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Instability of a homeotropic nematic subjected to an elliptical shear : theory
E. Dubois-Violette, F. Rothen
To cite this version:
E. Dubois-Violette, F. Rothen. Instability of a homeotropic nematic subjected to an elliptical shear : theory. Journal de Physique, 1978, 39 (10), pp.1039-1047. �10.1051/jphys:0197800390100103900�.
�jpa-00208842�
LE JOURNAL DE
PHYSIQUE.
INSTABILITY OF A HOMEOTROPIC NEMATIC SUBJECTED
TO AN ELLIPTICAL SHEAR : THEORY
E. DUBOIS-VIOLETTE and F. ROTHEN (*)
Laboratoire de Physique des Solides, Université Paris-Sud, 91405 Orsay, France (Reçu le 17 janvier 1978, révisé le 26 mai 1978, accepté le 29 juin 1978)
Résumé. 2014 Nous étudions l’instabilité qui se produit quand un nématique homéotrope est soumis
à un cisaillement elliptique alternatif. Pour cela nous considérons des fluctuations indépendantes
du temps. Nous obtenons le seuil de l’instabilité et déterminons dans quelle direction la structure
en rouleaux apparait.
Abstract. 2014 One studies the instability appearing when a homeotropic sample is submitted to an
alternatif elliptical shear. For that, we consider time independent fluctuations. We calculate the
instability threshold and estimate in which direction the roll structure appears.
Classification Physics Abstracts
61.30 - 47.20
1. Introduction. - Experimental observations of
high frequency shear (few kHz to 10 kHz) instability
in a homeotropic sample of MBBA have been first
reported by Scudieri, Bertolotti, Melone and Albertini [1]. Similar experiments on a homeotropic sample have been done by Pieranski and Guyon [2].
Actually it does not seem very clear that the instabi- lities are the same in both cases. Pieranski and Guyon
noticed the necessity of an elliptically polarized shear
to induce the instability. No such ellipticity effect has
been observed by Scudieri.
In the experiments, two plates are moved in two perpendicular directions (x, y) at a frequency v = mJ2 n.
This corresponds to displacements of the form
The regime we shall describe in this paper corresponds
to the where the penetration depth b = (r¡/ pro) 1/2
is large compared with the sample thickness d (where 1
is some averaged viscosity). Then the sample is
submitted to two uniform shears
Flc.1. - Experimental set up. A nematic is sandwiched between two
rectangular plates. The thickness of the cell is d. The two plates are
moved at a frequency v = co/2 n by imposing displacements of the
form X(t) = Xo cos rot, Y(t) _ - Yo sin wt.
In the following we shall consider that the upper plate
alone is submitted to an elliptical motion defined by { X(t), Y(t)} contrary to the experimental set up.
These two situations are equivalent in the limit à > d.
The purpose of this paper is to develop a detailed
theoretical analysis of the results found by Pieranski
and Guyon [2]. Their observations were made on a
flow cell schematically shown on figure 1.
where
(*) Permanent address : Institut de Physique Expérimentale de l’Université, CH 1015, Lausanne-Dorigny, Suisse.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197800390100103900
1040
The main experimental results are the following :
For low shear rates one observes a uniform regime corresponding to a flow and a distortion uniform in the x-y plane. As the shear exceeds a certain threshold a stationary periodic pattern appears.
A detailed presentation of the experimental obser-
vations is given in reference [2]. However some important points will be emphasized.
1) Below the threshold the director moves, at
frequency w, on an elliptical cone having the axis perpendicular to the glass plates.
2) The threshold appears when the dimensionless number N = Xo Yo coy 1 IK reaches a critical values Ne (K is a typical elastic constant, Y1 the rotational
viscosity, Ne is a complicated function of the various
viscosity ratios). Let us point out that Ne is independent
of the sample thickness.
3) When Xo and Yo are different, one observes,
at threshold, a periodic pattern of rolls similar to Williams domains [13]. The diffraction pattern of a laser beam is one-dimensional : one observes a row
of diffraction spots. This allows to define the angle ç between the rolls axis and the y axis of the ellipse.
The value of ç only depends on the sign of Yo/Xo -1.
For Yo > Xo, qJ = qJo and for Yo Xo, qJ = ço +n/2.
Let us summarize the assumptions we shall make
in the following analysis :
1) The amplitude of the plate oscillations Xo, Yo,
the thickness of the sample d and the hydrodynamic penetration depth à satisfy the inequalities :
2) The elastic constants Kl, K2, K3 are taken to be equal
3) The frequency v of the plate motion corresponds
to the limit ro’rR 1 (where ’rR = y1 d2/K is the
elastic relaxation time).
4) In our analysis, we restrict ourselves to the imme- diate neighbourood of the threshold ; i.e. we make
a linear analysis of the instability.
