Three Dimensional Finite Element Modelling of Liquid Crystal Electro-Hydrodynamics by Eero Johannes Willman A thesis submitted for the degree of Doctor of Philosophy of University College London Faculty of Engineering Department of Electronic & Electrical Engineering University College London The United Kingdom
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Three Dimensional Finite ElementModelling of Liquid Crystal
Electro-Hydrodynamics
by
Eero Johannes Willman
A thesis submitted for the degree of Doctor of Philosophy of
University College London
Faculty of Engineering
Department of Electronic & Electrical Engineering
University College London
The United Kingdom
I, Eero Johannes Willman, confirm that the work presented in this thesis is my own.
Where information has been derived from other sources, I confirm that this has been
The principal axes of anchoring (e, v1, v2) are the easy direction and two mutually
orthogonal unit vectors respectively, so that e = v1 × v2. Equation (4.14) can be
directly discretized for implementation, but is here expanded in an analytical form in
order to show how meaningful values can be assigned to the scalar coefficients as, W1
and W2. e is the easy direction only when both W1 and W2 are positive scalars. If
Wi = 0 and Wj > 0, the anchoring becomes degenerate in the (e, vi)-plane. Setting
55
W1 or W2 to a negative value minimises Fs in the direction of v1 or v2, and e loses
its physical meaning as the easy direction. The types of anchoring achieved by using
negative coefficients are equivalent to a rotation of the principal axes when using
positive W1 and W2. For this reason, only cases of non-negative anchoring strength
coefficients are considered in what follows.
Without loss of generality, the geometry can be defined locally: (e, v1, v2) are cho-
sen to coincide with the (x, y, z) coordinates. The traceless Q-tensor, when including
biaxiality of LCs, is written as:
Qij =S
2(3ninj − δij) + P (kikj − lilj), (4.15)
where S is the scalar order parameter and P the biaxiality parameter. n, k and l,
are the director and two vectors that define the direction of nematic order in three
dimensions and δij is the Kronecker delta.
The three orthogonal unit vectors n, k and l can be written in terms of the three
angles α, β and γ, where α is the angular deviation of n from the (e, v2) plane (local
twist), β is the angular deviation of n from the (e, v1) plane (local tilt) and γ is a
rotation of k and l around n determining the orientation of the plane of biaxial order:
n =
cos(α) cos(β)
− sin(α) cos(β)
sin(β)
, (4.16)
k =
sin(α) cos(γ) + cos(α) sin(β) sin(γ)
cos(α) cos(γ)− sin(α) sin(β) sin(γ)
− cos(β) sin(γ)
, (4.17)
56
l =
sin(α) sin(γ)− cos(α) sin(β) cos(γ)
cos(α) sin(γ) + sin(α) sin(β) cos(γ)
cos(β) cos(γ)
, (4.18)
so that when α = β = γ = 0 , (n, k, l) = (e, v1, v2). Equation (4.14) can then be
written in terms of S, P , α, β and γ as:
Fs = as
(3
2S2 + 2P 2
)
+ W1 F1S (S, α, β) + F1P (P, α, β, γ)
+ W2 F2S (S, β) + F2P (P, β, γ) , (4.19)
where F1S and F2S are:
F1S(S, α, β) =S
2
(3 sin2 α cos2 β − 1
)
=3S
2(n · v1)
2 − S
2, (4.20)
and
F2S(S, β) =S
2
(3 sin2 β − 1
)
=3S
2(n · v2)
2 − S
2. (4.21)
In the limit of constant uniaxial order F1P , F2P and the isotropic part of Fs are
constants and can be ignored, and (4.14) reduces to the sum of (4.20) and (4.21)
multiplied by W1 and W2 respectively. In this case (4.14) is equivalent to expression
(4.9) of Zhao et al., with anchoring strength coefficients related by a factor of 3S/2.
Figure 4.2 shows the angular variation of the anchoring energy density for different
values of the polar to azimuthal anchoring ratio, R = W2/W1 when order variations
57
(a) (b)
(c) (d)
Figure 4.2: Normalised anisotropic parts of the anchoring energy density for a surfacewith e = [1, 0, 0], v1 = [0, 1, 0] and v2 = [0, 0, 1]. (a) R = 1. (b) R = 3. (c) R = 0.(d) R = ∞. (R = W2/W1)
are not considered.
4.4.1 Determining Values for the Anchoring Energy Coeffi-
cients
Without the simplification of constant uniaxial order, the preferred surface order and
biaxiality parameters Se and Pe, that minimise (4.14), are determined by the relative
values of W1, W2 and as. The two constants W1 and W2 define the anisotropic
azimuthal and polar anchoring strengths and the value of as determines the resulting
58
easy surface order. The preferred surface order occurs when n = e, i.e. α = β = 0.
Equation (4.19) then simplifies to:
Fs = as
(3
2S2 + 2P 2
)
+ W1
(2 cos2 γ − 1
)P − 1
2S
+ W2
(−2 cos2 γ + 1)P − 1
2S
. (4.22)
The value of as which minimises Fs for a given value of the surface order parameter,
Se, can be found by minimising (4.22) w.r.t. S, giving:
as =W1 + W2
6Se
. (4.23)
The resulting biaxiality parameter distribution as function of γ in the plane of
l and k is found in a similar fashion by minimising (4.22) with respect to P and
substituting as from (4.23) giving:
Pe =1−R
1 + R
1− 2 cos2 γ
3
2Se. (4.24)
Alternatively, in terms of the three eigenvalues of Q, expression (4.14) is minimised
when the eigenvalue in the direction of e is λe = Se, and the difference between the
two remaining eigenvalues is λv1−λv2 = 2Pe. Figure 4.3 shows the three eigenvalues of
a Q that minimises the surface energy density of (4.14) as a function of R, normalised
for Se = 1. Two cases can be identified from the figure.
