Dynamics and Stability of Systems, Vol. 14, No. 3, 1999
The Freedericksz transition as a bifurcation
problem
G. I. Blake1 , T. Mullin 1 and S. J. Tavener 2
1
Department of Physics and Astronomy, Schuster Laboratory, Manchester University,
B r unswick Street, Manchester M13 9PL, UK
2
Department of Mathematics, Penn State University, University Park, PA 16802, USA
(Received August 1998; ® nal version Febr uar y 1999)
Abstract. We consider the eþ ects of the application of electric and magnetic ® elds to a
homogeneously aligned nematic liquid cr ystal which lies between parallel plates. The
average alignment of the ¯ uid molecules is known as the director and it is set parallel to
the top and bottom bounding surfaces. At a critical ® eld strength a Freedericksz transition
occurs which results in a distortion of the director ® eld. We discuss this phenomenon from
the point of view of bifurcation theor y and use numerical bifurcation techniques to explore
it. Imperfections in the cell, either in the applied ® eld or the director orientation at the
boundaries, are also considered. Finally, we investigate the eþ ect of ® nite aspect in a twodimensional version of the problem.
1 Introduction
The application of bifurcation theory to a variety of problems in physics and
applied mathematics has led to a more complete understanding of how complicated
non-linear behaviours arise in these systems (Golubitsky & Schaeþ er, 1985). This
approach has been successfully applied to a wide variety of problems in both ¯ uid
and solid mechanics and to chemically reacting systems. An example of this is the
so-called Taylor± Couette ¯ ow of an isotropic, Newtonian ¯ uid between concentric
cylinders, for which quantitative as well as qualitative agreement between experiment, numerical computations and predictions based on bifurcation theory has
been obtained, as reviewed by Mullin and Kobine (1996). Using the governing
equations for the ¯ ow of such ¯ uids, the Navier± Stokes equations, and numerical
techniques developed at AEA, Harwell, it proved possible to calculate solutions
and bifurcation points which occurred as the control parameters of the problem
were varied. Not only was precise quantitative agreement obtained between calculation and experiment, but also, the numerical techniques were used to uncover
unstable and therefore unobservable solutions. These in turn were shown to have
a direct bearing on the origins of more complicated behaviour including lowCorrespondence to T. Mullin, Department of Physics and Astronom y, Schuster Laboratory, Manchester
U niversity, Brunswick Street, Manchester M13 9PL , U K.
ISSN 0268-111 0 (print)/ISSN 1465-338 9 (online)/99/030299-3 3
€ 1999 Taylor & Francis Ltd
300
G . I. BLAK E ET AL.
dimensional chaos, and the combination of theory, calculation and experiment
proved to be a signi® cant one. For many other physical situations, although the
governing equations are known, they may be extremely diý cult to solve in practice.
Bifurcation theory may still provide insight in such cases, though it may be only
qualitative and localized in nature.
In the present study we apply these powerful techniques to a nematic liquid
crystal aligned uniformly between parallel plates, to which either a magnetic or
electric ® eld is applied. A nematic liquid crystal can be considered as a ¯ uid
which contains rod-like molecules which have orientational alignment without
any positional order. The dielectric and magnetic properties of the material are
anisotropic and we will be concerned with materials where the diþ erence between
the susceptibilities parallel and perpendicular to the molecules is positive. The
average alignment of the molecules is called the director and is prescribed to be
parallel to the top and bottom bounding surfaces in our work. This form of
alignment is called homogeneous, whereas when the average molecular alignment
is normal to the boundary, it is called homeotropic. As is well documented (see,
e.g. de Gennes & Prost, 1993), at a critical value of the strength of the magnetic
or electric ® eld, a static distortion of the nematic occurs and this phenomenon is
often referred to as a Freedericksz transition. It is this problem which we discuss
from the perspective of bifurcation theory and demonstrate that these methods can
also be applied to an anisotropic, non-Newtonian ¯ uid.
The Freedericksz transition that occurs when a magnetic ® eld is applied across
a thin layer of a nematic con® ned between parallel plates has attracted considerable
attention, partly due to its utility as a means of measuring elastic constants. By
considering extrema of an integral expression for the free energy of the sample,
both Saupe (1960) and later Dafermos (1968) determined the critical magnetic
® eld strength above which non-trivial solutions can exist for planarly aligned
nematic liquid crystals. They were also able to show that above this critical ® eld
strength the non-trivial solution in which the directors are no longer uniformly
parallel to the plates is the free energy minimizer. The related problem, in which
the nematic is initially aligned normal to the walls and a magnetic ® eld is applied
parallel to the walls, was shown to diþ er only in that the roles of the bend and
splay elastic constants are reversed. Leslie (1970) con® rmed these results by posing
the problem in terms of the conservation of linear and angular momentum, and
then applied this new approach to two diþ erent con® gurations. He considered the
Freedericksz transition which occurs in a nematic con® ned in the annular gap
between concentric cylinders in which the directors are initially aligned tangentially
and to which a radial magnetic ® eld is applied. He further considered the transition
which occurs when a nematic is con® ned in a wedge with the directors initially
aligned radially and to which a tangential magnetic ® eld is applied. In each case he
determined a critical ® eld strength above which a non-trivial solution can exist and
showed using the free energy of the sample, that above the critical ® eld strength
the distorted con® guration is the energy minimizer. Once again, the heterogeneously aligned problem, with the magnetic ® eld at right angles to the initial
director con® guration, was shown to be equivalent once the roles of splay and bend
elasticity were reversed. Derfel (1988) showed that the Freedericksz transition that
occurs when a magnetic ® eld is applied normal to a nematic sample con® ned
between parallel plates arises at a pitchfork bifurcation point. He then considered
the qualitative aspects of the unfolding of this pitchfork bifurcation produced by
imperfect alignment at the plates and when the magnetic ® eld is applied at an
FR EEDERICKSZ TR ANSITION
301
angle other than 90 deg to the plates. Leslie (1970) completed the trio of
Freedericksz transitions between parallel plates by considering the transition which
occurs when a magnetic ® eld is applied in such a manner that its tendency to align
the molecules is balanced by the elastic resistance to twist. Schiller (1989) developed
a perturbation expansion for the solutions which arise beyond transition in this
scenario.
The Freedericksz transition which occurs when an electric rather than a magnetic
® eld is applied across a planarly aligned nematic sample con® ned in the narrow
gap between two parallel plates has been examined by both Deuling (1972) and
by Gruler and Meier (1972). Both studies looked for solutions which minimized a
free energy and both neglected the conductivity of the sample. They not only found
a critical value for the electric ® eld above which the trivial solution loses stability
to a non-uniform con® guration, but successfully compared calculations of the
director angle as a function of the applied ® eld with experimental measurements
of the shift in the birefringence pattern. The experimental measurements were seen
to depart from the theoretical values for large voltages and for large values of the
dielectric anisotropy. Gruler and Meier conjectured this departure to be the result
of the non-uniform ® eld which they could only approximate accurately for small
deformations of the director ® eld. They present plots of the non-linear electric
® eld between the plates which are qualitatively similar to those resulting from our
own ® nite-element computations. In common with the magnetic case, the roles of
the dielectric constants parallel and perpendicular to the director and the splay and
bend elastic constants are reversed when the nematic is aligned normal to the
plates and the electric ® eld is applied parallel to the plates. Deuling (1978)
completed the trio of standard scenarios between parallel plates when he considered
the case in which the aligning eþ ects of an electric ® eld are balanced by the elastic
resistance to twist.
