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The thermal oscillatory instability in a homeotropic nematic : an inverse bifurcation
E. Dubois-Violette, M. Gabay
To cite this version:
E. Dubois-Violette, M. Gabay. The thermal oscillatory instability in a homeotropic ne- matic : an inverse bifurcation. Journal de Physique, 1982, 43 (9), pp.1305-1317.
�10.1051/jphys:019820043090130500�. �jpa-00209509�
The thermal oscillatory instability in a homeotropic nematic :
an inverse bifurcation
E. Dubois-Violette
Laboratoire de Physique des Solides (*), Université Paris-Sud, Bâtiment 510, Centre d’Orsay, 91405 Orsay, France
and M. Gabay (**)
Laboratoire de Physique de l’Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France
(Reçu le 25 janvier 1982, révisé le 6 mai, accepté le 18 mai 1982)
Résumé. 2014 Dans le cadre d’un modèle libre-libre, au moyen d’approximations usuelles, nous montrons que l’instabilité oscillante d’un cristal liquide homéotrope chauffé par le bas est une bifurcation inverse. Ceci confirme la prédiction faite par Lekkerkerker concernant ce système. Nous étudions également l’influence d’un champ magnétique vertical sur la transition et la durée de vie de l’état métastable qui peut s’établir au-delà du seuil linéaire.
Abstract. 2014 By means of standard approximations, using free-free boundary conditions, we show that the oscilla- tory instability of a homeotropic liquid crystal heated from below is an inverse bifurcation. This confirms Lekker- kerker’s prediction for this particular system. The influence of a vertical magnetic field on the transition is also considered as well as the lifetime of the metastable state which may develop above the linear threshold.
Classification
Physics Abstracts
47.20 - 47.25Q
1. Introduction. - Thermal instabilities in Nematic
Liquid Crystals (NLC) have been investigated in detail
both experimentally and theoretically [1]. The novel feature, when comparing with the Rayleigh-Benard
case in isotropic liquids, is that the instability mecha-
nism in NLC is dominated by the director behaviour
(mean orientation). An outstanding consequence of this fact is that stationary convection is induced by heating an homeotropic NLC sample from above (molecules initially perpendicular to the plates) [1].
Furthermore in the absence of a magnetic field, the
orientation perturbations relax much more slowly than
thermal perturbations do.
This situation may be compared with thermal instabilities in binary mixtures where the Soret effect
causes the denser component to move to the warmer
boundary [2]. In that case too, stationary convection is induced by an adverse thermal gradient and the desta-
bilizing process (concentration gradient) has a much longer relaxation time (TJ than the stabilizing process
(thermal gradient) T, :
When heating these mixtures from below the thermal effect is now destabilizing whereas the concentration effect becomes stabilizing; Lekkerkerker [3] has then
shown that, because these competing effects are govern- ed by the inequality (1), an oscillatory instability (overstability) belonging to the class of inverse bifur- cations (finite amplitude convection) may set in with characteristic period T such that
It is then natural to expect similar behaviour for
homeotropic NLC heated from below, where the
orientation relaxation time To plays the role of T,,,.
Lekkerkerker and other authors [4, 5] have thus
exhibited an oscillatory solution to the hydrodynamic equations of a homeotropic NLC and experiments
tend to support the existence of overstability in these
substances [6] ; they also seem to reveal metastability
close to the threshold which would indicate that, just as for binary mixtures, the inequality (1) is respon- sible for both an oscillatory and an inverse bifurcation in the system. It is the purpose of the present article to show that one indeed deals with an inverse bifurcation (*) Laboratoire associ6 au C.N.R.S.
(**) Permanent address : Laboratoire de Physique des Solides, Universit6 Paris-Sud, Centre d’Orsay, 91405 Orsay,
France.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043090130500
in that case and to study some properties of this instability.
In the first part we briefly recall the general charac-
teristics of bifurcations in fluids.
In the second part, given some approximations, a
linearization of the hydrodynamic equations of a NLC
allows us to rederive known results for the homeo-
tropic configuration : indeed, when the sample is
heated from below, two modes are recovered corres- ponding to either an oscillatory or a stationary convec-
tion: The effect of a magnetic field H normal to the sample’s plates on these modes is such that below
some cross-over value Ho the linear oscillatory
threshold ATC(H) is first reached. Above Ho the system switches to a regime where the linear stationary threshold is first obtained (Fig. 2).