1. R = 1, the two anchoring strength coefficients are equal, (W1 = W2) and λv1 =
λv2 = −λe/2, so that Q at the surface is uniaxial with a positive order parameter
S = λe = Se and n = e.
2. R < 1 or R > 1, the two anchoring strength coefficients are not equal. As R
varies from 1 to 0 or from 1 to ∞, the surface order undergoes a transition from a
59
10-4
10-2
100
102
104
-2
-1.5
-1
-0.5
0
0.5
1
R.
Eig
en
va
lue
s.
λ e
1
v2
λλ
v
Figure 4.3: Eigenvalues of a Q-tensor that minimises the surface energy density as afunction of R, when Se is unity.
positive uniaxial order to a negative uniaxial order through a biaxial state. In the
limits of R = 0 or R = ∞, when either W1 or W2 is zero, the anchoring is planar
degenerate with a uniaxial negative scalar order parameter of value S = −2Se, with
n parallel to the unit vector corresponding to the non-zero anchoring coefficient.
However, a more complete description of the surface order needs to include the
bulk energy density terms, which in the standard Landau-de Gennes theory for ne-
matic liquid crystals favour a uniaxial Q-tensor with a positive scalar order parameter
S = S0. The resulting Q at the surface then describes a state that minimises the
combination of the surface and bulk terms.
Figures 4.4 and 4.5 show the calculated variation in order for various anchoring
conditions when the bulk thermotropic coefficients for the 5CB liquid crystal (see
appendix A) are used with the single elastic coefficient approximation and K = 5pN.
When the anchoring energy is low the bulk terms dominate and Q at the surface is
close to the bulk equilibrium value for all R. Figures 4.4a and 4.4b show the variation
of the order parameter (S = λe) and biaxiality parameter (P = (λv1 − λv2)/2) with
R and the distance to the surface when W2 ≈ 5 × 10−5 J/m2. A small degree of
biaxial order is induced at the surface when R > 1, resulting in a decrease in S. The
variations in order are contained within about a ten nanometre thick transition region
60
00.01
0.02
100
102
104
0.6236
0.6237
0.6238
0.6239
Dist.R.
S
00.01
0.02
100
102
104
0.005
0.01
0.015
0.02
Dist.R.
P
(a) (b)
Figure 4.4: (a) Scalar order parameter S and (b) biaxiality parameter P as functionsof the distance from the surface (in µm) and the ratio R between W2 and W1.
near the surface. Figure 4.5 shows the eigenvalues of Q at the surface, normalised
by S0, as functions of W2 for R = 1, 3 and ∞. For comparison, the eigenvalues
corresponding to a linear surface energy density (a = 0) when R = 1 are also shown
(marked with circles). The influence of increased anchoring strengths can be observed
in the eigenvalues. The surface energy becomes comparable to the bulk energy in the
region around W2 = 10−3 to 10−1 J/m2, where a reduction in λe can be observed.
As the anchoring strength is further increased, the surface anchoring becomes the
dominant energy term, and the eigenvalues converge towards those that minimise the
surface energy as shown in figure 4.3.
4.5 Numerical Results
Results of numerical simulations using the weak anchoring expression of (4.14) are
presented next. First, results of simulations of the switching of a twisted nematic cell
using the Landau-de Gennes and the Oseen-Frank theories with weak anchoring are
shown. Then, the effect of anchoring induced biaxiality and order variations on the
effective anchoring strength is investigated in the Landau-de Gennes theory.
The numerical simulations are performed using the finite elements discretisation
61
10−6
10−4
10−2
100
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
R
R
R
W2 [J/m2]
Eig
enva
lues
λe
λ1
λ2
Figure 4.5: Normalised eigenvalues of Q at the surface as a function of W2 for R = 1, 3and ∞, when a is set according to expression 4.23 (no markers) and for the linearcase as = 0 and R = 1 (circles).
of the Landau-de Gennes theory described in chapter 5 and a previously developed
finite elements implementation of the Oseen-Frank theory [98]. In both cases the weak
surface anchoring energy densities are modelled by (4.14) and (4.9) respectively.
The simulations are performed using a finite elements mesh of dimensions 0.002×0.002× 1.0µm., with periodic x and y side boundaries. In practice this is equivalent
to a one dimensional case.
The values of the thermotropic energy coefficients are for 5CB in both cases (see
Appendix A), with (T−T ∗) = −4 giving an equilibrium order parameter S0 ≈ 0.624.
4.5.1 Comparison between the Landau-de Gennes and Oseen-
Frank Models
Two cases are considered for the comparison between the Oseen-Frank and Landau-
de Gennes models. First, the switching of a twisted nematic cell (with 90 twist
throughout and 5 pre-tilt) is compared for a constant ratio of the polar and azimuthal
anchoring strengths, with R = 3, as a function of the applied voltage. Both the mid-
plane and surface tilt, and the surface twist angles are obtained using both theories
and plotted in figures 4.6a−4.6c. Then, a constant 1.5 V is applied, but R is varied
62
from 1 to 1× 104. Again, the mid-plane and surface tilts and the surface twist angles
are recorded and plotted in figures 4.6d−4.6f.
In both cases the polar anchoring strengths are kept constant at B2 = 8×10−4J/m2
and W2 = 2B2/(3S0), whereas the azimuthal anchoring strengths are set as B1 =
B2/R and W1 = W2/R. Furthermore, in the Landau-de Gennes theory, the isotropic
surface energy density coefficient a is determined by equation (4.23), assuming Se =
S0. Values for the three elastic coefficients and dielectric anisotropy for the 5CB liquid
crystal are used (see. appendix A).
The two simulations yield slightly different results, but this is to be expected
since the Zhao et al. expression does not allow for order variations occurring both at
the surfaces due to the anchoring and close to the surfaces where the director field
undergoes rapid distortions due to the electric field.
