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On the theory of Rayleigh-Bénard convection in homeotropic nematic liquid crystals
Q. Feng, W. Decker, W. Pesch, L. Kramer
To cite this version:
Q. Feng, W. Decker, W. Pesch, L. Kramer. On the theory of Rayleigh-Bénard convection in homeotropic nematic liquid crystals. Journal de Physique II, EDP Sciences, 1992, 2 (6), pp.1303- 1325. �10.1051/jp2:1992202�. �jpa-00247729�
J. Phys. II France 2 (1992) 1303-1325 JUNE 1992, PAGE 1303
Classification Physics Abstracts
47.25 47.20 61.30G
On the theory of Rayleigh-B4nard convection in homeotropic
nematic liquid crystals
Q.Feng *, W. Decker, W. Pesch and L. Kramer
Physikafisches Institut, Universitit Bayreuth, D-8580 Bayreuth, Germany
(Received 19 February 1992, accepted 23 March 1992)
Rdsumd, Nous reconsiddrons la description th40rique du comportement au seuil et prls du seuil de convection de Rayleigh-Bdnard dans un cristal liquide n6matique en gdometrie hom£otrope. Lorsque le systbme est chaufl6 par le haut et soumis h un champ magn6tique faiblement d4stabilisant, une analyse de stabilit4 des rouleaux explique leur transition dans les 4chantillons rectangulaires connus exp4rimentalement. Dans la m4me configuration mars avec
un champ vertical stabilisant, nous observons le passage d'une bifurcation directe h une bifurca- tion inverse. On trouve, dans un intervalle de grands champs magn4tiques, un regime oh deux
modes stationnaires convectifs avec difl6rentes longueurs d'onde et de symmetrie oppos4e ont le m4me seuil critique. Cela conduit h une comp4tition et une interaction r4sonnante. En cas de chauflage par le bas une instabilit4 oscillante se d4veloppe comme on salt. La nature de la
bifurcation est analys4e pour des rouleaux stationnaires et propagatifs. Une comparaison entre les th60ries pr6c4dentes et les exp£riences est conduite, lorsque c'est possible.
Abstract. A rigorous linear and weakly nonlinear analysis of Rayleigh-B6nard convection in
homeotropically aligned nematic liquid crystals is presented for both adverse convection (heating
from above) and normal convection (heating from below) allowing for the presence of magnetic
fields. In adverse convection we find a transition from forward to backward bifurcation with increasing vertical (stabilizing) field. The stability analysis of roll solutions explains the experi- mentally observed transition between roll and square patterns for weak horizontal (destabilizing)
fields. In a range of high vertical magnetic fields two steady convective modes with different wavelengths and symmetries become critical at nearly the same threshold giving rise to compe- tition and resonant interaction. For the oscillatory instability in normal convection the nature of the bifurcation for both traveling and standing rolls is determined. Comparison ,vith previous
theories and with experiments is made wherever possible.
* Present address: Progran~ in Appfied Mathematics, University of Arizona, Tucson, Arizona 85721, U-S-A.
1. Introduction.
The study of convective instabilities in nematic liquid crystal layers has experienced a revival
during the last years. Thi
reasons for this are mainly the progress in experimental tech-
niques (computer control, image processing), improved theoretical methods (analytic as well
as numerical) and the increased general interest in pattern formation and transition to weak turbulence. Most of the effort has been devoted to electrohydrodynamic convection (EHC) in
planarly aligned nematic layers (see e.g. [I]). Here one can have normal or oblique rolls at threshold (the latter are oriented obliquely with respect to the undistorted director), or in rare
cases a superposition of the two degenerate oblique roll systems which gives rise to rectangles.
It is also well known that in Rayleigh-B4nard convection (RBC) in nematics a number of in-
teresting scenarios are found (see e.g. [2, 3]). Recently RBC in planarly aligned nematics has received renewed attention [4] and experiments in this system are in progress [5].
In this work we investigate RBC in homeotropically aligned nematics. This case is interest-
ing, among others, because it belongs to a different symmetry class: the isotropy in the plane
of the layer makes it similar to classical RBC in simple fluids. Then one may have rolls, squares
or hexagons with arbitrary orientation at threshold, and additional destabilization mechanisms
come into play at a finite distance from threshold [6, 7]. In the homeotropic configuration it is in addition quite simple to break the isotropy by applying a horizontal magnetic field. Sim-
ilar to the planar case convection is enhanced substantially by the heat focusing mechanism
through the anisotropy of the heat conductivity leading even to the c~se of adverse convection
(heating from above) first predicted by Dubois-Violette [8, 9] and observed subsequently [10].
