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Theory of instabilities of nematics under A. C electric fields : Special effects near the cut off frequency
E. Dubois-Violette
To cite this version:
E. Dubois-Violette. Theory of instabilities of nematics under A. C electric fields : Spe- cial effects near the cut off frequency. Journal de Physique, 1972, 33 (1), pp.95-100.
�10.1051/jphys:0197200330109500�. �jpa-00207231�
THEORY OF INSTABILITIES OF NEMATICS UNDER A.
CELECTRIC
FIELDS :SPECIAL EFFECTS NEAR THE
CUTOFF FREQUENCY
E. DUBOIS-VIOLETTE
Laboratoire de
Physique
desSolides,
Faculté desSciences,
91 2014Orsay,
France(Reçu
le 9juillet 1971)
Résumé. 2014 Nous étudions numériquement les instabilités électro-hydrodynamiques d’un cristal
liquide nématique soumis à un champ électrique. Nous déterminons la transition entre les régimes
basse fréquence et haute fréquence dans le cas où le champ appliqué a la forme d’un créneau et dans le cas où il est de forme sinusoïdale. Dans les deux cas, la courbe donnant la tension de seuil en fonc- tion de la fréquence a une forme en S près de la fréquence critique : d’où la possibilité de certains
effets d’hystérésis.
Abstract. 2014 We have obtained numerical solutions for the coupled equations describing the charge density and the curvature in a nematic system under oscillating electric fields (rectangular pulses or sine waves). In both cases, the resulting curve of threshold voltage versus frequency has
an S shape in the vicinity of the cut off frequency. This should lead to observable hysteresis effects.
Classification Physics Abstracts :
14.82, 14.88
1. Introduction. - The effects of
alternating
electricfields on nematic
liquid crystals
withnegative
dielectricanisotropy (811
- 81. = e.0)
have been described in apreceding
paper[1].
This paper discussed the threshold Vc for the onset of a cellular type ofhydro- dynamic
motion. The mainequations
of the model are summarised in section II.They
gave :a)
Aconducting regime (at frequencies
below acertain cut off
v c)
where thecharges oscillate,
while themolecular distortions are static.
b)
A dielectricregime (v
>vc)
where thecharges
aretime
independent,
but the distortions of the molecularalignment
oscillate at thedriving frequency.
FIG. 1. - A nematic crystal is, at zero electric field, aligned along x direction. An A. C electric field E(t) is applied along
z direction. rp is the fluctuation angle of the director around
x direction. d is the sample thickness.
The
regime (a)
could beanalysed accurately
in all theinterval 0 v Vc. On the other hand we were
able,
to derive the features of
regime (b) only
when v wassignificantly larger
than v,,. The aim of the present paper is tobridge
this gap ; since the numericalanalysis
israther
heavy,
webegin by
aslightly simplified
case,where the electric
field,
instead ofbeing
a sine wave, isa succession of
rectangular pulses (Fig. 2).
We thenFIG. 2. - Form of the periodic applied field a(O).-10 is the period.
find an
unexpected
Sshape
for the curve of thresholdversus
frequency (section III).
This then induces us to repeat the calculation for the(more usual)
case of asinusoidal field : the results are derived in section IV :
they again
show the Sshape, although
lessconspicuously,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197200330109500
96
II.
Electrohydrodynamic équations.
- We considera nematic slab of thickness d submitted to an alterna-
ting
fieldE(t) along
Z(see Fig. 1).
If the
sample
isparallel
to the xyplane
with themolecules
along x
in theunperturbed
state, we assume that allphysical quantities depend only
of x notof y
or z. We
study
thestability
of the system under purebending
fluctuations(these
purebending
modes are theonly
ones to induce shearflows)
where the « director »(preferred
direction of themolecules)
is in the(x, z) plane making
a smallangle qJ(x, t)
with the x direc-tion
(Fig. 1).