4.5.2 Effect of Order Variations on the Effective Anchoring
Strength
In section 4.4.1, a proportionality relationship with a factor of 3S/2 between the
anchoring strength coefficients Wi of (4.14) and Bi of (4.9) was established in the
limit of constant uniaxial order. However, when R 6= 1 this assumption is not true
implying that the anchoring energy density will be different from (4.9), and the actual
effective anchoring strength, Weff , acting on the director will differ from the expected
value of Wi used in expression (4.14). In order to investigate this, the torque balance
[90, 91] method described earlier in section 4.2.3 is used in conjunction with modelling
results of the Q-tensor distribution [99] to find Weff acting on the director.
The azimuthal anchoring strength is found by considering a twisted cell with zero
tilt (90 twist, 0 tilt). The polar anchoring strength is found by considering a cell
with equal but opposite amount of pre-tilt on both surfaces (±45 tilt) without twist.
The latter configuration produces a constant splay deformation through the cell.
63
0 0.5 1 1.5 2 2.50
10
20
30
40
50
60
70
80
90
Voltage
Mid
pla
ne ti
lt an
gle,
deg
rees
LdGOF
(a)
0 0.5 1 1.5 2 2.50
5
10
15
20
25
30
Voltage
Sur
face
tilt
angl
e, d
egre
es
LdGOF
(b)
0 0.5 1 1.5 2 2.50
1
2
3
4
5
6
Voltage
Sur
face
twis
t ang
le, d
egre
es
LdGOF
(c)
100
101
102
103
104
16
16.5
17
17.5
18
18.5
19
Polar to azimuthal anchoring strength ratio, R.
Sur
face
tilt
angl
e, d
egre
es
LdGOF
(d)
100
101
102
103
104
73.5
74
74.5
75
75.5
76
76.5
77
77.5
Polar to azimuthal anchoring strength ratio, R.
Mid
−pl
ane
tilt a
ngle
, deg
rees
LdGOF
(e)
100
101
102
103
104
0
5
10
15
20
25
Polar to azimuthal anchoring strength ratio, R.
Sur
face
twis
t ang
le, d
egre
es
LdGOF
(f)
Figure 4.6: (a)−(c) Tilt and twist angles as a function of V , with a constant R = 13.
(d)−(f) Tilt and twist angles as a function of R, with a constant applied voltageV = 1.5.
64
Equation (4.3) is modified by including the proportionality factor of 3S0/2 as
explained in section 4.4, to give the effective azimuthal and polar anchoring strengths:
W1eff =2Kφ
3S0d sin(2∆φ), (4.25)
and
W2eff =2Kθ
3S0d sin(2∆θ). (4.26)
In (4.25) and (4.26) φ , θ , ∆φ, ∆θ and d have the same meaning as defined earlier
in section 4.2.3. A single elastic coefficient approximation K = K11 = K22 = K33 =
7pN. is used in both cases. The thermotropic coefficients for 5CB (see Appendix A)
are used.
For both cells, starting with values of W1 and W2, the distribution of Q over the
complete cell can be found by modelling using the Landau-de Gennes theory. Then,
using expressions (4.25) and (4.26), the effective anchoring strength coefficients are
calculated from the director profile obtained from the tensor field. The ratio between
Wieff and Wi is plotted in figure 4.7.
The azimuthal and polar anchoring strengths were set as W1 = 85 × 10−5 J/m2,
W2 = W1/R for R > 1 and W2 = 85 × 10−5 J/m2, W1 = W2R for R < 1. When
R is close to 1 and the order at the surfaces is uniaxial a good agreement between
Wi and Wieff is found. As R departs from 1, the effective anchoring strength in
the plane of increased biaxial order is reduced, whereas anchoring to the same plane
is increased. That is, when R < 1, W2eff < W2 and when R > 1, W1eff < W1.
It is then possible to define an effective anchoring anisotropy, Reff = W2eff/W1eff ,
which is greater than R when R > 1 and smaller than R, when R < 1. In general,
the difference between Wieff and Wi depends on the degree of surface biaxiality and
order parameter variation, so that the effective anchoring strength is a function of
65
10−4
10−2
100
102
104
0.9
0.95
1
1.05
1.1
RR
elat
ive
Anc
horin
g S
tren
gth
W2eff /W2
W1eff /W1
Figure 4.7: Ratio of the effective azimuthal anchoring strength coefficient and W1 asa function of R
both bulk and surface terms.
4.6 Discussion and Conclusions
A power series expansion in terms of the Q-tensor and two mutually orthogonal
unit vectors has been used to describe the anchoring energy density at the interface
between a solid surface and a liquid crystal in the Landau-de Gennes theory. This
expression allows for practical and flexible modelling of various weak anchoring types,
ranging from isotropic through anisotropic to degenerate anchoring.
The lower order terms of the expansion have been considered, resulting in a simple
expression with three coefficients, which in the limit of constant uniaxial order re-
duces to the well-known anisotropic generalisation of the Rapini-Papoular anchoring
expression of Zhao, Wu and Iwamoto [17, 18]. This allows the assignment of numer-
ical values with a physical meaning to the scalar coefficients of the expression. Both
the polar and azimuthal anchoring strengths can be independently defined, as well as
the value of the easy surface order parameter.
Inclusion of higher order terms may allow for an improved description of variations
in order or the anchoring energy when the angle between the director and the easy
direction is large, but this would introduce the disadvantage of added coefficients
66
(material parameters) whose values need to be known. Furthermore, simulations
using higher order expansions dramatically reduced the convergence of the numerical
scheme used. This is in accordance with [95], where it is reported that typically more
than thousand iterations of the Newton-Raphson method were needed to achieve
convergence using equation (4.11) as a surface term in a Landau-de Gennes model.
