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ELECTROHYDRODYNAMIC INSTABILITY IN PLANAR, POSITIVE DIELECTRIC ANISOTROPY
NEMATIC LAYERS AT D.C. EXCITATION
A. Petrov
To cite this version:
A. Petrov. ELECTROHYDRODYNAMIC INSTABILITY IN PLANAR, POSITIVE DIELECTRIC ANISOTROPY NEMATIC LAYERS AT D.C. EXCITATION. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-310-C3-313. �10.1051/jphyscol:1979359�. �jpa-00218755�
JOURNAL DE PHY SlQUE Colloque C3, supplkment au no 4, Tome 40, Avril 1979, page C3-310
ELECTROHYDRODYNAMIC INSTABILITY IN PLANAR, POSITIVE DIELECTRIC ANISOTROPY NEMATIC LAYERS
AT D.C. EXCITATION
A. G. PETROV
Institute of Solid State Physics, Bulgarian Academy of Sciences, Sofia 1 1 13, Bulgaria
RCsumC. - On a dtveloppe une gtnCralisation du modkle de de Gennes pour l'instabilitt a condi- tion d'une forte injection unipolaire. L'inclusion du torque Clastique dans l'tquation de la balance des torques donne aussi la possibilitt d'une description des mattriaux positifs. On a trouvt deux domaines des valeurs de l'anisotropie dielectrique (grande et petite) avec qualitativement diffkrente conduite de I'instabilite. On a consid6rC aussi le cas de manque d'injection. En conclusion on a discutk brikvement d'autres mtcanismes, actifs au cas d'une excitation D.C. et plus sptcialement d'effet flexo-tlectrique.
Abstract. - A generalisation of de Gennes model for the electrohydrodynamic instability a t strong unipolar injection is developed. Including in the torque balance equation the elastic torque gives the possibility for description of positive materials as well. Two domains of dielectric aniso- tropy values (small and big) are found with qualitatively different behaviour of the instability.
The case of no injection is considered as well. In conclusion some other mechanisms, active at d.c.
excitation are briefly discussed, especially the flexo-electric effect-gradient and linear one.
1. Theory. - The considerations given below ori- ginate from a model developed by de Gennes in 1970 [I]. Let a nematic, placed in a planar capacitor with interelectrode distance d and oriented parallel to the electrodes, is subjected to a strong unipolar injection by a d.c. voltage V, applied to the electrodes.
The condition for strong injection means that the density of injected charges go near to the injecting electrode (the cathode usually) is big enough to compensate a great part of the applied electric field in that region. Numerically it means that
The development of an instability will be studied on the basis of the charge balance equation. The time evolution of a charge density fluctuation is described by
where an assumption is made for the convective term at strong injection that
In the second term 7 is the dielectric relaxation time :
o is the mean value of the conductivity and the coef- ficient 2 arises because the conductivity itself is proportional to 69 [I].
The third term is due to the Carr-Helfrich effect.
The transverse current is determined by the angle of deviation cp of the director from the Ox axis. For
small angles this connection is
In order to express the fluid velocity v, via 69 the Stokes equation is used (with neglecting of the inertial effects) :
q : mean viscosity of the nematic.
In the case of spatial variations of v, in the following form (determined by the presence of electrodes) :
2 nx
v, = v t
)
cos, (
cos(7)
, (6)Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979359
ELECTROHYDRODYNAMIC INSTABILITY IN POSITIVE NEMATICS C3-311
it follows from (5) that
In the original version of the model the deviation angle cp was calculated from a torque balance equation containing only dielectric (re) and hydrodynamic (T,) torque with neglecting of the elastic one. The complete equation should read
Here rk,,, the y-component of the elastic torque, in isotropic elasticity approximation (K,, = K,, = K) is
The last result can be obtained if the same form as for v, is adopted for the spatial variation of 9 :
(P = cp, cOS
(T)
cos(T) .