To model the behaviour of the liquid crystal we use the static continuum theory
of Ericksen and Leslie (see, for example, Leslie (1992)), which we outline in the
next section. In Section 3 we discuss qualitative aspects of the Freedericksz
transition and in Section 4, and Appendix A, we describe the application of the
® nite-element method to the Freedericksz transition with an applied magnetic ® eld.
In Section 5 and Appendix B, we extend these numerical techniques to examine
the analytically less tractable situation when an electric ® eld is applied to the cell.
We extend previous analyses to include the eþ ect of mobile ions in the sample and
the coupling between dielectric and conductivity anisotropies, and we investigate a
fully two-dimensional version of the problem.
2 Static continuum theory for nematics
Cartesian tensor notation is used throughout, where double subscripts denote sums
and commas denote diþ erentiation with respect to the relevant spatial variables.
A vector n is introduced to describe the orientation of the anisotropic axis. This
vector, commonly referred to as the director, is assumed to lie parallel to the
average, local orientation of the molecules and is subject to the constraint
ni ni 5
1
(1)
where n is physically indistinguishable from 2 n for the non-chiral nematics
considered here. Using a balance of work equation, Leslie (1992) formulates
concisely the dynamic continuum theory for these materials, from which it is
302
G . I. BLAK E ET AL.
relatively straightforward to deduce the static version. In this case, we simply have
the equation
( )
­ W
­ n i, j
2
,j
­ W
+ G i + c ni 5
­ ni
i, j 5
0,
1, 2, 3
(2)
where W is the local stored energy function, G is the external orientation body force
per unit volume and c is a Lagrange multiplier arising from the constraint (1). This
equation governs the orientation of n subject to elastic and external ® eld eþ ects.
The local stored energy function W is introduced to model elastic eþ ects and is
assumed to depend solely upon the direction n and its gradients. By assuming that
W depends quadratically upon the gradients of n one arrives at (Frank, 1948)
2W 5
K 1 (n i,i ) 2 + K 2 (n i e ijk n k,j ) 2 + K 3 n i,p n p n i,q n q + (K 2 + K 4 ) (n i, j n j,i 2
(n i,i ) 2 )
(3)
where the elastic coeý cients, K i , are constants at ® xed temperatures and satisfy
the inequalities (Ericksen, 1958)
K 1 > 0,
K 2 > 0,
K3 > 0
(4)
The constitutive equation for G has diþ erent forms depending upon whether a
magnetic or electric ® eld is applied to the liquid crystal. We consider both situations
and then G takes the form
Gi 5
D v 5
D v H j nj H i ,
v
k
2
v
(5)
^
in the case of an applied magnetic ® eld, whilst for electric ® elds we have
Gi 5
D e 5
D e Ej nj Ei ,
e
k
2
e
(6)
^
The applied magnetic and electric ® elds are represented through the vectors H
and E, respectively. The constants D v and D e measure the diþ erences between the
magnetic and dielectric susceptibilities parallel to the director, v k and e k , respectively, and perpendicular to the director, v ^ and e ^ , respectively. In general, and
for all the cases considered here, D v and D e are both positive.
For applied magnetic ® elds, equation (2) is suý cient to describe the behaviour
of the nematic as the magnetic anisotropies of nematics are very small in practice,
and as a result it is reasonable to assume that the eþ ect of the material on the
applied magnetic ® eld is negligible. On the other hand, D e is not generally small,
so that a coupling between the liquid crystal and the applied electric ® eld arises.
Thus the liquid crystal distorts the applied electric ® eld pattern in a way which
must be calculated. Further, nematic liquid crystals possess free charges (de
Gennes & Prost, 1993), so we must also take the charge density, q e , of the material
into consideration. We therefore need to introduce two more equations when we
consider electric ® elds, these being Poisson’ s equation
e
0
D i,i 5
q
(7)
e
and the continuity of charge equation
ji,i 5
0
(8)
where e 0 is the permittivity of free space, and the electric displacement D and the
current density j are given by
Di 5
e
^
Ei + D e Ej nj ni ,
ji 5
r
^
Ei + D r Ej nj ni ,
D r 5
r
k
2
r
^
(9)
FR EEDERICKSZ TR ANSITION
303
where r k and r ^ are the conductivities of the material parallel and perpendicular
to the director, respectively (see Kaiser & Pesch, 1993). The electric ® eld is of
course subject to the constraint
e ijk E k,j 5
0
(10)
where the third-order tensor e is the alternator.
Our approach extends that of previous studies (for a review see Dunmur &
Toriyama, 1998) which do not allow for the presence of mobile ions in the
sample and assume instead that D i ,i 5 0. This simpli® cation allows the free-energy
minimization problem to be reduced to a suitably scaled version of the problem
when a magnetic ® eld is applied (see, e.g. de Gennes & Prost, 1993, p. 135). Our
numerical techniques permit the exploration of the coupling between the director,
the local charge density and the electrical ® eld.
3 The Freedericksz transition
We consider a nematic liquid crystal con® ned between parallel plates and uniformly
aligned so that the director is everywhere perfectly parallel to the surfaces, as shown
schematically in Fig. 1.
It is well known that if a magnetic ® eld is applied perpendicular to the plates, or
a potential diþ erence is applied across them, a distortion of the nematic alignment
occurs above a critical magnetic or electric ® eld strength provided the magnetic or
dielectric anisotropies are positive. In other words, below the critical ® eld strength
the director pro® le is unperturbed and remains uniform, but above this value a
distortion occurs. This eþ ect is due to a competition between restoring elastic
forces induced by the alignment at the boundaries and destabilizing torques
produced by the external ® eld. The external ® eld produces destabilizing torques
since the nematic materials we consider have positive magnetic or dielectric
anisotropies, promoting the director to lie parallel to the direction of the applied
® eld. Below the critical ® eld strength the destabilizing torques are insuý cient to
overcome the elastic forces and the director pro® le remains uniform. However,
above this critical ® eld strength, the torques arising due to magnetic or electric
anisotropies overcome the elastic forces and the director pro® le becomes distorted.
This phenomenon is commonly known as a Freedericksz transition.
The Freedericksz transition can be considered as an example of a symmetry
breaking (pitchfork) bifurcation. The symmetry being broken is a re¯ ectional
(anti)symmetry about the mid-plane. Other examples of such bifurcations are
discussed in Mullin (1995). In our problem, let the angle which the director in the
Fig. 1. Schem atic for the Freedericksz transition.
304
G . I. BLAK E ET AL.
Fig. 2. A pitchfork bifurcation diagram.
centre of the cell makes with the plates be denoted by h 0 and the ® eld strength by
k . Then h 0 5 0 for k < k c , say, as shown schematically in Fig. 2. At k 5 k c two nontrivial solutions appear corresponding to clockwise and anticlockwise rotations of
the director, both of which are stable. The trivial solution becomes an unstable
solution of the equilibrium equations at this point and a distortion in the director
® eld occurs. In other words, as we increase the parameter k steadily from zero, the
trivial solution exchanges stability with one or two possible non-trivial solutions at
k 5 k c . Note that due to the symmetry of this idealized, perfect cell about the
central plane parallel to the bounding surfaces, either of the clockwise or anticlockwise distortions are equally likely to occur, and the bifurcation is a pitchfork.
Figure 2 illustrates the exchange of stability between the undistorted and distorted
states, the dashed line representing the fact that the situation h 0 5 0 is still a possible
con® guration for k > k c , but an unstable one. The distorted solution branches
approach either p /2 or 2 p /2 in the limit as k ® ` .
In the following sections we show numerically that the qualitative description of
the Freedericksz transition applies equally to the magnetic and electric ® eld cases.