On the contrary, here, using the lower order terms of expression (4.14), the rate of
convergence of the numerical scheme is practically unaffected by including the surface
energy term to the model as compared to strong anchoring conditions where the Q-
tensor is simply fixed at the surfaces.
Results of numerical simulations of the switching characteristics of a twisted test
cell under various anchoring conditions and applied electric fields, using a finite el-
ement discretisation of (4.14) in the Landau-de Gennes theory, compare well with
those using (4.9) in the Oseen-Frank model. The resultant tilt and twist angles differ
typically by less than 2 and this can be explained by the fact that biaxiality and
order variations are not considered in the Oseen-Frank formulation.
The effect of varying the anisotropy of the anchoring and the magnitude of the
anchoring strength are also investigated. As the anisotropy of the surface anchoring
is increased from R = 1 to R = ∞, the surface order undergoes a transition from a
uniaxial positive ordering to a uniaxial negative order through a state of biaxial order
(see figure 4.3). The tendency for this to happen depends on the relative magnitudes
of the anchoring energy and the thermotropic energy of the Landau-de Gennes theory,
which favours a positive uniaxial order (see figure 4.5).
It was found by applying the torque balance method to results of simulations that
the anchoring induced order variations at the surfaces also change the effective an-
choring strengths. As the surface becomes biaxial, the effective anchoring strength
is increased to the plane of biaxial order and decreased in the same plane. In other
words, surface biaxiality induced by the anisotropy of the anchoring energy density
67
further increases the anisotropy of the anchoring, so that Reff > R. The practical
implications of this in a simulation is that the ratio of W2 and W1 can be underes-
timated to achieve a desired effective anchoring anisotropy. However, in order to do
this accurately it may be necessary to measure the effective values of the anchoring
strengths (e.g. by simulating the torque balance method, as done here) since these
also depend on the properties of the bulk thermotropic energy.
68
Chapter 5
Finite Elements Implementation
69
5.1 Introduction
In this chapter, the methods used in this work for obtaining a numerical solution
to the coupled equations governing the liquid crystal physics and the electrostatic
potential are presented. The equations to be solved are partial differential equations
(PDE) that must in practice be solved numerically due to the complexity of the
problem.
A number of different methods for solving PDEs on a computer exist, e.g. the finite
differences, finite volumes, finite elements and various mesh free methods. The finite
elements method is chosen for this work for three reasons: 1. Complex geometries pose
no problems for the method. 2. Unstructured meshes allow for local refinement of
the spatial discretisation making accurate three dimensional modelling of LC devices
with defects computationally feasible. 3. Implementation of boundary conditions is
efficient and relatively straightforward.
Broadly speaking, two different situations are considered: The solution sought
describes either the LC dynamics or the steady state.
The dynamic case describes the time evolution of the LC orientation and order
distribution. This can be used e.g. for describing the switching between ‘on’ and
‘off’ states of a pixel in a LC display device. The dynamic behaviour is found by
repeatedly solving the equations (3.47) and (3.48), giving the time rate of change of
the Q-tensor and updating it accordingly.
The steady state situation describes the static LC orientation and order distribu-
tion when time →∞. This solution corresponds to the case when the Euler-Lagrange
equations (3.22) and (3.23) are satisfied. It is possible to obtain this solution by
simply performing a sufficiently long dynamic simulation (in practice, it is not nec-
essary to simulate until time → ∞, but some tens of milliseconds usually suffice).
However, other computationally more efficient methods can be used for solving the
Euler-Lagrange equations in cases where only the final LC configuration is of interest.
70
5.2 The Finite Element Method
In the finite differences method, variables and their spatial derivatives are represented
by interpolation of values on a (usually regular) grid. These can then be directly
substituted into the PDEs that are to be solved. This is known as the strong solution.
In the finite elements method, however, an indirect approach of seeking a solution
satisfying some conditions which simultaneously satisfy the original problem is taken.
The solution obtained in this way is known as the weak solution (but despite its name
it is by no means less correct).
In order to obtain the weak solution, the strong form of the problem (the PDEs)
must be re-written in a weak form. Two commonly used methods for obtaining the
weak formulations of a problem are the weighted residuals method and the variational
method. When the problem is self-adjoint, the two approaches result in identical FE
formulations.
Before describing the procedure of obtaining a weak formulation, some definitions
that are needed in the process are presented.
The Boundary Value Problem
In general, the problem that is to be solved using the FE method is defined within a
region Ω with boundaries Γ. This can be written in terms of PDEs as:
Lu(x) = s(x) in Ω (5.1)
Bu(x) = t(x) on Γ (5.2)
where L and B are linear operators, u(x) is the unknown sought function of spatial
coordinate x and s(x) and t(x) are some known functions.
Boundary conditions (5.2) must be imposed in order for a unique solution to
exist. Different types of boundary terms exist, e.g. B = 1 results in fixed or Dirichlet
71
boundaries, where the value of u is known on Γ, and B = η ·∇ in Neumann boundaries
where the gradient of u normal to the boundary is known.
Inner Product
The inner product of two functions f(x) and g(x) is defined as:
〈f, g〉 =
∫f(x)g(x) dx. (5.3)
When 〈f, g〉 = 0 for any and all choices of g, it must follow that f = 0. This
property is used later in the FE formulation to minimise an error residual.
Spatial Discretisation
The finite element method is a technique for obtaining a numerical approximation
to some unknown function u(x). The exact function is approximated by forming the
expansion:
u(x) ≈ u(x) =n∑
j=1
ujbj(x), (5.4)
where bj(x) are known basis functions (e.g. sinusoidals or polynomials) and uj are
scalar coefficients. The task of trying to find the exact function u(x) in an infinite
dimensional search space is then reduced to calculating n discrete values that produce
the best approximation of the solution. This process is explained sections 5.2.1 and
5.2.2.