(9)With de Gennes' results for both other torques the torque balance equation reads
A differentiation of this equation with respect to x together with de Gennes' estimation :
and the result from (7)
gives finally
Comparing (7) and (13) we see that all terms in the right hand side of (2) are expressed by Sq. Therefore the charge balance equation takes the following general form :
- at
a
S q = A(V, I:, Ao, E, AE ...). S q.
(14) The solution of this equation is
It demonstrates that when A < 0 the initial fluctuation S q , falls and the system is stable, while at A > 0 the
fluctuation is growing. So the instability condition is A = O o r
2. Solutions and comparison to the experiment. -
2 . 1 NEGATIVE AND ZERO DIELECTRIC ANISOTROPY. - The equation (16) is in fact an equation for the threshold voltage Vtr at which the instability arises.
If we consider at first the negative anisotropy case we can notice that at small K and big V,, the term 8 n2 K/V2 in the denominator can be neglected. In such a case the approximate de Gennes' formula can be obtained :
where V, is the Felici instability threshold in an isotropic liquid :
and p = o/qo is the mean mobility of the injected carriers.
Equation (17) shows that Vt, is always smaller than Vi and if Ao
-
o, V,,/& is essentially smaller than 1.At bigger values of the elastic constant the schematic graphical solution of (16) in the form
shows that taking into account the stabilizing elastic torque gives higher value of the threshold voltage than that one calculated by de Gennes (Fig. 1). On figure 1 the solution of the case AE = 0 (i.e. d = 0) is represented as well. In this case
FIG. 1. -Graphical solution of equation (19) from the text (schematic). V,, : threshold voltage at A& = 0 ; V l : with neglecting the elastic constant K i.e. c = 0 (de Gennes) ; V , : threshold voltage with K 7 0. Note that V, 7 V l ; V, : Felici instability threshold
in the isotropic phase (Ao = 0).
C3-312 A. G. PETROV
2.2 POSITIVE DIELECTRIC ANISOTROPY. - Essen- tially new results can be obtained in the case of positive dielectric anisotropy. Here the solution is more complicated due to the fact that the expression 8 n2 K/V2 - As14 n can reverse its sign from positive to negative depending on the voltage. The reversal voltage Vrev is
where VF is the Freedericksz threshold. In fact at V = VF the planar initial orientation becomes unstable due to the dielectric torque only. This static instability is the well known Freedericksz transition and at V
2
VF the spatial distribution of q takes the formq = q", cos
( y ) .
At V = V,,, =
$
V, this distribution can already obtain a x-dependence as well - the form (9). Let us consider the following subcases separately :2.2.1 Small dielectric anisotropy :
So, while at As < 0 the instability condition is satisfied for each voltage V > V,, here only a narrow voltage range exists, where the instability is possible.
This is in accordance with the experiment : in the nematic DIBAB (dibutylazoxybenzene) with A s = 0.2 hydrodynamical domains at d.c. voltage are registrated as a transient effect only slightly above V, [2]. At a constant voltage above V, these domains change into loop domains that disappear after some time. In the nematic DIHOAB (diheptoxyazobenzene) the' domains arise slightly below V, [3]. In this case they also disappear at a definite voltage above
5,.
Ae ofDIHOAB is rather small -
+
0.05 [8].The second threshold value V: corresponds already to an instability existing for any voltage above it.
In this sense it is analogical to V,, for A s < 0. But here this threshold voltage is higher than the Felici threshold (Fig. 2) because of the stabilizing action of the Carr-Helfrich effect [I]. That's why at voltages V
2
Vi in the isotropic phase an instability can be observed, but in the nematic - not [2].2.2.2 Big dielectric anisotropy. - In the equa- tion (23) we will have already f c 0. Then graphical solution will look like that on figure 3. One can see that at low Felici threshold the expression
E, mean value of the dielectric constant.
The graphical solution of the equation
is bigger than zero for each voltage bigger than V,,,.
b - a~ =
f/(s
- d) (23) So, V,,, plays the role of an instability threshold in the nematic phase. The instability exists at eachwhere voltage above V,,,.
is shown on figure 2. It can be seen that two threshold voltages exist already -
V,:
and V:. The first of them - VA, is bigger or smaller than VF depending on the value ofF.