As shown
by
recentexperiments,
thisapproximate
treatment is
qualitatively
correct. Theonly
way we take into account boundaries conditions is to say that the fluctuationwavelenght,
in the xdirection,
cannot bemuch
larger
than thesample
thickness.Electrohydrodynamic
effects are then described[1]
by
twocoupled equations
for thecharge density q
andthe
curvature t/J
these
equations
are obtained bywriting charge
balanceequation,
acceleration and torqueequations [1]. They
are
explicitately
1 is a dielectric relaxation time defined
by :
aH is an apparent
conductivity
ail, 8.l, UII’ Ql are the dielectric constants and conducti- vities
parallel
andperpendicular
to the molecules.T(t)
is a molecular relaxation time,depending
onthe field
strength,
definedby :
with
we use for the
viscosity
coefficients the notation of[2].
Eô
takes into account elastic torque(eventually magne-
tic
torque).
where k is the fluctuation wave vector defined
by
K33
is the Frank elastic constantcorresponding
to abending
deformation[3]. Equation (II.1)
does not takeinto account diffusion currents.
They
are in factimportant only
in thehigh frequency regime
where k becomes of the same order as theDebye
wave vector qD defined
by qD
=1/iDjj (D :
diffusionconstant).
Here we are more
specially
concernedby
the transi-tion between low and
high frequency regimes :
in sucha case the
wavelengths
of interest 2n/k
arecomparable
to the
sample
thickness and muchlarger
than thescreening
radius. Thus we may omit the diffusion currents for our present purposes(1).
III.
Instability
underrectangular
fieldpulses. -
We discuss the case where the
applied
fieldE(t)
has theform shown on
figure
2. This case isexactly
soluble.Let us first rewrite the system
(II
1,2)
in asymmetrized
form and introduce dimensionless parameters.
then
equation (II. 1)
and(II.2)
become :It is convenient to rewrite
equation (III.1)
and(III.2)
in the formwhere the vector (~) = «P2) (~1~2) )
and 3C is the matrixdefined with use of Pauli matrices
^ 1 is the unit matrix
(1) The effects of the diffusion currents in the high frequency limit will be discussed in a separate paper.
STABILITY OF THE SYSTEM. -
Je(t)
isperiodic
andcommutes with the evolution operator in
one period (or
translations operator
U)
definedby
where 0 is the
period.
These two operators may be
simultanuously diago- nalysed.
Then we canexpand
the solutions of the system(III.3) (which
form a two dimensional linearspace)
on theeigenfunctions (Wi)
of the operator UThe
eigenfunctions (Wi)
of the translation operator Uare of the form :
where
a,(O)
andbi(O)
areperiodic
functionsof période, Al and Â2 being
thecorresponding eigenvalues
of theoperator U.
The system is instable when
s+ is the
eigenvalue
which possesses thehighest
realpart.
The
stability
of the system is then determinedby knowing
theeigenvalues
of the translation operator Uas functions of the parameters
(a
andao).
EVOLUTION OPERATOR. - The evolution operator 0 is defined
by :
13 is the
chronological product :
The translation operator U may be
decomposed
as :U+ and U - are the evolution operators in the two half
period
u =e/2
=nlp (see Fig. 2). They
areeasely
calculated for the present
problem
because in each half-period Je(t)
is constant.Using
the usual notations, V.6= Li i Vi - - ui, U may be
written :
with
The matrices
Tt
are calculated with use of Pauli matrices commutation rules(2)
where
U is then calculated
by application
of theidentity (3)
for
evaluating
theproduct
T+ . T-.the two
eigenvalues Â+
and Â- arealways
real andgiven by :
The system is instable when the greatest
eigenvalue Â,
is
equal
to one. Let us defineF(a, ao)
as :F(a, ao) = À+ -
exp (2 Au) .Then the system is instable if there is one
pair (a, ao)
such as
F(a, ao)
> 0.THRESHOLD CONDITION. - The threshold field is obtained
by determining
the greatest domain where the system is instable. Thisgives
the two conditions :with
a(ao)
minimum for the low andhigh frequency regime,
a(ao)
maximum for the transitionregime.
Equations (III).8) (III.9)
have beennumerically
solved on a computer with the
supplementary
condi-tion
a2> a 2 .