The accuracy of the approximation depends on the form of the chosen basis
functions bj(x), and the number of terms used in the expansion. In general, as
n →∞ , u(x) → u(x).
72
5.2.1 Weighted Residuals Method
The weighted residuals method is a systematic method for obtaining the weak form
of PDEs. Starting from the general PDE given in (5.1), an error residual r(x) can be
defined as:
r(x) = Lu(x)− s(x). (5.5)
The task is then to find u(x), such that the error is zero everywhere. This is equivalent
to requiring that the inner product between r(x) and any possible test function h(x)
vanishes, i.e.:
〈r(x), h(x)〉 = 〈Lu(x)− s(x), h(x)〉 = 0. (5.6)
The test function h can now be approximated by the expansion:
h(x) =n∑
i=1
ciw(x), (5.7)
and substituted into (5.6) giving:
〈r(x), h(x)〉 =n∑
i=1
ci〈r(x), wi(x)〉 = 0. (5.8)
Since 5.6 has to be satisfied for any h, it is sufficient to write
〈r(x), h(x)〉 = 〈r(x), wi(x)〉 = 0 for i = 1...n (5.9)
73
Similarly, expanding u(x) in terms of basis functions gives:
〈r(x), wi(x)〉 = 〈Lu(x)− s(x), wi(x)〉
= 〈LN∑
j=1
ujbj(x)− s(x), wi(x)〉
=N∑
j=1
uj〈Lbj(x), wi〉 − 〈s(x), wi(x)〉 = 0 for i = 1...n. (5.10)
In (5.10), only the values of the coefficients uj are unknown, and the expression
can be written in matrix form as:
Au = f , (5.11)
where
Aij = 〈wi(x),Lbj(x)〉 , uj = uj and fi = 〈s(x), wi(x)〉 for i = 1...n.
Many standard methods for finding the solution vector u on a computer exists. These
are outlined in section 5.2.4.
The basis and weight functions have not yet been defined, and different choices
are possible (see e.g. [100] p. 46). The Galerkin approach where the weighting
functions are chosen as the same set of functions used to expand the desired function
u is common in the FE method, i.e. bi(x) = wi(x).
5.2.2 Variational Method
Another way of obtaining a weak solution of (5.1) is using an appropriate variational
form. This is an integral expression Π that maps the sought function u(x) to a scalar
(i.e. it is a functional), and is stationary with respect to small variations δu when
74
u(x) is the solution to the problem:
Π =
∫
Ω
L(u,∂
∂xu, ...) dx +
∫
Γ
B(u,∂
∂xu, ...) dx. (5.12)
The solution to the problem is then obtained by requiring that the first variation
of Π vanishes:
δΠ = 0 (5.13)
Enforcing stationarity (5.13) implies the satisfaction of a partial differential equa-
tion, known as the Euler equation for the variational form and some boundary con-
dition, known as natural boundary condition. So, if the form is chosen such that its
Euler equation and the natural boundary condition correspond to (5.1) and (5.2), the
desired solution is found by enforcing (5.13).
It is possible to construct a variational expression in a systematic fashion starting
from the differential equations (5.1) and (5.2) (see e.g. [101, 102]). Alternatively, a
variational expression can be identified from the physics describing the problem. The
integral expression can e.g. be the total energy of the system that is modelled, and
is minimised for the correct solution u.
After a variational expression has been established, u can be approximated by the
expansion:
u ≈ u =n∑
j=1
ujbj(x) (5.14)
The sought approximation to the solution is then given by the set of discrete coefficient
values ui that render Π stationary, that is:
δΠ =∂Π
∂ui
= 0 for all i = 1...n, (5.15)
75
which is a system of n equations.
The process of seeking stationarity of the variational expression with respect to
the scalar coefficients of the expansion (5.14) is commonly known as the Rayleigh-Ritz
procedure.
If the functional Π does not contain terms higher than quadratic in u and its
derivatives, (5.15) results in a system of n linear equations and can be written in
matrix form as:
Ku = f . (5.16)
If the resulting equations are not linear in ui, some linearisation technique (e.g.
the Newton’s method described in section 5.6.1) can be used.
5.2.3 Enforcing Constraints and Boundary Conditions
In some cases the natural boundary condition is adequate and no action is required,
the function u that satisfies (5.13) will satisfy the desired boundary conditions. If the
natural boundary condition is not adequate, it is often possible to modify the func-
tional (variational form) to change this. If this is still not adequate and a boundary
condition or another constrain must be enforced explicitly, there are various tech-
niques to do so in the finite elements method. Some of these are described next. In
addition to boundary conditions (5.2) which must be satisfied, other constraints may
have to be imposed on a system. In this work, for example, the incompressibility of
the LC material must be maintained.
In general, the constraint which limits the unknown function can be written as
an additional differential relationship C(u) = 0. The equations can then be supple-
mented using this relationship as a Lagrange multiplier or as a penalty term [100].
76
Lagrange Multipliers
When constraints are enforced using Lagrange multipliers, the supplemented func-
tional describing the problem is written as:
Π(u, λ) = Π(u) +
∫
Ω
λC(u) dx, (5.17)
where λ is the Lagrange multiplier enforcing the constraint C(u) = 0. The final
discretized system of equations can be written in matrix form as:
K =
K C
CT 0
u
λ
=
f1
f2
. (5.18)
This approach increases the number of unknowns to be solved since in the finite
element method λ is discretized and its value must be found at each node where
the constraint is enforced. Furthermore, zeros are introduced along the diagonal
increasing the condition number of the matrix K, which may complicate the matrix
solution process.