We also discuss the more physically relevant case where the symmetry about the
central plane of the cell is broken. As is well known from singularity theory,
imperfections which arise naturally in physical systems can be modelled mathematically independently of the precise nature of the imperfection. In our computations
we consider the eþ ects of a non-perpendicular ® eld in the magnetic case and
imperfect anchoring in the electrical one.
4 Application of a magnetic ® eld
In this section we consider the liquid crystal cell just described, to which a magnetic
® eld is applied at a ® xed angle, w , to the normal of the plates. The direction of the
305
FR EEDERICKSZ TR ANSITION
® eld is assumed to lie in the plane perpendicular to the plates which contains the
undistorted director pro® le. That is
H5
p
2
H(sin w , 0, cos w ),
2
<w <
p
(11)
2
where the x-axis is de® ned to be parallel to the plates and the undistorted director
con® guration and the z-axis is perpendicular to the plates. Two cases will be
discussed: w 5 0 and w ¹ 0. In the ® rst case the problem has a re¯ ectional symmetry
about the mid-plane of the cell, but for ® nite values of w this symmetry is broken.
This latter instance is of more practical signi® cance, as in any experiment of this
type the magnetic ® eld will always be at some non-zero angle to the normal of the
plates.
We assume that distortions in the director pro® le caused by the application of
such a magnetic ® eld are entirely uniform in planes parallel to the plates and
therefore seek solutions of the form
n5
h 5
(cos h , 0, sin h ),
h (z)
(12)
The constraint (1) is satis® ed automatically. From (2), (3), (5) and (11) we
then have
2
K3
(
() )
(
( ) )
(
( )
2
d 2h
dh
sin h + 3
dz 2
dz
K1
d 2h
cos h 2
dz 2
+ K3
+D
cos h
dh
dz
sin 2 h + D v H 2 sin(h + w ) sin w
2
5
0
(13)
2
sin h
d 2h
sin h cos h +
dz 2
v H sin(h
+ c cos h
dh
dz
2
cos 2 h 2
+ w ) cos w + c sin h
5
( ) )
dh
dz
2
sin2 h
sin h
(14)
0
and upon elimination of c
K1
(
d 2h
cos h 2
dz 2
( ) )
dh
dz
2
sin h
cos h + K 3
(
d 2h
sin h +
dz 2
( ) )
dh
dz
2
cos h
+ D v H 2 sin(h + w ) cos(h + w ) 5
sin h
(15)
0
We specify that strong anchoring conditions hold at the boundaries, that is the
director is not free to move at the plates. Taking the x-axis to be equidistant and
parallel to the bounding surfaces and the gap between the plates as d, the boundary
conditions read
h (2
d /2) 5
We note for future reference that when w 5
h (z) 5
h (d /2) 5
0
(16)
0 the trivial solution
0
(17)
is a solution of (15) subject to (16) for all ® eld strengths H. However, there is no
such trivial solution for non-zero w .
306
G . I. BLAK E ET AL.
4.1 Numerical method
The results we present in the following sections have all been obtained using the
numerical package ENTW IFE (Cliþ e, 1996), which has been developed at AEA,
Harwell. The package utilizes the ® nite-element method to solve systems of nonlinear partial and ordinary diþ erential equations and implements modern numerical
continuation and bifurcation algorithms.
We ® rst recast (15) in weak form. Multiplying (15) by a test function H (z) which
vanishes at z 5 6 d /2, then integrating by parts over the interval [ 2 d /2, d /2] we
arrive at
ò {[
d /2
2
(K 3 2
d /2
K1 )
( )
dh
dz
]
2
sin h cos h + D v H 2 sin(h + w ) cos( h + w ) H
2
(K 1 cos2 h + K 3 sin 2 h )
d h dH
dz dz
}
(18)
dz 5
0
We non-dimensionalize this equation by introducing the new variable
zÄ 5
z
d
(19)
and multiplying through by d2 /K 1 , to obtain
ò {[
1/2
2
1 /2
(l 2
1)
( )
dh
dzÄ
]
2
sin h cos h + k sin(h + w ) cos( h + w ) H
2
(cos 2 h + l sin2 h )
d h dH
dzÄ dzÄ
}
(20)
dzÄ 5
0
where
l 5
K3
and k 5
K1
D v H 2d 2
K1
(21)
From (4), l is positive. We ® nd that this parameter does not have any marked
eþ ect on what follows, changing the quantitative but not the qualitative nature of
the solutions. As a result we assume that l is unity, i.e. K 1 5 K 3 , throughout.
There are now two main parameters in our problem which we are free to vary.
These are k and w . We vary k and associate an increase or decrease of k with a
corresponding increase or decrease of H 2 . We thus assume that D v , d and K 1 are
® xed. We also regard w as being free to vary, but usually consider a speci® c value
of this parameter. In the nomenclature of bifurcation theory, k and w are the
distinguished and secondary parameters, respectively.
A ® nite-element discretization of (18) as described brie¯ y in Appendix A, results
in a ® nite-dimensional algebraic system of equations
f(u, k ) 5
0,
f :RN3
Rj
RN
where u is the vector containing the coeý cients of the N basis functions w i(zÄ ) and
de® nes the ® nite-element approximation h h (zÄ ) of h (zÄ ) via equation (A16). If (and
307
FR EEDERICKSZ TR ANSITION
only if ), w 5 0 and the ® nite-element grid is symmetric about zÄ 5 0, this non-linear
system of equations is equivariant with respect to an orthogonal matrix SÃ , i.e.
f (SÃ u, k ) 5
SÃ f (u, k )
where SÃ ¹ I and SÃ 2 5 I. Details appear in Appendix A. The orthoganal matrix SÃ
induces a unique decomposition of R N into symmetric and antisymmetric subspaces
RN 5
R Ns
R Na
where
{u Î R N : SÃ u 5
R Ns 5
u}
and
R Na 5
{u Î R N : SÃ u 5
2
u}
At a simple symmetry-breaking bifurcation point (u c , k c )
u c Î R Ns
Null(f cu ) 5
span{a},
Range(f cu ) 5
{y Î R N : w
a Î R Na , a ¹
T
y5
0},
0
w
Î R N, w ¹ 0
where
­ f
(u c , k c )
­ u
f cu 5
The Freedericksz transition occurs at such a symmetry-breaking bifurcation point
and was computed as a regular solution of the extended system described by
Werner and Spence (1984), namely
( )
f
F(v) 5
fu a
l a2
T
5
(22)
0
1
where
vT 5
(u T , a T , k ),
F : R Ns 3
and l Î R N with l ¹
R Na 3
R1®
v Î (R Ns 3
R Ns 3
R Na 3
R Na 3
R 1)
R1
0.
4.2 A perfect pitchfork bifurcation
W hen w 5 0 the magnetic ® eld is perpendicular to the plates and the governing
equations are equivariant with respect to a re¯ ection about zÄ 5 0. In this case a
Freedericksz transition occurs at a critical strength of the magnetic ® eld at a
symmetry breaking bifurcation point. Equation (22) was solved using a ® niteelement mesh on the interval zÄ Î [ 2 1/2, 0] comprised of one-dimensional elements
with quadratic interpolation within each element. A convergence study is shown in
308
G . I. BLAK E ET AL.