Thiscorresponds
to the condition that the fluctuationwavelength
be smaller than thesample
thickness d.
Remembering
that :a5m,
obtained for k =re/d,
isproportional
tol/d2.
(2)
à, âk
= iL Gjkl;;1
where Gjkl is the usual completelyi=i
antisymmetric tensor with GXyz = 1.
98
Numerical results are shown on
figure
3 where the reduced field thresholda(ao)
isplotted
versus thereduced
frequency p
forcorresponding
to asample
thickness d =100 y
inM. B. B. A with
k33 L--
10-6 CGS, 1 = 5 x 10-3 sand
FiG. 3. - Instability threshold reduced field a is plotted versus the reduced frequency p.
Let us note that the diffusion currents are in fact
negligible
for not tohigh frequencies.
Moreprecisely
atthe transition :
aô
0.8 whichcorresponds
tofor a
sample
with D= 1
x 10-’ CGS.LOW FREQUENCY REGIME. - We are concerned with
frequencies
such as p 1. Then we shall show that the electric field threshold is small :a’ «
1and ao
1.Then,
assuming
thatB2/H2 N
1 - 4 Z2a2,
theequation (III . 8)
reduces to :The condition
(III.9) imposes
thataô
=ao^. Equa-
tison
(III.10)
then defines avoltage
threshold(ad) independent
of thesample
thickness. This is in a verygood
agreement with the numerical calculations. Let usremark that
equation (III .10)
leads to a «eut offfrequency »
Pc =relue
definedby :
The
study
of the transitionregime
shows that Pc is not a real « cut off »frequency
butonly
defines theupper limit of the
voltage
threshold branch.HIGH FREQUENCY REGIME. - We are concerned with
frequencies
such as p » 1.Neglecting
terms of firstorder in
1/p, equations (III.B)
and(III.9)
lead to :where
Ci
andC2
are two constantsindependent
of thesample
thickness. Thisregime corresponds
to a fieldthreshold
independent
of thesample
thickness.TRANSITION REGIME. - This
regime
hasonly
beennumerically
determined on a computer. The condi- tion(III.9)
leads toaô
=am.
The results are shownon
figure
3, thevalue P2
= 1. 21 is in agood
agreementwith p,,,.
There is aninteresting
intervalwhere two types of instabilities can be observed. In this
region :
- If we increase the field
amplitude,
from 0upwards,
at fixedfrequency,
we find thevoltage
threshold characteristic of the
conducting regime.
- If we start with a
high frequency
and a suitablefield
amplitude
and then decrease thefrequency
toreach the
regime
of interest, wemight keep
a(meta- stable) unperturbed
state. If we now raised, or decreas-ed,
the fieldamplitude,
we would find convectioninstability. Typically
in M. B. B. A. withthe transition width
Ap corresponds
to Av = 17 Hz.IV. Instabilities under a sinusoïdal field. - We now
consider the case where the
applied
fieldE(t)
is of theform
With use of dimensionless parameters p, 6,
z2,
thesystem
(11. 1, 2)
may be rewritten as :where
and
As shown in section III, the solutions of
(IV. 1)
are ofthe form :
where
(Wi(O»
andÂi
are theeigenfunctions
andeigen-
values defined
by (III). 5).
STABILITY OF THE SYSTEM. - The
instability
condi-tion is still
given by (III. 6).
The calculation of the translation operator U is much more difficult in the case of an
alternating
field.We shall assume that at the threshold
This seems to be a
quite
reasonableassumption : experimental
observations[4], [5], [6]
have shown that,in all known cases, at
threshold,
thehydrodynamic
cellular motions are
infinitely
slow. Also for the caseof
rectangular
fieldpulses
we could prove that theeigenvalues
were real and the two cases do not appear to bedrastically
différent. With thisassumption,
theinstability
threshold is definedby :
This is
equivalent
to findperiodic
solutions(f (0»
ofsystem
(IV. 1).
THRESHOLD CONDITION. -
Expanding
theperiodic
solution
( f(0))
in Fourier serieswe find after some calculations that the threshold field is defined
by
theimplicit equations :
with
e(eo)
minimum for the low andhigh frequency regimes, e(eo)
maximum for the transitionregime.