Penalty Terms
Alternatively, it is possible to enforce constraints by the addition of penalty functions
to the original equations. The functional can then be written as:
Π = Π + α
∫
Ω
C(u)C(u) dx, (5.19)
where α is a positive penalty coefficient. The resulting matrix system after FE dis-
cretisation can be written as:
Ku = (K + αKC)u = f , (5.20)
77
where KC contains the terms corresponding to the penalty functional. The value of
α determines the degree to which the constraint is enforced; the larger α is, the more
stringently the constraint is enforced. However if α is chosen too large (5.20) will
differ too much from the actual problem defined by (5.1) and (5.2).
Direct Enforcement of Boundary Conditions
In addition to using supplementary Lagrange multipliers and penalty terms, it is
possible to enforce some boundary conditions directly on the variables once the matrix
problem is assembled. This normally results in a rearrangement and elimination of
terms from the matrix system. The advantage with this approach is that the boundary
conditions are exactly enforced and the number of unknowns that are solved is reduced
without affecting the condition number of the matrix.
If the values of u are known at the nodes k, and unknown elsewhere, the (now
known) terms Kikuk can be passed to the right hand side, resulting in a transformation
of fi into fi−Kikuk and the elimination of the rows and columns k (since uk are not
unknown, there is no need to establish those equations). The unknown nodal values
are found from the solution to the reduced system Ku = f , where:
K = Kij,
u = uj,
f = fi, with i, j 6= k.
(5.21)
Periodic boundary conditions or any other situation where the value of u is con-
strained to be equal but free for a set of nodes, e.g. the electric potential on a
disconnected electrode that is left ‘floating’, can easily be enforced when the nodal
equivalencies are known, such that ul = uk.
78
In this case multiple nodal values are effectively represented by a single degree of
freedom in the matrix system. In order to take into account the contributions of the
nodal values uk, the matrix entries located at these rows and columns are added to
the corresponding rows and columns l and eliminated from the system.
After the reduced system is solved, the values uk are recovered from uk = ul.
5.2.4 Solution Process
The coupled equations presented in chapter 3 governing the LC physics consist of both
linear and nonlinear equations. The Euler-Lagrange equations for the Q-tensor are
nonlinear while the electrostatic potential is described by the linear Poisson equation.
Obtaining a numerical solution to nonlinear simultaneous equations typically con-
sists of an iterative linearisation process (see section 5.6.1) which involves solving lin-
ear systems multiple times. Whether the problem is linear or nonlinear, it is necessary
to solve linear matrix systems of the form:
Ku = f , (5.22)
where K is known as the stiffness matrix, u is the solution coefficient vector consisting
of the unknown nodal values and f is the source vector.
One way of solving (5.22) would be to invert the matrix K, and write:
u = K−1f . (5.23)
However, this is in general impractical for large, sparse systems, and much faster
algorithms such as Gaussian elimination, LU-decomposition or some variant of Krylov
subspace methods are used in practice (see e.g. [103], for details on these).
Typically, matrix solver routines can be categorised into direct and iterative meth-
ods. Direct methods are less affected by the matrix conditioning than iterative ones,
79
but they also require more computer memory, limiting the size of the problems that
can be solved. In this work, the solutions are obtained using routines included in the
MATLAB software package.
5.3 Shape Functions
Linear (first order) tetrahedral shape functions are used for the spatial interpolation
of the variables of interest in three dimensions and two dimensional linear triangles
for the surface terms. Other types of shape functions are possible, but tetrahedral
and triangular elements are in general better suited than e.g. quadrilaterals for the
meshing of complex geometries .
Higher order shape functions provide a more rapid convergence of the solution,
but introduce other difficulties: First of all, the programming of the finite element im-
plementation is more complex, especially when mesh adaptation is used (see chapter
6). Secondly, the resulting matrix bandwidth is increased due to the higher number of
interconnected nodes, making the matrix solution process slower. Thirdly, the matrix
assembly time is greatly increased due to the larger number of Gaussian quadrature
points needed for the exact evaluation of the integrals (see section 5.3.1).
Four shape functions, Ni, i = 1...4, one for each corner, are needed for first order
tetrahedral elements. For the purpose of simplifying the integrals that are essential
to the finite elements method, it is more convenient to express these in terms of local
80
coordinates r, s and t ranging from 0 to 1:
N1 = r
N2 = s
N3 = t
N4 = 1− r − s− t
(5.24)
The physical meaning of the local coordinates can be understood in terms of a ratio
of volumes. For example, the value of N1 for the tetrahedron shown in figure 5.1 at
any location P inside the element is ([100] p.187):
N1 = r =Volume(P,2,3,4)
Total Element Volume. (5.25)
The value of a variable u (or a global x, y or z coordinate) is interpolated within
the tetrahedron by:
u = N1u1 + N2u2 + N3u3 + N4u4, (5.26)
where ui are the four nodal values of u.
Gradients of the shape functions also need to be evaluated for the spatial deriva-
tives involved in the PDEs. In local coordinates this can be achieved by considering
the Jacobian matrix for the coordinate transformation between the Cartesian and
local coordinates:
J =
∂x∂r
∂y∂r
∂z∂r
∂x∂s
∂y∂s
∂z∂s
∂x∂t
∂y∂t
∂z∂t
, (5.27)
81
so that the derivatives can locally be expressed as:
∂Ni
∂x
∂Ni
∂y
∂Ni
∂z
= J−1
∂Ni
∂r
∂Ni
∂s
∂Ni
∂t
(5.28)
Figure 5.1: Local coordinates of a tetrahedron.
5.3.1 Analytic and Numerical Integration of Shape Functions
The finite element method relies on writing the equations in a form which involves
integrals over the domain Ω. This is performed on an element by element basis, taking
advantage of local element coordinates:
∫ ∫ ∫
Ωe
f(x, y, z) dx dy dz = |J |∫ 1
0
∫ 1−t
0
∫ 1−t−s
0
f(r, s, t) dr ds dt, (5.29)
where, in the case of linear tetrahedra, |J | equals six times the volume of the element
e over which the integration is performed.