From experiments one knows that without horizontal magnetic field the preferred pattern at threshold are stationary squares [10, 11]. If the field is turned on one finds instead rolls oriented
perpendicular to the field which in a secondary instability transform into rectangles [11].
For normal convection (heating from below) the situation is different. As first recognized by Lekkerkerker [12, 13], the primary instability should be oscillatory (Hopi bifurcation) which was indeed observed experimentally [14]. At high vertical magnetic fields the bifurcation becomes stationary [14]. Thus here one has a scenario reminiscent of binary-fluid convection [15], which
has attracted considerable interest in recent years. An advantage of the liquid crystal system is the possibility to change very easily the magnetic field which plays a role analogous to the
separation ratio in binary fluid mixtures.
The paper is organized as follows. In section 2 we briefly review the general set of equations together with the methods used for the linear and weakly nonlinear analysis. Results on adverse convection are discussed in section 3. We present a complete linear analysis allowing
for a vertical magnetic field (Subsect.3.I) as well as a weakly nonlinear analysis that gives the parameters of the Ginzburg-Landau (amplitude) equation (Subsect.3.2). In subsection 3.3 we show that rolls are unstable with respect to the cross-roll instability leading to a square pattern.
A weak horizontal magnetic field, however, stabilizes rolls near threshold and their stability
range is discussed. In section4 the results for the linear and weakly nonlinear analysis (complex Ginzburg-Landau equation) of normal convection are presented and discussed. Comparison
with previous theories and with experiments is made wherever possible. Section 5 is devoted to some concluding remarks. The material parameters for the nematic MBBA used in the
calculations are listed in the Appendix.
2. Formulation of the problem.
We consider a layer of nematic liquid crystal of thickness d confined between two infinite, hori- zontal plates with homeotropic configuration: the director at the plates is aligned uniformly in
N°6 RAYLEIGH-BENARD CONVECTION IN HOMEOTROPIC NEMATICS 1305
the vertical direction. The upper and lower plates are held at diRerent constant temperatures, Tu and 2j, respectively. The layer is either heated from below (2l > Tu, normal connection)
or from above (2j < Tu, adnerse connection). We choose the coordinate system (z, y, z) with the origin in the midplane of the layer and z-axis directed vertically upward. The convective
motions are described by the heat transport equation, torque balance equation and the velocity equation (generalized Navier-Stokes equation) together with the continuity equation and the constitutive relations. These equations are described in detail in references [16-18] and have been collected in an earlier paper [4]. The notations used there will be adopted. We shall apply the Boussinesq approximation in which all the material parameters (heat diffusivities
~cjj and
~c i, parallel and normal to the director; elastic constants kii, k22 and k331 viscosity coefficients al, a2,
,
a6) are assumed to be constant with the exception of the mass density p, whose temperature dependence, p = po[I a(T To)] with To = (2j + Tu)/2, is taken into account in the buoyancy force only. This approximation is reasonable provided that the temperature
diRerence across the layer is not too large.
In order to write the equations in a dimensionless form we measure length in units of the thickness d and time in units of the vertical heat diffusion time d2/~cjj. The heat diRusivities, the
viscosity coefficients and the elastic constants are scaled as
~c i = ~c[~cjj, ~ca = ~cjj ~c i = ~c[~cjj;
ai = a( (a4/2), I = 1,2,
,
6 (note that a4/2 is the isotropic part of the viscosity) kit = k(; k33, I = 1,2, 3. The conventional temperature scaling is given by # = #'(a4/2)~cjj/(agpod~) (g
=
gravitational acceleration) with # the deviation from the linear temperature profile in the
conducting state. One is then left with three important dimensionless parameters: the Rayleigh
number R, which is the dimensionless temperature difference between the upper and lower
confining plates and thus acts as the main control parameter, the Prandtl number Pr and a material parameter F which is essentially the ratio of the director relaxation and vertical heat diRusion times,
~ ~~j~~~~~cjj~~' ~ ~~~' ~ ~~~'~~~
~~'~~
R is negative for adverse convection. For the material MBBA which will be adopted as a
standard in this paper one has Pr
= 545 and F
= 476. An additional control parameter is provided by a magnetic field applied either in the vertical direction (Hz, stabil12ing the
unperturbed state) or in a horizontal direction (H~, destabilizing). When the field is applied in the horizontal direction, chosen parallel to the z-axis, an anisotropy is turned on that breaks the rotational symmetry around the vertical axis. The magnetic field H is scaled by the bend-
Frdedericksz transition field Hi
Hz(~) = hz(~)Hr, Hf =
jl. (2.2)
PoXa We shall always assume h~ < I.