Table 1. The location of the num erically determined bifurcation point k Å c , and the relative error E,
using N elements with quadratic interpolation within each elem ent
No. of elem ents, N
2
4
8
16
32
64
128
256
Bifurcation point, k Å
9.8746590 26
9.8699277 89
9.8696247 35
9.8696056 74
9.8696044 81
9.8696044 06
9.8696044 01
9.8696044 01
c
Relative error, E
5.1214 3
3.2766 3
2.0603 3
1.2897 3
8.0956 3
4.9647 3
1.0132 3
1.0132 3
10 2
10 2
10 2
10 2
10 2
10 2
10 2
2
10
4
5
6
7
9
10
11
11
Ratio, E(N) /E(N /2)
Ð
6.3979 3
6.2879 3
6.2598 3
6.2771 3
6.1326 3
2.0408 3
1.0
10 2
10 2
10 2
10 2
10 2
10 2
2
2
2
2
2
2
Table 1. The theoretical value is k c 5 p 2 (see de Gennes & Prost, 1993, p. 128)
and the relative error is de® ned as ½ k Å c 2 k c ½ / k c . It can be seen that the relative error
converges to zero with rate O (h 4 ) where h is the mesh spacing. A convergence
criterion of 102 12 was used.
The computed bifurcation diagram for the Freedericksz transition is shown in
Fig. 3, which is a plot of the angle of de¯ ection of the director with respect to the
plates in the centre of the cell against the parameter k . The upper branch
corresponds to an anticlockwise rotation of the director towards the applied ® eld
while the lower branch corresponds to a clockwise rotation. Note that as k increases,
the branches tend towards 6 p /2 as expected.
The calculated director pro® les h (zÄ ) corresponding to the upper branch of the
Fig. 3. Computed bifurcation diagram for an applied m agnetic ® eld with w 5
0.
309
FR EEDERICKSZ TR ANSITION
Fig. 4. Director pro® les h (zÄ ) on the upper branch: (a) k 5
10; (b) k 5
40; (c) k 5
90; (d) k 5
160.
bifurcation diagram are shown in Fig. 4 for k 5 10, 40, 90 and 160. These solutions
were calculated using a grid of 160 elements on the interval zÄ Î [ 2 1/2, 1/2]. The
parameter value k 5 10 is just past the bifurcation point and the associated solution
is a slight perturbation of the trivial solution and is essentially sinusoidal in shape.
The other parameter values correspond to approximately two, three and four times
the critical ® eld strength at which the Freedericksz transition occurs. It can be seen
that a greater proportion of the nematic is aligned in the direction of the ® eld as k
increases and the shape is far from sinusoidal.
4.3 A disconnected pitchfork bifurcation
The perfect pitchfork bifurcation described in the last section is of course a
mathematical abstraction which can never be realized in any practical situation, as
any experiment contains imperfections. These imperfections come from many
sources, for example, non-perfect alignment of the liquid crystal, non-parallel
plates and imperfections in the material. We have introduced the angle w , the
amount by which the direction of the applied magnetic ® eld deviates from the
normal to the plates, to model the eþ ect of imperfections on the behaviour of the
Freedericksz cell. The theory of imperfect bifurcations (Golubitsky & Schaeþ er,
1985) tells us that it is reasonable to model possibly multiple and complicated
imperfections in this manner.
The trivial solution h (zÄ ) 5 0 is now no longer a solution for k > 0 and a distortion
occurs for any value of k . The computed bifurcation diagram with w 5 0.02 rad,
approximately 1 deg, is shown in Fig. 5. The axes are the same as in the previous
bifurcation diagram. It is immediately obvious that the perfect pitchfork of the
previous section has become disconnected. The upper branch is known as the
primary solution branch (Benjamin, 1978) and indicates that there is a distortion
of the director pro® le even in the absence of an applied ® eld. For k increasing
steadily from zero, the primary solution, corresponding to anticlockwise rotation
310
G . I. BLAK E ET AL.
Fig. 5. Bifurcation diagram for an imperfect Freedericksz experiment w ith w 5
0.02.
of the director, is the only one available to the cell. This branch is straightforward
to calculate and is achieved by gradually increasing the value of the distinguished
parameter k . Two other solutions exist for k suý ciently large and cannot be reached
by a smooth increase in ® eld strength. The solution associated with the top part of
the lower, disconnected branch is unstable and therefore unobservable. The bottom
part of the lower, disconnected branch is a stable solution but one which can only
be reached by suddenly switching on the magnetic ® eld.
The disconnected branch can be computed by setting w to zero and increasing
the distinguished parameter to some value greater than k c , meanwhile ensuring
that the solution remains on the trivial branch. Upon increasing w from zero we
remain on the unstable part of the now disconnected bifurcation diagram. Keller
arclength continuation (Keller, 1977) can then be used to ® nd the other points on
the disconnected branch.
For negative values of w the situation is reversed and the primary solution branch
corresponds to clockwise rotations of the director and the disconnected branch to
anticlockwise rotations. The bifurcation diagram for w 5 2 0.02 is simply a re¯ ection of Fig. 5 about the k -axis.
The calculated stable director pro® les h (zÄ ) are shown in Fig. 6 for the connected
and disconnected branches when k 5 50 and w 5 0.02 rad. The disconnected
branch at this value of w only exists for k > k L P 5 11.21. Once the cell is in a state
corresponding to the stable part of the disconnected branch and the ® eld strength
is reduced continuously, the magnitude of the clockwise rotation decreases continuously until at the limit point where k 5 k LP the cell returns catastrophically to the
anticlockwise solution branch. As w is increased or decreased the disconnection
between the two solution branches increases or decreases, respectively, and the
position of the limit point also varies as a consequence. The computed locus of
limit points is shown as a function of the secondary parameter w in Fig. 7. The
location of the limit point can be computed as w is varied using the extended
system described by Moore and Spence (1980). The curve is a cusp which is
aligned with the k -axis with vertical asymptotes at w 5 6 p /2, and results from an
FR EEDERICKSZ TR ANSITION
Fig. 6. Director pro® les h (zÄ ) for k 5
50 and w 5
311
0.02: (a) prim ary branch; (b) disconnected branch.
unfolding of the pitchfork. The qualitative nature of the unfolding of the pitchfork
bifurcation for small imperfections has been described previously by Derfel (1988)
who performed a normal form analysis. A similar result to this has previously been
obtained in Rayleigh± Be nard convection (Cliþ e & Winters, 1984) in Newtonian
¯ uids which conform to the Boussinesq approximation, where a tilt of the convection cell provided the unfolding parameter.
5 Application of an electric ® eld
Now that we have illustrated how ideas from bifurcation theory can be applied to
the simple Freedericksz transition with respect to a magnetic ® eld, we move on to
discuss the more complicated situation where an electric ® eld is applied. In this
case, instead of a single variable, namely the angle the director makes with the
312
G . I. BLAK E ET AL.
Fig. 7. (a) Locus of lim it points k
LP
on the disconnected branch. (b) T ip of the cusp.
plates, the variation of the electric ® eld and the charge density in the sample must
also be taken into consideration. We also introduce a second spatial dimension into
the problem and demonstrate that the ideas we presented in the previous section
for an ODE apply equally well for a system of partial diþ erential equations. To this
end we allow the variables to depend upon x as well as z, the x-axis being associated
with the direction the director is pointing in before the electric ® eld is applied (see
Fig. 1). It is assumed that in the experiment, the nematic is a perfect insulator
containing mobile charges and that the sample is separated from the electrodes by,
for example, suitable insulating layers to eliminate charge injection.
To satisfy the constraint (10) we assume that E can be expressed as
E5
(
2
­ u
, 0, 2
­ x
)
­ u
,
­ z
u
5
u (x, z)
(23)
313
FR EEDERICKSZ TR ANSITION
and then the three variables in our problem are the electric potential, u , the charge
density, q e , and h , the angle the director makes with the plates. All these variables
are assumed to be functions of x and z. Again, elastic eþ ects contribute towards
the quantitative rather than the qualitative behaviour of the system and so we set
K 1 5 K 3 5 K.