It is possible to evaluate (5.29) either analytically or using numerical integration
techniques. However, the complexity of implementing analytic integration increases
82
with the number and order of terms that need to be evaluated. This is because terms
of equal order in the shape functions need to be collected and grouped together,
requiring extensive manipulation of the equation of the weak form. For this reason,
numerical Gaussian Quadrature integration whose complexity does not increase with
the equations is used in this work.
In Gaussian Quadrature, the integrals are evaluated by forming a weighted sum
of values of f at discrete sampling points:
∫ 1
0
∫ 1−t
0
∫ 1−t−s
0
f(r, s, t) dr ds dt ≈n∑
i=1
wif(ri, si, ti). (5.30)
Provided that a sufficiently large number, n, of sample points is used, the integrals
can be evaluated exactly. For example, if it is known that the value of a variable
changes in a linear fashion within an element, only a single Gauss point located at
the centre of the element is needed, i.e. n = 1, w1 = 1, r1 = s1 = t1 = 14. Similarly, if
the value is known to vary quadratically, four points are needed and so on.
The values of the weights and the locations of the integration points can be found
tabulated in many standard textbooks on the finite elements method and applied
mathematics, e.g [100, 104].
5.4 General Overview of the Program
Three sets of coupled PDEs are solved for the dynamic case and two for the steady
state. The steady state case requires solutions to the electric potential and the Q-
tensor field. In dynamic simulations, it is additionally possible to include the flow
field of the liquid crystal material and its effect on the Q-tensor field. The general
approach to solving these equations is given next.
Figure 5.2 shows a flowchart describing the basic structure of the solution process
for the dynamic case. Each time step involves finding an electric potential distribution
83
consistent with the Q-tensor field, and an optional flow solution. The flow field is
assumed to follow the liquid crystal [12], and is updated after the potential and Q-
tensor solutions for the time step are found .
The Q-tensor dynamics is calculated using an iterative Crank-Nicolson time step-
ping scheme described in more detail in section 5.6.2. This is indicated in figure 5.2
by the ‘Newton Iterations’ loop arrow. In practice, the execution time of this loop
takes up a major portion of the total running time of the program. The finite element
mesh may be refined at the end of each time step if necessary (see chapter 6 for more
details).
5.5 Electrostatic Potential
Externally applied electric fields are used for the switching of LC devices. The electric
field is given by the negative gradient of the electric potential φ which satisfies the
Poisson’s equation:
ε0∇ · (¯ε · ∇φ) = −ρ, (5.31)
where ε0 and ¯ε are the permittivity of free space and the relative permittivity tensor
respectively and ρ is a charge density. Inside the LC material, ¯ε is defined in terms
of the Q-tensor as:
¯εij = ε⊥δij + ∆ε
(2
3S0
Qij +1
3δij
). (5.32)
The charge density ρ may be due to ions in the LC material (not considered in
this work) or due to the flexoelectrically induced polarisation (see section 3.4.3).
The electric potential is approximated using the expansion φ ≈ ∑φjNj and an
84
Figure 5.2: Flowchart of the program execution.
inner product of expression (5.31) and Galerkin weight functions Ni is formed:
φj
∫
Ω
Ni∇ · (¯ε · ∇Nj) dΩ = −∫
Ω
Niρ dΩ. (5.33)
85
Integrating (5.33) by parts gives:
−φj
∫
Ω
∇Ni · (¯ε · ∇Nj) dΩ + φj
∫
Γ
Ni(¯ε · ∇Nj) · η dΓ = −∫
Ω
Niρ dΩ, (5.34)
where η is a unit vector normal to each element face. The boundary term reduces to
zero in internal elements that have no faces on external boundaries of the FE mesh
due to cancellation of the opposing directions of η in neighbouring elements, and can
be ignored. However, it must be taken into account in elements where the Neumann
boundary condition ∇φ · η = 0 is required:
−φj
∫
Ω
∇Ni · (¯ε ·∇Nj) dΩ+φj
∫
ΓN
η ·Ni(¯ε ·∇Nj +∇Nj) dΓ = −∫
Ω
Niρ dΩ. (5.35)
Here, the surface integral only need to be performed over Neumann boundaries
ΓN . The resulting matrix is:
Kij = −¯εkαβ
∫
Ω
Nk∂Ni
∂xα
∂Nj
∂xβ
dΩ +
∫
ΓN
ηαNi(¯εkαβNk ∂Nj
∂xβ
+∂Nj
∂xα
) dΓ, (5.36)
and the source vector is given by:
fi = −∫
Ω
Niρ dΩ. (5.37)
The Greek subscripts α and β refer to the Cartesian coordinates x, y and z. The
permittivity tensor is discretized as ¯ε ≈ ∑¯εkNk.
The FE discretisation of the Poisson’s equations results in a linear system of
equations, which is solved as described in section (5.2.4).
86
5.6 Q-Tensor Implementation
In order to solve the Euler-Lagrange equations that minimise the LC free energy,
the symmetry and tracelessness of the Q-tensor must be maintained. When these
conditions are satisfied, the Q-tensor represents five independent degrees of freedom:
Three rotational degrees of freedom and two for the LC order distribution.
It is possible to solve for each of the 9 tensor components while enforcing symmetry
and tracelessness using Lagrange multipliers. However, it is computationally more
efficient to write Q in a five dimensional subspace [105] as:
Q =5∑
i=1
qiTi, (5.38)
where
T1 = (3ez ⊗ ez − I)/√
6,
T2 = (ex ⊗ ex − ey ⊗ ey)/√
2,
T3 = (ex ⊗ ey + ey ⊗ ex)/√
2, (5.39)
T4 = (ex ⊗ ez + ez ⊗ ex)/√
2,
T5 = (ey ⊗ ez + ez ⊗ ey)/√
2,
where ex, ex and ex are unit vectors in the x, y, and z directions respectively.