The
only
non zero elements of the infinite matrixM(e, eo)
are :As in section III,
equations (IV. 4)
and(IV. 5)
havebeen
numericallly
solved on acomputer with
the condi-tion
3
Numerical results are shown on
figure
4. We find forlow and
high frequency regimes
the same results asthose obtained in
[1] («
conduction » and « dielectric »regime).
As in section III, diffusion currents are
negligible
atthe transition
where eô
0.8. Thiscorresponds
tofor a
sample
with .FIG. 4. - Instability threshold reduced field e is plotted versus
the reduced frequency p for z2 = 3.05 and
Let us recall first the
asymptotic
behaviour derived in[1].
LOW FREQUENCY REGIME
(«
CONDUCTION REGIME»).
- This
corresponds
tofrequencies
p g 1 and field such ase2 1, e2/p 1.
Then the threshold isgiven by :
Equation (IV. 7)
defines a cut offfrequency
pc :As in the case of the
rectangular
fieldpulses,
we shallsee that Pc is not a real cut off
frequency
butonly
defines the
voltage
threshold domain.Equation (IV. 5)
leadsto eô
=eôm
=agm/2
which isproportional
toIld 2
Then
equation (IV. 7)
defines avoltage
threshold(ed)
HIGH FREQUENCY REGIME
(«
DIELECTRIC REGIME » - Thiscorresponds
tofrequencies p >
1.There is a field threshold
independent
of thesample
thickness defined
by :
where
Ci
andC2
are two constantsindependent
of thesample
thickness.100
TRANSITION REGIME. - The condition
(IV. 5)
leadsto :
The results are shown on
figure
4, the value p2 = 1.42 is in agood
agreementwithpc.
Here
again,
the threshold curve has an Sshape
with acritical interval
Ap
= P2 - p 1 = 0.12. With the sameexperimental
conditions as in section IIIthis
corresponds
to a transition width Av = 4 Hz.The same
qualitative hysteretic
features are stillexpected.
To make the interval Avlarger,
it ispossible
in
principle
to usedoped samples (with
ionicimpuri- ties)
since Av isproportional
to theconductivity.
V. Conclusion. - The
stability
of a nematicliquid crystal
submitted torectangular
fieldpulses
or to asinewave field is determined
by
avoltage
threshold forlow
frequencies.
But there is a certain interval(vi
v V2 where both thresholds may beobserved,
the choice between them
depending
on the detailedmethod used to
change
theamplitude
or thefrequency
of the
applied
field. The interval V2-Vl islarger
in thecase of
rectangular
fieldpulses
but in both cases,hysteresis
effects could be seen inprinciple.
Of course, there may be some difficulties in
observing
these effects : to reach the metastable static
region,
weneed to decrease the
frequency
at constant fieldampli-
tude. If this decrease is « adiabatic », the process should be successful. But it is not
entirely
obvious thatwe will retain the metastable state if the decrease is achieved
by
anabrupt jump.
However, astudy
of themetastable range appears to be worthwhile :
experi-
ments in this direction are
currently
under way[7].
Acknowledgements.
- It is apleasure
to thank Pro-fessor P. G. de Gennes for
stimulating
discussions.I am very
grateful
to A. M. Esteve and F. Fiorentinofor
help
in numerical calculations.References
[1] DUBOIS-VIOLETTE (E.), DE GENNES (P. G.), PARODI (O.),
to be published in J. Physique.
[2] PARODI (O.), J. Physique, 1970, 31, 581.
[3] FRANK (F. C.), Discussions Faraday Soc., 1958, 25,
19.
[4] RONDELEZ (F.), Thesis III cycle, Orsay, 1970, see also
ref. (5).
[5] Orsay Liquid Crystal Group, Molecular Crystals
and Liquid Crystals, 1971, 12, 251.
[6] Orsay Liquid Crystal Group, Phys. Rev. Let., 1970, 25, 1642.
[7] GALERNE (Y.), Private Comm.