With (12) (where now h 5 h (x, z)) and (23) as our ansatz for n and E, we obtain
the governing equations
2
K
(
) (
­ 2h
­ 2h
+ 2 +D e
2
­ x
­ z
2
e
2
(
^
D e
2
r
2
)
­ 2u
­ 2u
+ 2 2
2
­ x
­ z
­
­ z
(
^
D e
)
­ z
((
­
­ x
((
D r
­
­ x
­ u
cos h
­ z
­ u
­ u
cos h +
sin h
­ x
­ z
) )
sin h
5
0
(24)
cos h
5
e
e
0
­ u
­ u
cos h +
sin h
­ x
­ z
) )
)
(25)
q
sin h
((
­ u
­ u
cos h +
sin h
­ x
­ z
­ u
sin h 2
­ x
) )
­ u
­ u
cos h +
sin h
­ x
­ z
­ 2u
­ 2u
2
+
­ x2 ­ z2
­
D r
((
)(
­ u
­ u
cos h +
sin h
­ x
­ z
) )
cos h
(26)
5
0
from (2), (3), (6), (7) and (8).
We assume the plates to have length l in the x-direction and that a constant
potential diþ erence is applied at the plates which are perfect conductors, and hence
u (x, 2
d /2) 5
2
u
0
/2, u (x, d /2) 5
u
0
/2, q e (x, 2
d /2) 5
q
e
(x, d /2) 5
0, 2
l /2 <
x<
l /2
(27)
where u 0 > 0 is the voltage. (The motivation for this particular choice of boundary
conditions for u will become apparent in Section 5.1 where we show that this
choice supports a `trivial’ solution for which the electric potential u (x, z) is an odd
function of z, and the trivial solution is therefore symmetric with respect to the Z 2re¯ ection symmetry de® ned in equation (B3)).
Initially we assume that the director is strongly anchored at the boundaries so
that it is parallel to the plates. That is
h (x, 2
d /2) 5
h (x, d /2) 5
0,
2
l /2 <
x<
l /2
(28)
First we impose stress-free boundary conditions of the form
­ h
­ x
5
(2
­ h
­ x
l /2, z) 5
(l /2, z) 5
­ q e
(2
­ x
l /2, z) 5
­ q e
(l /2, z) 5
­ x
­ u
(2
­ x
l /2, z)
­ u
(l /2, z) 5
­ x
(29)
0
for 2 d /2 < z < d /2.
This boundary value problem is again recast in weak form by ® rst multiplying
314
G . I. BLAK E ET AL.
each of (24), (25) and (26) by test functions H (x, z), W (x, z) and U (x, z) which
satisfy
H (x, 2
d /2) 5
W (x, 2
H (x, d /2) 5
d /2) 5
U (x, 2
W (x, d /2) 5
for 2 l /2 < x < l /2, and integrating over the (x, z) Î [ 2
ducing the non-dimensional variables
x
,
l
xÄ 5
z
,
d
zÄ 5
uÄ
u
5
,
u
e
0
q
u
0e ^
U (x, d /2) 5
[2
l /2, l /2] 3
d2
e5
qÄ
d /2) 5
0
(30)
d /2, d /2]. Intro-
(31)
e
0
and the parameters
e
k 5
^
2
0
u
K
e 5
,
e
e
k
2
r
r 5
1,
r
^
2
k
C 5
1,
^
d
l
(32)
and integrating by parts as appropriate, we ® nd the weak forms of (24), (25) and
(26) to be
ò ò [(
1/2
2
1/2
1/2 2
­ h ­ H
­ h ­ H
+
­ xÄ ­ xÄ
­ zÄ ­ zÄ
2
C
1/2
+
1/2
2
e k
(
(
1/2
ò ò [
1/2 2
2
qÄ eW
1/2
2
(
ò ò [
1/2
2
e
1 /2 2
2
C
­ u ­ W
­ u ­ W
+
­ xÄ ­ xÄ
­ zÄ ­ zÄ
­ u
­ u
cos h +
sin h
C
­ xÄ
­ zÄ
1/2
­ u ­ U
­ u ­ U
+
­ xÄ ­ xÄ
­ zÄ ­ zÄ
2
C
1/2
+
r
(
)(
­ u
­ u
cos h +
sin h
­ xÄ
­ zÄ
2
C
)
2
C
) (
)
2
e
C
­ W
sin h
­ zÄ
) (
+
­ u
sin h 2
­ xÄ
r
­ u
­ u
cos h +
sin h
C
­ xÄ
­ zÄ
2
C
)
2
­ u
cos h
­ zÄ
)]
H
­ u
­ u
cos h + C
sin h
­ xÄ
­ zÄ
]
)
(33)
dxÄ dzÄ 5
0
­ W
­ xÄ
cos h
(34)
dxÄ dzÄ 5
0
­ u
­ u
cos h + C
sin h
­ xÄ
­ zÄ
­ U
sin h
­ zÄ
]
)
cos h
­ U
­ xÄ
(35)
dxÄ dzÄ 5
0
The boundary conditions (27), (28) and (29) become
u Ä (xÄ , 2
1/2) 5
­ h
­ xÄ
(2
2
1/2,
1/2, zÄ ) 5
5
u Ä (xÄ , 1/2) 5
h (xÄ , 2
1/2) 5
­ qÄ e
(2
­ xÄ
1/2, zÄ ) 5
­ qÄ e
(1/2, zÄ ) 5
­ xÄ
q Ä e (xÄ , 2
1/2,
h (xÄ , 1/2) 5
­ uÄ
(2
­ xÄ
1/2) 5
0
0
(36)
(37)
1/2, zÄ ) 5
­ uÄ
(1/2, zÄ ) 5
­ xÄ
q Ä e (xÄ , 1/2) 5
­ h
­ xÄ
(1/2, zÄ )
(38)
0
315
FR EEDERICKSZ TR ANSITION
where 2 1/2 < xÄ < 1/2 and 2 1/2 < zÄ < 1/2. The `natural’ boundary conditions (38)
arise from the integration by parts used in deriving (33), (34) and (35).
This boundary value problem was solved using the ® nite-element method on a
grid comprising quadrilateral elements with biquadratic interpolation of all the
variables within each element. We regard k as the distinguished parameter and e as
the secondary parameter. Symmetry considerations are presented in Appendix B.
5.1 Pitchfork bifurcation
The set of boundary conditions (27), (28) and (29) allows a trivial solution
h (x, z) 5
0, u Ä (x, z) 5
z, q Ä e (x, z) 5
2
0,
l /2 <
x<
l /2,
2
d /2 <
z<
d /2
(39)
for (24), (25) and (26) for all values of k . This solution has re¯ ectional (anti)symmetry about the line z 5 0 and we found the behaviour of the system to be similar
to that of the corresponding magnetic ® eld problem of Section 4.1. That is, by
setting e , r and C to some ® xed, physically meaningful values, we found that for k
less than k c the trivial solution given by (39) is stable, but at k c a pitchfork bifurcation
occurs and this solution exchanges stability with two non-trivial solutions. Again
these non-trivial solutions correspond to clockwise and anticlockwise rotations of
the director towards the direction of the applied ® eld. A bifurcation diagram was
plotted and found to be qualitatively similar to Fig. 3. Our non-dimensionalization
introduces a known relationship (see de Gennes & Prost, 1993, p. 135) between
the value of k c and e , that is
k
c
5
2
p
e
(40)
The inclusion of sample conductivity and the presence of free ions does not aþ ect
the critical ® eld strength for the Freedericksz transition due to a decoupling which
occurs when linearizing about the trivial solution. In Fig. 8 the computed values
of k c at discrete values of e are shown as solid circles and the theoretical value is
given as the solid curve. The close correspondence between the numerical and
analytical results provides con® dence in our computational procedures.