In the third part, the effect of non linearities on the
oscillatory mode are taken into account. The system is still placed in a field with the same geometry as discussed above. Non linearities induce a correction AT (2) (H) to the linear value Tc(H) which we have analytically calculated. For H less than a cross-over
value Hj, AT (2) 0 so that the bifurcation that sets in is of the inverse type. For H larger than Hl, AT (2) > 0 and the system undergoes a direct bifurca-
tion. No simple argument allows us to relate Ho to
’H 1 (our numerical results seem to yield HI =1= Ho).
Finally, for H H1, we estimate the metastability
effects close to the non linear threshold.
2. Survey of thermal instabilities in the Landau
theory. - Convective instabilities in the case of NLC
are usually described in terms of some hydrodynamic
variables such as the velocity field v, the director orien- tation n, the local temperature T. These quantities are
connected by non linear equations which generally
cannot be solved exactly. However, close to the threshold, they are linearized as a zero step approxima-
tion so that v, n, T will be analysed in a Fourier series
expansion [7]. Non linearities are then treated as
perturbations and couple these eigenmodes. Indeed if
one considers for instance the fluctuation of the ver-
tical component of the velocity field (in the z direction
normal to the plates) we may write
11
which inserted into the hydrodynamic equations yields :
When the temperature difference across the sample AT
is varied,
Re (aq) 0 for AT AT,(q) and the fluctuation is damped,
Re ( 69) > 0 for AT > âTc(q) and the fluctuation
develops in the system with a frequency given by
Im (uq)-
In the Landau theory one only considers the value qo of q for which ATc(q) is minimum and accordingly
the amplitude Aqo associated to it, which we now call A.
The critical behaviour is then exhibited if one separates in (4) the slow time part governed by Re ( a qo) from the
fast, governed by Im (aqo)’ Equation (4) now reads :
I A stands for the module of A and - b = ,uqoqoqo. The absence of a term like À-qOqO Aio is connected with the fact that it only contributes to the fast varying part.
(5) is also written as
where
a familiar expression in the theory of phase transi- tions ; (To is a model dependent constant).
9 If b > 0, an instability develops above AT, with amplitude
One deals with a direct bifurcation (2nd order transi-
tion).
. If b 0, the fluctuation would grow uncontrolled if one did not add a term in (7) of the form + c 6I A 6.
In that case fluctuations of finite amplitude Ao may
develop for AT ð.Tc if - b >> cA 0 2 and - bA 0 2 >>
I aqO I. One deals with an inverse bifurcation (1st order transition). Stability of these solutions are investigated by linearizing (5) and the conclusions are summarized in figure 1. In the next section we will make these statements more quantitative.
Fig. 1. - Evolution of an initial perturbation Ao as a function of AT : a) b > 0 the bifurcation is direct (2nd order transi- tion) ; f3) b 0 the bifurcation is inverse (1st order transi- tion).
3. Linear hydrodynamic theory of the convective oscillatory instability. - The hydrodynamic equations
for a NLC laid by Ericksen and Leslie [8] describe the
coupling between v and n. They allow to construct the appropriate form of the Navier-Stokes equation, the torque balance equation, the heat conduction equation which, together with the continuity equation Vv
determine n, v, the local pressure p and the local tempe-
rature fluctuation 0. In order to extract simple analy-
tical expressions for these quantities close to the
convective instability one usually does the following assumptions :
a) the Boussinesq approximation allows to neglect
the temperature dependence of viscosities, specific heat, heat conductivities and density (except in the buoyancy term of course). It is justified if the sample
thickness d is large enough such that AT is not too
large. AT = TL - TU where TL (TU) is the tempera-
ture of the lower (upper) boundary.
b) The linearizing scheme allows to find the thres- hold as the solution of a compatibility determinant.
Given the initial rotational invariance about the z axis the convective structure will consist in one dimensional rolls in the (xy) plane (for convenience we will choose their axis to be perpendicular to the x direction). To
that order the underlying assumption is that one deals
with a direct bifurcation and only non linear terms will
allow to determine whether or not this is actually the
case.
c) The free-free boundary conditions (BC) further simplify the analysis.