5) We do not compute the exact z-dependence of
the physical quantities involved (see Fig. 1 for the
definition of z). We assume for all quantities a sinu-
soïdal dependence which cannot fit realistic boundary
conditions. This is the so-called one-mode approxi-
mation which greatly simplifies the computation.
In this paper, we give first a description of the
uniform regime. When the uniform regime becomes unstable, a so-called roll regime develops. We compute the instability threshold and the spatial periodicity
of the rolls and a discussion about the direction of the roll pattern is presented. For the hydrody-
namic description of nematics see Frank [3], Leslie [4],
Ericksen [5], Parodi [6] or the recent text book [13].
2. Uniform régime. - When one applies two alter-
native shears S.,(t) and SY(t) to a homeotropic sample,
the director does not remain aligned along the initial direction Oz, but follows the excitation. The conser-
vation of the angular momentum implies that the
total torque F exerted on the director
is zero [13].
This leads to the two conditions on r x and T y :
As indicated above, we assume isotropy in the
elastic properties (typically K ~ 10-6 CGS for MBBA
at 25 °C).
In the middle of the sample the director follows the excitation but on the bounding plates it remains
fixed due to the strong anchoring at the plate.
Then there are two different regions in the sample :
- in the middle of the sample the director does not depend on z and moves periodically :
- near the plates the orientation depends on z
and an elastic boundary layer of thickness ç appears :
ç2 = K/(Y1 co) (where Y1 is a rotational viscosity) .
This boundary layer may be compared to the sample thickness :
where iR = Y1 d 2/K is a characteristic elastic relaxa- tion time of the director.
(J)’t"R will appear to be a pertinent parameter of the problem. For the frequencies of the applied shear the
elastic boundary layer is negligible since corr » 1.
Typical values in MBBA are : rR - 10 s for d = 100 u and (J) in the range 5 x 102 to 3 x 103 Hz.
The solutions, uniform in space, of eqs. (2.1), (2.2)
are :
n3 -
where
where Q = a2/(a3 - a2) is of order - 1 for MBBA.
The solutions (2.4) and (2.5) appear as an expan- sion in terms of Slco = X jd which will be proved to
be of order (W’tR)-1/2 1 at threshold. In what follows we shall always develop the solutions in powers of X/d and only keep the first relevant non
trivial terms. Then a good description of the uniform
regime is given by the first term of the right hand side of eqs. (2.4), (2.5) and (2.6). In this approximation
the director moves on an elliptical cone as observed experimentally. The higher order terms only slightly modify this motion but will be important to get the angle ç of the rolls.
3. Roll régime. - 3.1 HYDRODYNAMIC EQUATIONS.
Just above a given threshold, the uniform regime
becomes unstable and, as discussed in the introduction,
a periodic pattern of rolls appear in the sample. This
array is stationary ; it corresponds to a distribution of velocities and director orientations which are
periodic in space and do not vanish when averaged
over a period of the plate oscillation. We shall study
this roll regime using standard linear stability ana- lysis [12]. The physics of the effect will be discussed further in this section.
Let us now consider small fluctuations bnx, bny, bnz
of the director around the value na corresponding to
the uniform regime, called in the following unper- turbed state. These director fluctuations will induce small fluctuations of the velocity. Since for the
unperturbed flow only the velocity gradient Sx and Sy play a role, we shall use the following notation : Sx, S,,
for the unperturbed velocity gradients and vx, Vy, V_, for the velocity fluctuations. Stroboscopic observa-
tions [2] indicate that the director distortions ônx and ôny are not time dependent. We shall then consider time independant fluctuations vx, vy, Vz, bnx, bny and
a time-dependent fluctuations bnz. This last point
results from the unitary of n which leads to the relation :
In order to find the relation between the different static fluctuations one can use the force and torque equations averaged over a period of the excitation.
As already mentioned, the instability is due to bulk
forces and torques having a non vanishing component when averaged (noted in the following) over
one period of the excitation. This is expected as the
viscous stress tensor a depends on the velocity gradient
S as well as on the director n. Then, even for shear S
with (5’)=0, (c) may not vanish since n also is time dependent.
Let us develop a linear analysis of the instability
as given in ref. [8] by retaining first order terms in the equations. Prior to this a number of general consi-
derations and order of magnitude estimates are pertinent. The convective terms, of order pSo V, in
the force and torque equations can be neglected compared with the viscous one of order r¡ V/d2.
Indeed the ratio of the two terms is of order dXo/b2
which is much smaller than one in the limits we
consider Xo « d « b. On the other hand, as indicated
in section 2, we shall expand all quantities in power of X/d, keeping only the relevant lower order terms.
The velocity fluctuation, expressed in a dimensionless form as V/rod, is of order (Xold)’ bni (1) (as will be
seen below). Then the consistency in the arguments demands that terms like
should be retained.