The calculations of the symmetry breaking bifurcation point were performed on
the grid (xÄ , zÄ ) Î [ 2 1/2, 1/2] 3 [ 2 1/2, 0]. Solutions along the stable branches for
k > k c were calculated (xÄ , zÄ ) Î [ 2 1/2, 1/2] 3 [ 2 1/2, 1/2] using a grid containing
two elements in the xÄ -direction and 96 in the zÄ -direction. Only two elements were
required in the xÄ -direction since the boundary conditions (27), (28) and (29) allow
quasi-one-dimensional solutions in which all xÄ derivatives of the solution vanish.
Of the 96 elements in the zÄ -direction, 32 were placed in the interval
2 0.5 < zÄ < 2 0.3, 32 in the interval 2 0.3 < zÄ < 0.3 and 32 in the interval
0.3 < zÄ < 0.5. This grid re® nement was used since the charge density can possess
steep gradients close to the bounding surfaces.
The computed solution for C 5 1, r 5 0.5, e 5 0.3 and k 5 115 is shown in Figs
9(a)± (c) where we have taken a slice through our two-dimensional domain along
the line xÄ 5 0. All other parallel slices are identical. For this value of e , k c » 32.9.
In Fig. 9(a) we have plotted the director angle h against zÄ . It can be seen that h is
symmetric about the mid-plane of the cell and the distorted solutions are qualitatively similar to the solutions obtained in the previous section, when a magnetic
rather than an electrical ® eld was applied. We have chosen the branch for which
316
G . I. BLAK E ET AL.
Fig. 8. Computed critical ® eld strength k
c
against scaled dielectric susceptibility anisotropy e . The solid
line is k 5 p 2 / e .
the director is rotated towards the bottom plate. The plot of h against zÄ along the
other stable branch for the same values of the parameters can be obtained by
re¯ ecting Fig. 9(a) about h 5 0. The charge density q Ä e is plotted against zÄ in Fig.
9(b). The charge density has extrema close to the boundaries of the cell where
there is a build-up of charge, positive at one plate, and negative at the other. The
rate of change of the electric potential in the zÄ -direction (­ u Ä /­ zÄ ) is plotted against
zÄ in Fig. 9(c). Plots of the charge density and rate of change of the electric potential
along the other stable branch for the same values of the parameters are identical.
The solution for the electric potential u Ä against zÄ is only a slight perturbation from
the linear pro® le that exists before the bifurcation occurs and is therefore not
shown.
The pro® les of h , q Ä e and ­ u Ä /­ zÄ along xÄ 5 0 are shown in Figs 10(a)± (c),
respectively, for C 5 1, r 5 0.5, e 5 0.7 and k 5 75 ( k c » 14.1). The director rotation
is again symmetric about zÄ 5 0 and we have again chosen the branch along which
the director is rotated towards the bottom plate. The director rotation along the
other stable branch can be obtained by re¯ ection in h 5 0. Notice that the sign of
the charge density near the upper and lower plates in Fig. 10(b) is reversed
compared to the case when e 5 0.3 shown in Fig. 9(b), while Figs 9(c) and 10(c)
are qualitatively similar. Plots of q Ä e and ­ u Ä /­ zÄ against zÄ along the other stable
branch at the same values of the parameters are identical to Figs 10(b) and (c).
For such quasi-one-dimensional solutions, equations (25) and (26) (after division
by e ^ and r ^ , respectively) become
2
e
­
­ z
(
­ u
sin 2 h
­ z
)
5
q
­ 2u
2 +
e
­ z
0e
e
^
317
FR EEDERICKSZ TR ANSITION
and
2
r
­
­ z
(
)
­ u
sin 2 h
­ z
5
­ 2u
­ z2
Hence
e
1+
5
r
q
e
0e
e
u
^
(41)
zz
and immediately
q
q
e
e
u
5
e
0 if
q
e
1
e
> 0 if > 1
r
zz
u
5
r
e
< 0 if < 1
r
zz
This behaviour is independent of h and is observed in Figs 9(b) and (c), and in
Figs 10(b) and (c). In Fig. 9(b), q Ä e is positive for zÄ < 0 and negative for zÄ > 0,
while in Fig. 9(c), ­ u Ä /­ zÄ is a decreasing function for zÄ < 0 and an increasing
function for zÄ > 0. Hence q e and u zz have opposite sign, as is appropriate when
e < r . A comparison of Figs 10(b) and (c) shows that q e and u zz have the same
sign, to be expected with e > r . W hen e 5 r , the computed charge density was
identically zero as required.
5.2 Imperfect anchoring conditions
To introduce an imperfection into the system when an electric ® eld is applied, a
diþ erent approach is required from the magnetic ® eld case. The simplest way to
account for imperfections when an electric ® eld is applied to the nematic slab is to
assume that the alignment of the director is not perfect at one or both of the
boundaries, and we introduce another parameter a to describe the misalignment
at the bottom boundary. The boundary conditions at the bottom plate then read
h (xÄ , 2
1/2) 5
a ,
2
1/2 <
xÄ <
1/2
(42)
These boundary conditions do not support Z 2 -invariant solutions, and in particular (39) is no longer a solution. The introduction of imperfection into the system
leads to a disconnection of the pitchfork bifurcation and one obtains a diagram
similar to Fig. 5. As the parameter a is increased or decreased the disconnection
becomes larger or smaller accordingly and upon following the position of the limit
point on the disconnected branch a cusp qualitatively similar to that of Fig. 7 is
achieved. The position of the tip of the cusp, which occurs when a 5 0, varies with
the value of e and the cusp is again aligned with the k -axis with vertical asymptotes
at a 5 6 p /2.
5.3 Two-dimensional solutions
We now consider a truly two-dimensional version of the problem in which the gap
has length l in the x-direction and in which we apply strong anchoring on the end
walls such that there is homeotropic alignment at x 5 6 l /2. In doing so we
318
G . I. BLAK E ET AL.
319
FR EEDERICKSZ TR ANSITION
Fig. 9. Quasi-one-dimensional solution for C 5 1, r 5 0.5, e 5 0.3 and k 5 115: (a) director angle h (zÄ );
(b) charge density q Ä e (zÄ ); (c) rate of change of electric potential (­ u Ä /­ zÄ ) (zÄ ).
destroy the quasi-one-dimensional behaviour observed above. We retain boundary
conditions (36) and (37), but on xÄ 5 6 1/2 we apply
h (2
­ qÄ e
(2
­ xÄ
1/2, zÄ ) 5
5
­ uÄ
(2
­ xÄ
1/2, zÄ ) 5
­ qÄ e
(1/2, zÄ ) 5
­ xÄ
1/2, zÄ ) 5
­ uÄ
(1/2, zÄ ) 5
­ xÄ
h (1/2, zÄ )
(43)
0
for 2 1/2 < zÄ < 1/2. These boundary conditions support Z 2 -invariant solutions of
the governing equations and the Freedericksz transition again occurs at a Z 2symmetry breaking bifurcation point. The critical value of the electric ® eld at
which the transition occurs is now a function of the aspect ratio C , and in Fig. 11
for r 5 e 5 0.5, we see that k c approaches the value p 2 /e (the dashed line in Fig.
11) as C increases, as is expected.