By requiring
Tu+TL
where n = (- sin 0, 0, cos 0) and T(z) =
2 -
(AT/d) z + 0, one may look for solutions of the type
(p and vx are deduced from (10) with use of the Navier- Stokes and continuity equations respectively). We will
discuss below the consequences of this approximation, given that the actual BC should be rigid-rigid :
but would not allow for simple analytical solutions of the form (10). Given a), b), c), we show in Appendix A
that VZ’ 0, 0 satisfy the following linear equations
relevant to the homeotropic configuration :
where the characteristic times are
and
(the (+) or (-) sign corresponds to a field along the z or x direction respectively)
Inserting (10) into (12) yields :
where
This equation yields two classes of solutions :
o stationary convection corresponding to Q = 0 in (10), so that D = 0 and B > 0, or
o oscillatory convection, first studied by Lekkerkerker [3], is obtained by seeking (14) under the form
(U2 + wõ)(a + WI) = 0 where (t), 1 > 0, so that BC = D and B > 0, C > 0, or
with
Threshold values A T, are obtained for the wave vector qo which minimizes AT in expressions (15)
and (16).
1) In the absence of a magnetic field, typical values
for d = 0.5 cm sample thickness are :
(17) can be fairly simplified to yield
Using typical values for MBBA at 25 °C and for
d = 0.5 cm, AT. = 3.1 °C, wo rr 4.35 x 10-2 s
(To - 144 s) and the wave vector of the structure is
qo - 0.7. It is quite apparent from (16) and (18) that
the inequality (2) is indeed satisfied.
2) When a magnetic field is applied perpendicu- larly to the plates, for strong enough values of H, the
orientation of the director is blocked so that To - 0
and oscillations are no longer possible. The stationary
convection that sets in is characterized by (15) and tends, when H - oo, to the Rayleigh-B6nard value of
an equivalent isotropic fluid.
We find that mo(Ho) = 0 for Ho = 676 gauss and ðTc(Ho) 7.5 OC. Furthermore (o(H) has a maximum
as a function of H for HM ’" 400 G, ’" 1.4 wo. All these features are presented in figure 2. These curves
are in good agreement with those obtained from theo- ries including rigid-rigid BC [5] and those obtained
from experiments [6]. However the values AT,(O) --
3.1 -C, AT,,(HO) --- 7.5°C are only half of those
predicted by the above mentioned authors; this is not
too surprising since free-free BC only poorly describe
the actual profile of vx which is needed, via the torque
Fig. 2. - Threshold, frequency and wave vector of the oscil-
latory convection as a function of the vertical applied field.
The dashed line, in A Tr ,(H) versus H, represents the statio- nary convection curve obtained when heating from below;
it intersects the overstability curve for Ho - 676 gauss (see text).
acting on the director, to obtain an accurate theore- tical prediction of the threshold.
To conclude this section let us return to the phy-
sical process, already described in [4, 6] of the oscilla- tory instability in the absence of a held ; because of the relative orders of magnitude of T,, Tt, Ta, To (Eq. 16)
and because equation (2) is satisfied (12 y) may be
simplified to yield :
and
This shows that the velocity field and temperature perturbations are in phase, but in quadrature with the
orientation perturbations. Now this latter effect is
responsible for the heat focusing effect [1] which has a stabilizing influence when heating from below; there-
fore oscillations eliminate this effect while retaining
the destabilizing buoyancy influence.
4. Perturbation expansion about the linear solution.
- As mentioned in § 1 (Eq. 7) the sign of the coefficient of the first non zero non linear term determines the nature of the bifurcation. It is thus necessary to intro- duce the non linear terms in the Navier-Stokes (A. 1) torque balance (A. 4, A. 5) and h6at conduction (A. 6) equations.
In Appendix A, equation (10) and equation (12) were
written as
where Cc was the differential linear operator (A .10) and
where q., = qo nld is the critical wave vector at the linear threshold.
Including non linearities will lead to the formal equation :
where the r.h.s. is a vector the components of which are given by the non linear terms off
Above the threshold, ’U ) (Eq. A. 9) is given by the expansion :
A 7". the threshold for non linear (finite amplitude) instability, by :
and L by :
Much in the same way,
where the terms inside brackets in (23) denote the vectors the elements of which enter the non linear terms to
various orders.
This method has already been used by E. Dubois-Violette and F. Rothen in the context of thermal instabi- lities in planar NLC [9].
Identifying the coefficients of the successive powers of i,- in (22) and (23) gives
with ’I t c I ’ given by (A. 9) and (17)
with
Let us designate by I LfL (’) > = c 0 the eigenvector of the operator t ’, adjoint of the operator C,,,, satisfying
(9.)
cwith the free-free BC requirement :
One has
and
where
and qo and co are obtained from equation (16).
Projecting equation (25) onto I "a(’) > yields :
where
Equations (10) and (30) clearly tell that åT(1) = 0.