In the numerical analysis it is convenient to represent the velocity v by two scalar potentials f and g according to
v =
if + ig, (2.3)
where the operators I and I
are defined by I
= (0~0z, 0y0z, -0( 0(), I = (by, -0~, 0).
This representation ensures automatically the incompressibility V v = 0. Applying the oper-
ators I and I to the velocity equation yields two equations for f and g where the pressure is
eliminated. The two relevant equations for the director u
= (n~, ny, nz) are obtained from the torque balance equation by projecting the torque r onto a convenient local coordinate system
F.jfxn)=0, F.j-(n.f)uj=0, (2.4)
where f is the unit vector in the y direction. Together with the heat equation we are then left with 5 partial differential equations for the 5 variables: #, nz, ny, f and g. The z-compmient of the director, nz, results from the normalization relation n~
= I. This general set of equation is always satisfied by the homogeneous solution # = n~ = ny = f = g = 0. The equations can
then be expanded and written symbolically in the form
LV + N2(V(V) + N3(V[V[V) + = [Bo + Bi(V) + 82(V(V) + ~, (2.5)
where V denotes the column vector V
= (#, n~,ny, f, gl'~ (the superscript T denotes trans-
position). The components of vector operators N2 (N3) are quadratic (cubic) in V and its
spatial derivatives, whereas the quaniities L and Bi are matrix differential operators of the order in V indicated. The expansion has been done with the help of MACSYMA. The explicit form of the equation (2.5) is too lengthy to present here and we shall give the linear part only (see below). In the following the primes denoting the dimensionless quantities will be left out.
The equations are to be supplemented by the boundary conditions
#=n~=ny=f=0zf=g=0 at z=+1/2. (2.6)
We shall mostly use these realistic rigid boundary conditions and only sometimes refer to results based on the stress-free boundary conditions
#=nz=ny=f=0jf=0zg=0 at z=+1/2. (2.7)
The first step of the study involves the linear analysis which yields the growth rate ofmodes.
One examines the linearized version of equation (2.5),
L(0~,0y,0z; R)V = Bo(0z,by,0z)01V. (2.8)
Written out in detail it reads (in dimensionless units)
(-0t + ~c1A2 + 0j)# ~caR0znz ~caR0yny RA2f
" 0,
(~g) (ki10] + k220( + 0j F710t + h] hj)nz + (kii k22)0~0yny+
F(a3A2 a20j)0zf Fa20y0zg
= 0, (2.10)
(kii k22)0~0ynz + (k220( + ki10( + 0j F710t h])ny+
F(a3A2 a20j)0y f Fa20~0zg
= 0, (2.ll)
[-V~0t/Pr + ) ((a6 + 2 + a3)A2 + (as + 2 a2)0j)V~ +
al A20j]A2f
-A24 + (a20j o3A2)o~otnz + (a20j a3A2)oyotny
= 0, (2.12)
a20y0z0tnz a20~0z0tny + [-A201/Pr + Al + )(as + 2 a2)A20j]g
= 0, (2.13)
where A2 + 0] + 0( and V~ % 0] + 0( + 0j.