The stress tensor components of A are :
one obtains
Note for example that, with the hierarchy defined above, terms such as oVx 6vx
have been neglected , above, terms such as ny a
GY’ n GX
have been ne g lectedy 7y 1 x cl
in (An)x.
To the same order one gets :
Looking for example at the time averaged stress
tensor one sees that two kinds of terms including the
director fluctuations may have non vanishing values.
They are easily located if one considers for example
(1) Where i stands for x or y.
1042
the contribution of the viscosity coefficient a2 to the stress tensor :
the two kinds of contributions may be noted as
one also expects symmetric contributions such as :
the force and torque equations can now be expressed
in terms of (3.1) to (3.4)
where
YI a and nb are the Miesowics viscosities [7]
in a similar way one gets the torque equations :
Let us recall that the instability is characterized
by the presence of rolls having an angle ç with the y
axis. New coordinates (ç, n, z) as shown on figure 2
are more convenient to use, with 1 is parallel to the
Fm. 2. - Above the threshold one observes a pattern with rolls
corresponding to a sinusoidal distortion in the ç direction. (p is the angle between the axis of the rolls and the y direction.
roll axis. With these coordinates all quantities will depend on ç, z but not on il. The linear stability of the
system is analysed in a classical way [12J by considering
normal modes of the form :
wa(ç, z, t) = wa exp 1(qj + kz + rot)
where wa stands for bnn, bnç, vç, vtl, vz.
We moreover assume that the periodic regime is stationary so that w is purely imaginary : then the
threshold is obtained for (J) = 0 (exchange of stability).
This assumption does not seem to contradict expe- rimental observations [2]. A few relations between the quantities S and N introduced in éqs. (3 .1 ) to (3. 4)
should be noted
where
Notice that the quantities characterized by a A
are smaller by a factor (X/d)2 than the other ones.
As we shall see below, one will only need to consider
the lower order terms in X/d to estimate the threshold.
This corresponds to the neglect of all quantities
labelled A in eqs. (3. 10) to (3.13) and the retention
of Q1 as the only relevant parameter. However, higher order terms such as AQ and Ah will be necessary to determine the angle ç.
Let us now give the expressions of the force and torque components in the new coordinates.
3.2 INSTABILITY THRESHOLD. - In order to deter- mine the threshold one uses the simplified eqs. (3.10)
to (3.13) obtained for AQ = Ah = 0
Then the force (eqs. (3.5) to (3.8)) and torque (eqs. (3. 8) (3 . 9)) equations read in the new coordinates
where
Before going on let us do some comments on the
torque equations. In addition to the classical elastic and viscous torques one sees new contributions
F’ > and IGn > to the torques due to th6 rotation
(characterized by Q 1) of the director :
This is similar to a gyroscopic effect due to the rotation of the director : a fluctuation bn4 of the
direction induces a torque r# ) which in turn
creates a fluctuation bnn of the director perpendicular
to bnç (Fig. 3). Due to the symmetry of the problem,
a fluctuation bnn also creates a fluctuation ân,. This
process, considered alone, is stabilizing. The desta- bilizing effect is induced by the viscous torque depend- ing on the viscosity coefficient (X2. For a nematic
with x3 1 1 OC2 I the gyroscopic effect will only give
a small correction to the viscous torque depending
on OC2.
FIG. 3. - Stabilizing gyroscopic effect. Due to the rotation of the director on a elliptic cone with the axis along z : a fluctuation bn, of
the direction induces a fluctuation bn, perpendicular to bnç; the
fluctuation bn, induces a fluctuation ôn, in a direction opposite
to the initial one.
In order to give the rationale for the instability
let us consider a director fluctuation of the form
bn,, - cos (qç) cos (kz). Conditions for non trivial solutions of eqs. (3.18) ... (3.22) yield
Due to the coupling Ql, the motor of the instability,
it can be deduced from eq. (3.18) and (3.20) that the
fluctuation bn" induces velocity fluctuations
Analogously (eq. (3.19)), bnç induces a velocity
fluctuation
This velocity field creates viscous torques (eqs. (3.21)
and (3.22)) which reinforce the initial fluctuation as
it is shown on figure 2.
1044
The mass conservation equation
permits elimination of vç in the force equations (3.18
to 3.20). After elimination of the pressure one obtains the relation between the director fluctuation
8nn and the velocity fluctuation v., as a function
of a = q2/k2 :
where and
Thus the velocity fluctuations are of order na Xo wbn
or na Yo màn, as indicated earlier in this section.
From the torque eqs. (3.21), (3.22) the velocities
can also be eliminated. We get :
’
where
The compatibility of this system leads to the threshold expression :
., A _r")’" ,
At this stage the exact procedure would be the same as the one used in the study of simple shear flow [8].
