higher voltages ^a system ^{of surface}

disclinations-images calculated by ^R. Meyer [75]

and by ^{V. Vitek and M. Kléman} [77] :

where

where

and P ^{2 is an unknown} parameter.

disclination, ^{as well as its} image, ^{enter in the} expression

of the _energy of interaction between the wall and the

disclination.)

With these simplifications ^and having ^{the above}

in mind we can write for the disclinations the enthalpy

per cm’ for the splay ^elastic energy arising ^{at the}

formation of the domains alone :

where t is the depth ^of integration along ^z, E, ^sin (Jo,

cos 00 ^{are the mean values of the} electric field and the tilt in this interval, p is the period ^{of the domains,} WSlp ^sin’ 00 ^{sin’ 9 is a surface energy} of tilted bulk

layers along lfJ [83], K 11 ^and eiz ^are the splay ^elastic

coefficient and splay flexocoefficient respectively.

In (7) ^the disclinations separate ^the regions ^{with the}

next deformation :

The reversal of the sign ^{of the} applied electric field

or of the flexoelectric coefficient e lz requires ^a change

in the sign ^{of the} initial tilt 00.

This problem ^here is solved in two ways :

a) ^The disclinations participate only ^{in the energy}

balance and consequently ^{in the} determination of the

period (the ^usual procedure of several authors [46], [48], [54], [61]).

b) The disclinations also participate in the boun-

dary conditions (when ^the disclination _energy ^{is a} known function).

The calculations showed that only ^{way a) is correct.}

The disclinations _appear in the interrupted 9 regions taking part only ^{in the} energy balance.

The minimization of (7) ^{leads to the} following

differential equation :

and to the following boundary conditions :

The next result for the deformation was obtained around the threshold from (8) ^and (9) :

and for the integration ^{constant :}

Finally substituting (10) ^and (11) ⁱⁿ (7) ^{and mini-} mizing the latter with respect ^{to the} period p of the domains we obtained the following transcendental

equation (similar ^{to that obtained} by ^{De Gennes} [48] ⁱⁿ

the similar surface problem) :

where K,,IW,, ^{is the} extrapolation length ^introduced by ^{De Gennes}

The equation (12) determined the period ^{of the}

domains :

and the threshold :

The values of plt (if K111 WSQ&#x3E; f"O.oI ^{t) and} Uth ^were

obtained from the extrapolation ^{of the} experimental

curves (see Fig. 16) ^{for MBBA. The next} range of the flexoelectric coefficient _elz was calculated from (14) :

or

The following extrapolation length

(see Fig. 16) is in accordance with the thickness of the disclinations for thinner cells.

The periodicity ^{of the} domains is determined by ^the type of the surface energies : along Y, parallel (PSIFED) domains; along X ^and Y, ^cross-like domains (CLD) ^and intermediate domains with different periods along X ^{and Y.}

The flexoelectric torques ^are strongest ^when

Oo - 450. On the other hand the saturation value of the electric double layer ^after A. Sussman [84]

is a few volts. Consequently only ^weak anchoring along ⁶ (A. Rapini ^{and M.} Papoular [85], ^{H. P. Hinov} [86], ^J. Nehring, ^A. Kmetz and T. Scheffer [87])

may ^ensure such angles (in ^the experiment performed

the 6 deformations start at 1.5 V).

In thick cells (al oz =fi 0) ^the splay deformation is

acompanied by twist and bend deformations. This fact leads to a larger period (smaller elastic deforma- tion energy). ^{At a} certain thickness the negative

flexoelectric deformation _energy and the positive

elastic deformation _energy ^are equal ; ^{above this}

thickness the domains disappear.

The formula (13) shows the importance ^{of the}

thickness of the electric double layer. ^{After G.} Sprokel [88] ^{the double} layer width is of the order of 500 À,

after A. Sussman [84], ^0.1 gm. This question is difficult and depends ^on conductivity coating ^{of the} glass,

treatment of the glass, ^on boundary conditions etc.

As noted a full investigation of the surface discli- nations has been performed in the latest M. Kléman’s and V. Vitek’s [77 ^and 79] papers. These authors found theoretically ^the influence of the disclination

on the bulk orientation of the liquid crystal ^molecules.

This was confirmed by ^our experiment ^{with the}

formation of _very stable walls (W. ^Helfrich [89],

M. Kléman and C. Williams [90], ^A. Stieb, ^{G. Baur} and G. Meier [91], etc.) and demonstrated by ^the electrohydrodynamic ^{behaviour of the} adjacent

domains (especially ^{in the LC} 440).

The model developed ^for parallel ^domains sepa- rated by disclinations (PSIFED) confirmed the next very important experimental ^{facts :}

1) ^The period of the domains cannot change ^with

the external forces on account of attractive forces

existing between the disclinations and the wall and

repulsive ^forces existing between the disclinations which also stabilize the domain system.

2) ^The voltage ^{threshold obtained} (the appearance of the domains bounded by disclinations) may change

with the influence of external forces ^or with change ^of

the tilt 6o deformation. This is confirmed by ^the experiment.

3) ^The disappearance ^{of the} domains in thicker cells is due to disclinations whose _energy becomes

large ^{and the total} deformation _energy is positive.

(Here ^{we must mention that in this} energy balance an

important part ^is played ^also by the surface energies,

related to the important parameters Ws8 ^d, WSQJ d

where d is the thickness of the cell, ^which may become

larger ^{than the} flexoelectric ones.)

4) The appearance of these domains depends strongly ^{on the} formation of ^a double electric layer

extended in the bulk.

This theory ^cannot explain ^the domains observed

further in MBBA without disclinations (Figs. 17a-d)