N°6 RAYLEIGH-BENARD CONVECTION IN HOMEOTROPIC NEMATICS 1307
The above system of equations is solved by a Galerkin method. Due to translational invari-
ance in the z-y-plane and the boundary conditions (2.6), we choose the trial functions in the
following form
4nsn(Z)
M nznsn(Z)
V = £ nynsn(z) e"+'q'~
% Vo(q, z)e~'+~~'~, (2,14)
n=I fncn(Z) gusn(Z)
with x
= (z,y), wave vector q = (q,p) and truncation parameter M. The Sn(z) are taken
as trigonometric functions and Cn(z) as the Chandrasekhar functions satisfying the boundary conditions Sn = Cn
= 0zCn
= 0 at z = + 1/2:
Sn(Z) " Sillil~~(~ + ~/2)1
cash (z>(n+i~j~) cos (z>~n+i~j~)
~~~ ~ ~~~
~~~~~ ~1~~~((~~~~~~~in ~ij~~~~~~~~~~
Slnh ('~n/2) Slll ('~n/2) ~~~ ~ ~~~~
The parameters ii, pi are given in the appendix V of jig]. With (2.14) equations (2.9-2.13)
are transformed into an eigenvalue problem
L(q,0z; R)Vo = lBo(q>0z)Vo. (2.15)
We then multiply equations (2.9-2.13) with Sm, Sm, Sm, Cm and Sm, respectively, for m =
I,..., M and integrate in z across the layer. One obtains a homogeneous 5M x 5M algebraic system for the expansion coefficients which is solved by standard eigenvalue packages.
When the Rayleigh number R is increased at some fixed q, the real part a of one of the
eigenvalues
I(q, R) = a(q, R) + iw(q, R). (2.16)
becomes positive while the real parts of all other eigenvalues remain negative. The homoge-
neous rest state V
= 0 then loses stability with respect to convective motion character12ed by the eigenmodes
Vo(q, z)e'~'~~~~~~~'
The neutral surface R = Ro(q) is given by the condition of vanishing growth rate a(q, R)
= 0.
Minimizing Ro(q) with respect to q gives the threshold Rc = Ro(qc) with the critical wave vector qc = (qc,pc) and the critical frequency wc = w(qc), which is zero for a stationary bifurcation while nonzero for a Hopf bifurcation. Note that in the isotropic case (h~ = 0) the
direction of qc is undetermined.
Once the linear problem is solved, one proceeds with the weakly nonlinear analysis for R
slightly above Rc. Here we consider the case of roll solutions where only a single wave vector q (chosen in the z-direction, I-e- p
= 0) contributes. We will, however, also be dealing with linear disturbances of rolls and with more general slow modulations expressed conveniently in the framework of the Ginzburg-Landau equation formalism [20, 21]. The solution is expanded
in the complete set of the linear eigenvectors:
V = / dql(q, t)Vo(q, z)e~~'~+'~~~~' + c-c- + h-o-t- D+
% dqA(q, t)Vo(q, z)e'~'~ +
c-c- + h-o-t-, (2.17)
~+
where Vo(q, z) denotes the linear eigenvector corresponding to the eigenvalue with the largest
real part for given q and R and A(q,t) is the amplitude. The integration domain D+ is a small area centered at qc determined by the condition a(q, R) m 0 which need not be specified
in detail. The contributions of all other eigenvectors are considered as higher-order terms
(h,o.t.). The expression (2.17) represents a solution Ansatz for traveling waves. The standing
wave solution can be readily constructed by a superposition of two waves with equal amplitudes travelling in opposite directions.
Before projecting onto the subspace spanned by the linearly unstable or weakly damped modes, one needs to define a scalar product and to construct an adjoint eigenvalue problem.
By a straightforward partial integration of the original eigenvalue problem one can find the explicit form of the matrix in the adjoint problem and the boundary conditions for the adjoint eigenvector. In the framework of the truncated Galerkin-expansion the adjoint problem is
obtained by taking the transposition and conjugate of the coefficient matrix of the original
one. The projection procedure corresponds to a reduction to the center manifold and leads
directly to the generalized Ginzburg-Landau equation in the leading order.
~~~~~~~~'~~ ~~~~~~~~'~~~
(2.18) fD dqi fD dq2 a3(q> qi, q2)A(qi,f)A(q2>f)A(q qi q2,f),
where D
= D+ U D- with D- being an analogous small area centered at -qc.