The following boundary conditions will be considered :
However, this procedure is involved and instead of doing it one can use a one mode approximation
which leads to a good estimate of the threshold in this kind of problem [9, 10]. Instead of taking explicitly
into account all boundary conditions one considers
only one mode in z with k = n/d. This corresponds
to satisfying boundary conditions on vZ, bnn, bnç but
not on vç and vl,. k being fixed, the threshold corres-
ponds to the value of a that minimizes Q 1 given by
eq. (3.25). The plot of the reduced function (QJIKk2)2
in terms of a is given on figure 4. Solutions of eq. (3.25) (Dl Kk2)2 > 0) are obtained only in a certain
range of a values. Typically al a a2, where ai = 1.085 and a2 = 36.64 have been calculated for the MBBA at 25 °C with ce, = 0.065, a2 = - 0.785,
a3 = - 0.012, a4 = 0.832, as = 0.463, fia = 0.238,
lb = 1.035, 11H = 0.040 9, flF = - 0.169 6, in C.G.S.
units.
Fic. 4. - Reduced threshold (QdKk2)2 as a function of a reduced
wave vector a = (q/k). q is the wave vector in the ç direction.
k is the wave vector in the z direction and is taken equal to n/d.
The threshold only exists for wave vectors corresponding to a in the
range (a 1, a2). The threshold is obtained for a,, = 1.75 corresponding
to a wavelength Âx ~ 1.5 d.
z
The threshold corresponds to
and
The numerical value (eq. (3.27)) of the threshold
depends drastically on the different viscosity coeffi-
cients in such a way that the uncertainty on the
threshold value is too large. Typically, using the
values of viscosities and estimated errors for MBBA
[13] one obtains Ne in the range
and
One sees on figure 4 the range of wave vectors i.e. the
wavelength of the distortion corresponding to a
solution. At threshold the wavelength of the rolls lx = 2 dIVa, is in the range (1.52 d to 1.49 d). The
rolls are not completely circular since circular rolls would correspond to lx = 2 d.
Also on figure 2 the shape of the flow lines can be
seen which are not perpendicular to the q direction (roll axes). The distortion of the director is essentially
determined by the viscous torques induced by the
flow velocities v,, and v4.
Remarkably enough the threshold value expressed
in terms of plate displacements is independent of the sample thickness (see eq. (3.27))
.
This relation is in agreement with experimental
results [2] since the mean displacement 10 = (Xo Yo) - 1/2
varies as W-l/2. With use of eq. (3.27) one obtains,
for typical values for MBBA at room temperature
10 in the range (5 y to 8 y) for v = 270 Hz.
This is of the same order of magnitude than expe-
rimental results 10 - 20 y for v - 270 Hz [11].
In the case of circular polarization (Xo - Yo) one
observes experimentally a square pattern at threshold.
One can obtain an estimate of this threshold using
a linear analysis extensively described in [12], similar
to the one used to describe the rolls. For this one
considers fluctuations of the type
The main result (Eq. (3.25)) holds if q is now defined
as :
this implies that, for a square pattern, the edge length 1. is larger than the distance between the rolls
previously described 1,. More precisely
3 .3 ROLL ORIENTATION. - The expression of the
threshold (3.27) has been obtained by solving the dynamical eq. (3.5) to (3.9) with use of the simplified
relations (3.15) to (3.17). The fact that the angle rp did not appear in the new force and torque
eqs. (3.18),..., (3.22) resulted from the antisymmetry properties (3.15),..., (3.17) of S x, and N xy characte-
ristic of a first order approximation in X/d. Since we
are now concerned with the determination of the
angle ç, we need to consider higher order terms and
consider the relations (3.10) to (3.13) where we can
omit the corrective term Ah in the definition of Q since it would only change the threshold expression
in a negligible way. The essential modification
corresponds to the AQ term in eq. (3 .11 ). This will
affect the dynamical eq. (3.18) to (3.22) by intro- ducing the angle 9. Let us write the new forces and torques equations as :
where ( pi’) > and ( ri(1) > are given by eqs. (3.18) to (3.22).
The corrections bf and brl read : b/ç = f/l(COS (2 ({J) a ânlêz +
As in the preceding section, after elimination of the velocities one obtains the relation between the director fluctuations :
where
The compatibility condition of eqs. (3.36) (3.37)
leads to the threshold equation :
where
Here again we shall not solve the exact problem taking
into account all boundary conditions. As in the pre-
ceding section, we shall consider a one mode approxi-
mation corresponding to a fixed wave vector k. Then
the threshold is obtained for the q or a and ç values
which minimize Q.
Since A Q « Q the threshold is still obtained for Q2 = Q’ i.e. for a = ae (eq. (3.28)) the condition