Indeed, I XCe’l1I» > only contains even harmonics of
7c/d and qx so that the r.h.s. of (32) is zero. However a
term such as §
a
is non zero and is obtained in the lh.s.of
( 2)via
the definition of C1 ; thusåT(1) = 0.
The nature of the bifurcation is therefore deter- mined by the sign of å T(2). A negative sign corresponds
to an inverse bifurcation since the thermal gradient is
reduced with respect to å Teas the perturbation deve- lops.
To order B3,
AT(") is then determined by projecting (33) onto
we have evaluated ] JV£l’iL[) ), ] A"£l’iL[, (.1.1(2» )
and l ’iL ) analytically. Their expressions are given
in Appendix B. Finally the value of A T(2) was calcu- lated numerically and yields :
From the previous discussion we therefore conclude the oscillatory instability of a homeotropic NLC
heated from below is an inverse bifurcation.
To be able to plot the amplitude of say Vz as a function of AT would require calculations to order e5
Fig. 3. - a) LB T(2) versus H in the oscillatory instability.;
b) Phase diagram AT, H, A (vz for instance).
’
and have not been undertaken. We therefore only
obtain the metastable branch of figure 1. The effect
of a vertical magnetic field on the transition has also been considered. Results are plotted on figure 3. They
show that the bifurcation becomes direct (2nd order transition), A T(2) > 0, for H > H1 = 630 G.
To conclude this section let us estimate the life time of the fluctuation, close to A T,, when the initial
amplitude is located on the unstable branch (Fig. 1).
Incorporating a stabilizing c A 16 term in (7) yields
for (6) :
where
For Re (a) - 0,
h
Setting I A12 = A 2 (1 + À) yields
;
2 A 2 1 b H,whence a characteristic life time given by
For A T Z A T,, is it easily seen that
so that
5. Conclusion. - Given simplifying assumptions we
have confirmed Lekkerkerker’s prediction according
to which oscillatory instabilities and inverse bifurca- tions in homeotropic NLC heated from below occur as a consequence of competing effects with very different relaxation times. Unfortunately, unlike binary mix-
tures where Lekkerkerker has exhibited this property in a very simple manner, no dominant contribution to
ð T(2) in our approach allows to display this competi-
tion in a clear cut, elegant way.
On the other hand we are quite confident that taking
proper BC into account will not change our conclu-
sions. Indeed the nice thing with a homeotropic confi- guration is that BC do not modify the symmetry of the system and thus preserve, except close to the plates,
the behaviour predicted by a free-free theory. Still, including rigid-rigid BC would yield quantitative
results.
Acknowledgments. - We would like to thank H. Lekkerkerker for fruitful discussions.
Appendix A. - Using standard notations, (8) (10) the relevant equations describing the coupling between
the velocity field, the director orientation and the thermal gradient are : a) The Navier-Stokes equations :
where Fi is the ith component of a force acting on the system (i = 1, 2, 3 correspond to the x, y, z direction respectively), ðiz is the Kronecker delta symbol and 7 is the stress tensor such that
The components of the Ericksen tensor arE satisfy
K1, K2, K3 being the splay, twist and bend constants respectively.
The components of the viscous tensor a, are such that
b) The torque equations
where r is a torque acting on the director n and
c) The heat conduction equation
where avs is the symmetric part of the viscous tensor and
d) The continuity equation
Following (1. c) we introduce the quantities V A (V A F), V A r and 0 which allow to eliminate p and vx ; their
explicit expressions are obtained by taking the relevant derivatives in (A. 1)-(A. 7) ; close to the threshold these
equations may be linearized to yield :
Here we have set F = r = 0 and
with 04 = r1 a4 + ’12 a;4 + ,u a2z2 and the + sign in the magnetic term corresponds to a field applied in the
Oz ( - ) or Ox ( + ) direction.
Looking for solutions of the form (2) with boundary conditions (3), (A. 8) reduces to equation (4) of the
text. The compatibility equation reads
Requesting oscillatory convective solutions yields equation (16) and
we have normalized Vz to unity.
Appendix B.2013 1)
a) Stress tensor components I
b) Torque acting in the y direction
c) Heat conduction equation
2) Evaluating T(2). From Appendix B. .1, one has
with
then I ’U(’) > is unambiguously determined by
with
where i stands for 1 our J
To evaluate c 91(2» > we only retain terms which will effectively contribute to ð T(2) after projection onto I (a(’) c > (e.g. we neglect third order harmonics in q.,, q _, and co)
where
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