A solution constructed from a finite width wave packet centered around qc is usually repre- sented as a modulation of the periodic solution of wave vector qc. The modulation amplitude A(z,t) is defined through the following Fourier transformation:
A(x,t) = / dqA(q,t)e~~'~e~~~c'~+~~C' (2.19)
D+
Approximating a3(q,qi,q2) by its value evaluated at threshold and transforming equation (2.18) back into real space, we obtain the well-known Ginzburg-Landau equation for a traveling
wave (in the one-dimensional case)
To(01 n~0~)A = f(I + ice)A + (~(l + ici)0]A + c3(A(~A (2.20)
with
~
~ Rc' ~ ~~
~i~
~
' ~~ ~'
~~ ~i~
~° ~~ ~i' ~~ ~jc ~j~~' ~~ (2 ~'
C3 "
( [a3(~c>~c>~c) + a3(~c>~c>~~c) + a3(~c> ~~c> ~c)] (~'~~)
~l ~c
The derivatives in (2.21) can be expressed in terms of ai(q) and a2(q), and are evaluated at threshold. The sign of the real part of c3 determines the bifurcation type: one has a forward bifurcation if Re(c3) < 0 and a backward bifurcation if Re(c3) > 0.
In the presence of right- and left-traveling waves one has two coupled Ginzburg-Landau equa-
tions which also allow to describe standing waves (see Sect. 4.2). The generalization of equa- tion (2.20) to two dimensions is obtained in the isotropic case (hz = 0) by the replacement[20]
°z ~ °z + I°(/(2qc).
In the cases of a steady forward bifurcation we analyze the stability of roll solutions with
respect to arbitrary disturbances. The lowest-order stationary amplitude As(q) with a single
N°6 RAYLEIGH-BENARD CONVECTION IN HOMEOTROPIC NEMATICS 1309
wave vector q is obtained from equation (2.18).
'~~~~~'~ "
a3(q, q, q) + a3(/()~q)
+ a3(q, -q, ql' ~~'~~~
The roll solution Vr is determined up to second order in the amplitude
Vr = Vir + V2r> (2.24)
~~~~~
Vlr " As(~)~O(~> Z)~~~~ + ~'~" ~~'~~~
and V2r denotes the solution of the order O((As(~) which is obtained from equation (2.5) (one
has to solve for V2r an inhomogeneous linear equation with N2(Vir(Vir) as the inhomogeneity):
V2r = v2(2q,z)e~~~'~ + v2(-2q, z)e~~~~'~ + v2(0, z). (2.26)
Then we superimpose an infinitesimal disturbance onto the stationary roll solution
V = Vir + bvi + V2r + b'T2, (2.27)
where
bvi " e~~ [sivo(q + «, z)e~(~+"l'~ + s2Vo(-q + «, z)e'(~~+"l'~)
,
(2.28)
~v ~At jj~ ~i(2q+«).X ~ j~ ~i(-2q+«).X ~ ~£ ~i«.Xj (~~g)
with « = (ax,ay) being the modulation wave vector. bV2 also depends on z and has the same symmetry as V2r. It includes among others large scale disturbances (in the term To when « is small. Linearization of the general equation (2.5) with respect to the disturbances leads to a
homogeneous system which is solved numerically by the Galerkin method for the eigenvalues I(q, f,«). Positive Re(I) signalizes instability. For details see reference [4].
3. Adverse convection.
3.I THRESHOLD BEHAVIOR. When the nematic layer is heated from above, we always
found
a steady bifurcation in agreement with previous work [9, 22] (we used a standard set of parameters for the material MBBA, see Appendix). The neutral surface R = Ro(q) is then
given by the solvability condition for equation (2.15) with 1
= 0. We shall consider the case of vanishing horizontal field, h~ = 0 but allow for finite stabilizing field hz # 0), except in section 3.3 below where a weak magnetic field is applied along the z direction. So here the system has rotational symmetry around the z-axis. The analysis is therefore simplified considerably by choosing q in the z-direction (p = 0), and this leads to a two-dimensional
calculation with ny = g = 0.
In figure I the critical Rayleigh number Rc and critical wavenumber qc are shown as functions of the stabilizing magnetic field hz. There exist two classes of eigenmodes: one is symmetric (marked by "s" in the figure) and the other antisymmetric (marked by "a") with respect to
reflection in the midplane z = 0 of the cell. In fact from equations (2.9-2.13) it is evident that
(#n, n~n, fn) with odd n (symmetric mode) and with even n (antisymmetric mode) enter the
equations separately (in the case p # 0, ny would
come in with the same symmetry as n~, while g with the opposite), hence the equations give rise to symmetric and the antisymmetric modes, respectively. At not too high magnetic fields the threshold of the symmetric mode (Rc = -7.64