Plots of the director orientation, normalized charge density and electric potential
are shown for C 5 1, r 5 0.5, e 5 0.3, k 5 97.5 and C 5 1, r 5 0.5, e 5 0.7, k 5 38.5
in Figs 12(a)± (c) and 13(a)± (c), respectively. These solutions were computed using
32 3 32 quadrilateral elements with biquadratic interpolation of the angle, charge
density and electric potential on each element. In both ® gures, the contours of h
are equally spaced between 0 and 0.7, the contours of q Ä e are equally spaced between
2 0.35 and 0.35, and the contours of u Ä are equally spaced between 2 0.5 and 0.5.
By a similar argument to that above
r
e
5
1+
e
q
0e ^
(u
e
xx
+u
zz
)
(44)
320
G . I. BLAK E ET AL.
FR EEDERICKSZ TR ANSITION
321
Fig. 10. Quasi-one-dimensional solution for C 5 1, r 5 0.5, e 5 0.7 and k 5 75: (a) director angle h (zÄ );
(b) charge density q Ä e (zÄ ); (c) rate of change of electric potential (­ u Ä /­ zÄ ) (zÄ ).
Fig. 11. Critical ® eld strength k
c
against aspect ratio C for r 5
k 5 p 2 /e .
e 5
0.5. The horizontal dashed line is
322
G . I. BLAK E ET AL.
FR EEDERICKSZ TR ANSITION
323
Fig. 12. Two-dim ensional solution for C 5 1, r 5 0.5, e 5 0.3, k 5 97.5, and (xÄ , zÄ ) Î [ 2 1/2, 1/2] 3 [ 2 1/
2, 1/2]. (a) Director angle h (xÄ , zÄ ). Contours are equally spaced between 0 and 0.7. (b) Charge density
q Ä e (xÄ , zÄ ). Contours are equally spaced between 2 0.35 and 0.35. (c) Electric potential u Ä (xÄ , zÄ ). Contours
are equally spaced between 2 0.5 and 0.5.
hence
q
q
(u
+u
xx
zz
)
e
0 if
5
r
1
> 0 if > 1
r
e
e
xx + u
5
e
e
q
(u
e
zz )
< 0 if < 1
r
Any perturbations to the boundary conditions which do not allow symmetric
solutions disconnect the pitchfork bifurcation point as for the one-dimensional case.
6 Conclusions
We have successfully applied ideas from singularity theory coupled with numerical
bifurcation methods to explore the static instability known as the Freedericksz
transition in nematic liquid crystal. The eþ ects of both magnetic and electric ® elds
have been considered and we have also investigated the physically relevant situations
where imperfections are present. This approach has allowed us to uncover important details of the mechanisms involved such as the charge distributions. These are
diý cult to explore in the laboratory because of the small length scales involved
and have de® ed previous analytic approaches.
The detailed numerical investigation of such complex materials is of signi® cant
324
G . I. BLAK E ET AL.
FR EEDERICKSZ TR ANSITION
325
Fig. 13. Two-dim ensional solution for C 5 1, r 5 0.5, e 5 0.7, k 5 38.5, and (xÄ , zÄ ) Î [ 2 1/2, 1/2] 3 [ 2 1/
2, 1/2]. (a) Director angle h (xÄ , zÄ ). Contours are equally spaced between 0 and 0.7. (b) Charge density
q Ä e (xÄ , zÄ ). Contours are equally spaced between 2 0.35 and 0.35. (c) Electric potential u Ä (xÄ , zÄ ). Contours
are equally spaced between 2 0.5 and 0.5.
practical relevance since the Freedericksz transition is important in device applications. Hence we now have a powerful numerical tool which can readily be used to
investigate the eþ ects of material properties for example. We have already initiated
such an investigation and considered the case where the dielectric anisotropy
D e < 0. Here electrohydrodynamic convection occurs in the cell and the induced
¯ ow has been considered. The basic mechanism for this eþ ect is the motion of
ions through the cell. This situation is beyond the scope of this paper but our
ability to calculate the director pro® le, the electric potential and the distribution of
charge in the cell for D e > 0 is encouraging.
Acknowledgements
The authors are grateful to Andrew Cliþ e for useful discussions. GIB would like
to thank the Leverhulme Trust for ® nancial support. SJT acknowledges the ® nancial
support from NFS grant DMS-9704714. This work was undertaken whilst SJT
was visiting the Computing Laboratory at Oxford University.
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liquid crystals. Physical Review E 48, 451 0 ± 4528.
Keller, H. B. (1977 ) Numerical solution of bifurcation and nonlinear eigenvalue problem s. Applications
of B ifurcation Theor y (ed. P. H. Rabinowitz; Academic Press, New York), pp. 359 ± 384.
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Leslie, F. M . (1992) Continuum theory for nematic liquid crystals. Continuum M echanics Thermodynamics 4, 167 ± 175.
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Journal of Numerical Analysis 17, 567 ± 576.
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Journal of Numerical Analysis 21, 388 ± 399.
Appendix A. Freedericksz transition applying a magnetic ® eld
The problem described has a re¯ ectional (anti)symmetry about the mid-plane
parallel to the bounding surfaces when w 5 0. Consider the balance of torque
equation (15), which we write as
F( h ; H) 5
K1
+ K3
+D
(
(
d 2h
cos h 2
dz 2
d 2h
sin h +
dz 2
v H 2 sin(h
+
( ) )
( ) )
2
dh
dz
dh
dz
sin h
cos h
cos h
sin h
+
0
2
w ) cos( h
w )5
(A1)
327
FR EEDERICKSZ TR ANSITION
where h (z) has the necessary smoothness. We now show that for w 5
SF( h ; H) 5
0
F(S h ; H)
(A2)
for the re¯ ection operator
2
S h (z) 5
h (2
(A3)
z)
i.e. F is equivariant with respect to a symmetry operator S ¹
Note that
d
[S h ] 5
dz
d
[2
dz
h (2
z)] 5
d
[ h (2
dz
cos[S h ] 5
cos[ 2
h (2
z)] 5
cos[ h (2
d2
[S h ] 5
dz 2
z)],
sin[ 2
sin[S h ] 5
z)],
d2
[ h (2
dz 2
2
h (2
I such that S 2 5
I.
z)]
z)] 5
2
sin[h (2
z)]
hence
2
(
(
K3
( ) )
( ) )
d 2h
cos h +
dz 2
K1 2
F(S h ; H) 5
d 2h
sin h +
dz 2
dh
dz
2
sin h
cos h
2
dh
dz
sin h 2
cos h
D v H 2 sin h cos h
and
SF( h ; H) 5
2
2
K1
K3
(
(
() )
( ) )
d 2h
cos h 2
dz 2
d 2h
sin h +
dz 2
dh
dz
dh
dz
2
sin h
cos h
2
sin h 2
cos h
D v H 2 sin h cos h
Clearly
cos[S h + w ] 5
cos[ 2
h (2
z) + w ] ¹
cos[h (2
sin[S h + w ] 5
sin[ 2
h (2
z) + w ] ¹
2
sin[h (2
z) + w ]
z) + w ]
so equation (A1) is equivariant with respect to S if and only if w 5 0.
Boundary conditions (16) support Z 2-invariant solutions, i.e. solutions that are
invariant under the action of S.
Now consider the non-dimensionalized weak form (20) which we write as
ò {[
1 /2
a( h ; H , k ) 5
2
2
1/2
(l 2
1)
( )
dh
dzÄ
]
2
sin h cos h + k sin(h + w ) cos( h + w ) H
d h dH
(cos 2 h + l sin 2 h )
dzÄ dzÄ
}
(A4)
dzÄ 5
0
where h (zÄ ), H (zÄ ) Î V, the Hilbert space of functions whose (generalized) ® rst
derivatives are square integrable over the interval [ 2 1/2, 1/2] and whose function
values vanish at zÄ 5 6 1/2.