It corresponds to arising of a hydrodynamical type instability around the Free- dericksz threshold(c 3
VF), which however is sharply quenched at V,, because the whole expression A(V ...) in (14) will reverse its sign at that voltage.FIG. 3. - Graphical solution of equation (23) from the text with f < 0. Two cases corresponding to different values of Vi are shown : V i : no solution ; V; > V i : two solutions V; and V;.
0
I
I I
This conclusion is also in concert with the experi- ment : in the liquid crystal PEBAB-N-(p-ethoxyben- zi1idene)-p-aminobenzonitrile with Ae = 14 hydro- dynamical domains exist in the isotropic as well as in the nematic vhase r21.
FIG. 2. -Graphical solution of equation (23) from the text At higher ~ e l i c i thresholds it is possible eventually
with f > 0. V, : Freedericksz threshold, V,,, : reversal voltage, the instability condition to be disrupted in a voltage
V,: and V; : the two solutions. Note that Vz > Vi. range [v,',, V:] - see figure 3.
ELECTROHYDRODYNAMIC INSTABILITY IN POSITIVE NEMATICS C3-313
2 . 3 No INJECTION CASE. - This problem was rigo- rously considered in the one-dimensional case by Helfrich [4]. In our approach the threshold equation will be obtained putting in (16) q, = 0. Then (16) can be rewritten in the form
where the parameter
has a meaning analogical to that of the parameter
5,
introduced by Helfrich. From (24) it is evident that instability can be obtained only when
t
> 0. In this case the threshold voltage isThis simplified formula is analogical to that given by Helfrich. Its derivation*was not possible in the original version of the model [l] due to the neglecting of the elastic torque.
3. Conclusion. - The analysis of the instability conditions at d.c. excitation is of increased impor- tance today when the a.c. instability (without injec- tion) is relatively well understood. In the d.c. case
however much controversy exists and the experimental studies are relatively difficult. The simplified two- dimensional treatment presented here can help the elucidating of some threshold features depending on the material constants, e.g. the qualitatively different behaviour of small and big anisotropy materials.
It shouldn't be however considered as an adequate picture of the real situation even because in positive anisotropy case the domains are mostly parallel to the easy direction, with a periodicity of the orientation pattern along Oy axis and not along Ox, as it is postu- lated here. One simple reason for this can be that the inclination of the layer above Freedericksz threshold can start the Pikin-Indenbom mechanism [5]. But with this in mind only domain thresholds below V, can not be explained. Anyway, the rigorous theory predicting both the threshold and the domain pattern must be three-dimensional.
Another mechanism active at d.c. excitation and not included in the present treatment is the flexo- electric one with its both components - first one depending on the gradient of the electric field (Petrov, 1974 [6]) and the second one on the electric field itself (Bobilev-Pikin, 1977
[q).
They both lead to longitudinal domains again, not only in positive but in negative materials as well. Gradient flexoeffect itself must be very well pronounced in strong injection conditions.References
[I] DE GENNES, P. G., Comm. Sol. Stat. Phys. 3 (1970) 35. [5] PIKIN, S. A. and I N D E ~ O M , V. L., Kristallograjiya 20 (1975) [2] DE JEW, W. H. and GERRITSMA, C. J., J. Chem. Phys. 56 (1972) 1127.
4752. 161 PETROV, A. G., Ph. D. Thesis, Inst. Solid State Phys., Sofia,
1474
A < , ..
131 GRULER, H. and MEIER, G., Mol. Cryst. Li9. Cryst. 12 (1971) [7] BOBILEV, Yu. P. and PIKIN, S. A,, JETP 72 (1977) 369.
289. [8] DE JEU, W. H. and LATHOUWERS, Th. W., 2. Naturforsch. 29a
[4] HELFRICH, W., J. Chem. Phys. 51 (1969) 4092. (1974) 905.