328
G . I. BLAK E ET AL.
Applying the Riesz Representation Theorem (Brenner and Scott, 1994), we now
de® ne the operator
A : V3
Rj
V
by
A( h ; k ) 5
Î V
á Q, H ñ ; H
Q where a( h ; H , k ) 5
(A5)
and á ´, ´ñ denotes the inner product. At a solution of (A1), we have
A( h ; k ) 5
0
(A6)
Now, we more carefully de® ne the symmetry operator (A3) as
S :Vj
V
by
2
S h (zÄ ) 5
h (2
zÄ )
(A7)
from (A4)
a(S h ; H , k ) 5
a( h ; SH , k )
(A8)
0. Further, by de® nition of á ´, ´ñ
if and only if w 5
á Sa, bñ 5
á a, Sbñ ; a, b Î V
(A9)
From (A5)
SA( h ; k ) 5
SQ where a( h ; H , k ) 5
á Q, H ñ ; H
P where a(S h ; H , k ) 5
á P, H ñ ; H
Î V
De® ne
A(S h ; k ) 5
Î V
Now
Î V 5 a( h ; SH , k )
a(S h ; H , k ) ; H
á P, H ñ 5
by (A8), so
á P, H ñ 5
Î V 5 á SQ , H ñ
á Q , SH ñ ; H
(A10)
by (A9). Hence
P5
SQ
and we have
SA( h ; k ) 5
A(S h ; k )
That is, the operator A is equivariant with respect to S.
Finally, upon discretization of (20) we have
ò {[
1/2
ah (h h ; H
h
,k )5
2
2
(l 2
1/2
(cos 2 h
h
1)
( )
dh h
dzÄ
2
sin h
h
cos h
}
d h h dH h
dzÄ
+ l sin 2 h h )
dzÄ dzÄ
h
+
k sin(h
h
+ w ) cos(h
h
+
]
w ) H
h
(A11)
329
FR EEDERICKSZ TR ANSITION
where h h(zÄ ), H
h
(zÄ ) Î V h Ì
V and V h is ® nite dimensional. De® ne the operator
Ah : Vh 3
Rj
Vh
by
A h (h h ; k ) 5
Q h where a h ( h h ; H
h
á Qh , H
,k )5
h
ñ ; H
h
Î Vh
(A12)
and á ´, ´ñ is the inner product. At a solution of (A1), we have
A h (h h ; k ) 5
0
(A13)
Provided the ® nite-dimensional subspace V h is constructed so that the symmetry
operator S maps V h on to itself, that is
S : Vh j
(A14)
Vh
then
a h (S h h ; H
h
, H) 5
a h ( h h ; SH
h
, H)
A ® nite-dimensional subspace V h that is invariant under the re¯ ection operator S
can be ensured by constructing a ® nite-element mesh that is symmetric about
zÄ 5 0. Since
á SQ h , H
h
ñ 5
á Q h , SH
h
ñ
by repeating the previous argument we have
SA h ( h h ; k ) 5
A h (S h h ; k )
(A15)
The operator A h is also equivariant with respect to S.
Now let
h
h
(zÄ ) Î V h 5
R
n
ui w
i5
i
(zÄ )
(A16)
1
where w i(zÄ ), i 5 1, . . . , n form a basis for V h . Assuming that the space V h spanned
by w i(zÄ ), i 5 1, . . . , n is invariant with respect to S, i.e. that equation (A14) is
satis® ed, and numbering basis functions w i such that
Sw
then for any h
h
i
2
5
w
n + 12
i
Î Vh
R
h5
Sh
i5
n
2
u n + 12 i w
(A17)
i
1
De® ning the vector u by
u5
this can be written as
SÃ u 5
(
2
0
0
0
0
:
:
2
0
1
[(u i )] i5
¼
¼
2
0
2
1
1
0
:
:
¼
0
0
¼
0
0
1
0
1,. . . ,n
)
u
330
G . I. BLAK E ET AL.
where SÃ is a ® nite-dimensional representation of S on V h . De® ning
á A h ( h h , k ), w i ñ , i 5
fi 5
1, . . . , n
equation (A13) becomes
f (u, k ) 5
(A18)
0
Further, since
f i (SÃ u, k ) 5
á Ph , w i ñ 5
á Qh , S w i ñ
by (A10), then
á Qh , 2
f i (SÃ u, k ) 5
w
n + 12
2
ñ 5
i
f n + 12 i (u, k )
Thus
f (SÃ u, k ) 5
SÃ f (u, k )
(A19)
where SÃ ¹ I and SÃ 2 5 I. Therefore, we see that the non-linear algebraic system of
equations (A18) is also Z 2 -equivariant.
Appendix B. Freedericksz transition applying an electrical ® eld
The coupled system of equations governing the equilibrium of a nematic liquid
crystal in the presence of an applied electrical ® eld is also Z 2-equivariant. Let
( )
h (x, z)
u (x, z)
t(x, z) 5
q
e
(B1)
(x, z)
and write equations (24)± (26) as
( )
G 1 (t)
G(t) 5
G 2 (t)
G 3 (t)
5
0
We now show that
SG(t) 5
(B2)
G(St)
where S is the re¯ ection operator
( )(
2
h (x, z)
S u (x, z)
q
e
5
2
(x, z)
2
q
h (x, 2
z)
u (x, 2
z)
(x, 2
z)
e
or
St(x, z) 5
2
t(x, 2
z)
)
(B3)
331
FR EEDERICKSZ TR ANSITION
Note that
­
­ x
­
­ z
[St] 5
[St] 5
­
­ x
­
­ z
[2
t(x, 2
z)] 5
[2
t(x, 2
z)] 5
­
2
­ x
­
­ z
[t(x, 2
[t(x, 2
2
­
z)],
­ x
­
z)],
2
[St] 5
2
­ z
2
­ x2
­
2
[St] 5
2
­
2
2
­ z2
[t(x, 2
[t(x, 2
z)]
z)]
hence
2
G 1 (St) 5
K(2
+D
G 2 (St) 5
e
2
x
(h
xx
+
­
2
­ z
(h
r
cos h 2
u
D e
G 3 (St) 5
xx
e (2
^
^
h
zz
u
[(2
+
xx
h
zz
u
z
sin h ) ( u
­
[(2
­ x
D e
cos h 2
)2
zz
)
u
)2
x
h
­
[(2
­ z
D r
2
h
x
u
z
u
cos h 2
u
z
sin h 2
u
cos h 2
x
u
u
cos h 2
x
cos h )
z
sin h ) cos h ]
z
sin h ) sin h ] +
­
[(2
­ x
D r
x
q
e
e
0
u
z
sin h ) cos h ]
sin h ) ( u
x
sin h 2
sin h ) sin h ]
and
(SG) 1 (t) 5
K( h
(SG) 2 (t) 5
2
xx
e
e
­ z
2
(SG) 3 (t) 5
+D
(2
^
­
+D
r
r
+
^
h
zz
h
xx
[(u
(2
­
[(u
­ z
)2
h
zz
xx
x
2
h
x
cos h + u
)+ D e
cos h + u
x
h
2
D e (u
zz
z
where
z
x
cos h + u
sin h ) sin h ] +
)+ D r
cos h + u
­
[(u
­ x
z
­
­ x
[(u
x
q
z
h
z
cos h )
sin h ) cos h ]
e
e
0
cos h + u
z
sin h ) cos h ]
sin h ) sin h ]
( )
(SG) 1 (t)
SG(t) 5
(SG) 2 (t)
(B4)
(SG) 3 (t)
All the arguments applied to the previous scalar case can be suitably modi® ed
for this vector case.