The free energy described in section 3.4 is then written in terms of the modified
tensor Q. This results in five Euler-Lagrange equations that satisfy the tracelessness
and symmetry properties of the Q-tensor:
fi =∂F∂qi
− ∂k∂F∂qi,k
. (5.40)
87
Equations (5.40) are discretised using the weighted residuals approach with Galerkin
weight functions to obtain the weak form for the FE formulation. The resulting ex-
pressions are lengthy and in order to avoid human errors in the programming of these,
the symbolic algebra software Maple is used to generate the code.
5.6.1 Newton’s Method
Newton’s method is a well known iterative scheme for finding roots of nonlinear
equations (see e.g. [106] p. 270). It is based on a Taylor expansion of a function
f(u):
f(u + ∆u) ≈ f(u) + f ′(u)∆u + O(h2). (5.41)
Requiring that f(u+∆u) = 0 and rearranging gives ∆u, which is used to update the
value of u:
um+1 = um + ∆um = um − f(um)
f ′(um), (5.42)
where m is the Newton iteration number. Repeating the process in an iterative fashion
converges to the value of u that satisfies f(u) = 0, provided that the initial value u0
is sufficiently close to the solution.
When solving for the Q-tensor field that minimises the free energy, f(u) is replaced
by the vector obtained from the finite element discretisation of the five Euler-Lagrange
equations f = f1, f2, f3, f4, f5T and f ′(u) by the Jacobian matrix J:
J =
∂f1∂q1
· · · ∂f1∂q5
.... . .
∂f5∂q1
· · · ∂f5∂q5
. (5.43)
88
The nonlinear equations are then solved by successively solving the linear system
Jm∆qm = −fm and updating qm+1 = qm + ∆qm, until ∆q is smaller than some
tolerance value.
5.6.2 Time Integration
Time integration is needed for simulating the dynamics of a LC device. This is
performed using the finite differences method in time.
Explicit Time Stepping
A simple explicit time stepping algorithm giving the time evolution of the Q−tensor
can be constructed by considering:
qt+∆t = qt + ∆t qt, (5.44)
where the subscript denotes the time, q is the time derivative of the Q−tensor and
∆t is the size of the time step. As described in chapter 3, q is obtained from equation
(3.52) or (3.47). A finite element discretisation of this then results in the matrix
equation:
Mqt = −ft, (5.45)
where M is the mass matrix∫Ω
N iN j, and −ft is the right hand side vector resulting
from the discretised Euler-Lagrange equations. It is then possible to find the LC
dynamics by evaluating equations (5.44) and (5.45) successively. However, although
this approach is relatively simple, it is only first order accurate and also very unstable:
The size of the time step is limited by the Courant-Friedrichs-Lewy condition which
relates the maximum time step to the spatial discretisation [107].
89
Implicit Time Stepping
An improved time integration scheme can be devised by approximating the time
derivative using central differences in time and representing nonlinearities by r:
Mqt+∆t/2 + ft+∆t/2 = r. (5.46)
This scheme is known as the Crank-Nicolson time integration, and is unconditionally
stable for linear systems [108]. Nonlinearities, represented by r, in the time derivatives
can be taken into account by performing Newton iterations within each time step (this
is shown as the ‘Newton Iterations’ loop in figure (5.2)).
The central differences are written as:
Mqt+∆t/2 =1
∆tM(qt+∆t − qt), (5.47)
and
ft+∆t/2 =1
2(ft + ft+∆t) =
1
2(Atqt + At+∆tqt+∆t) +
1
2(gt + gt+∆t), (5.48)
where A and g correspond to non-linear and linear terms respectively in the free
energy. Using (5.47) and (5.48), expression (5.46) can be re-written as:
M
∆t+
At+∆t
2
qt+∆t +
At
2− M
∆t
qt +
1
2(gt + gt+∆t) = r. (5.49)
The goal is then to find qt+∆t such that r = 0. This can be achieved using
Newton’s method by writing:
Km∆qmt+∆t = rm, (5.50)
90
and
qm+1t+∆t = qm
t+∆t + ∆qmt+∆t, (5.51)
where the superscripts denote the Newton iteration number and K is the Jacobian
matrix:
Km =∂rm
∂qmt+∆t
=
M
∆t+
Jmt+∆t
2
, (5.52)
and J is as defined in (5.43). Iterations within each time step are performed until
∆qmt+∆t is deemed to be sufficiently small.
Additional loops may be needed in order to make sure that the electric potential
is consistent with the Q-tensor field both before and after the time step (see figure
5.2 ). This could be avoided by solving for the potential simultaneously with the
Q-tensor, but the solution vector would then be extended by the number of nodes.
Variable Time Step
The ability to automatically adapt the size of ∆t results in savings in computation
time: Longer time steps can be taken when the Q-tensor is changing slowly and
shorter steps when Q is changing rapidly. This can be achieved e.g. by writing [109]:
∆tnew =
(tolerance
error
)k
·∆told, (5.53)
where tolerance and k are user defined values (e.g. tolerance = 10−3 and k = 3) and
error is an error estimate on the time derivative. The error estimate can be calculated
in various ways, but in general it is related to the magnitudes of the corrections made
to qt+∆t during the Newton iterations in the Crank-Nicolson scheme.
91
5.7 Implementation of the Hydrodynamics
It has been previously explained in section 3.6 how the incompressible flow of the LC
material may be described by the generalised Navier-Stokes equations:
ρdv
dt= ∇ · σ −∇p,
∇ · v = 0, (5.54)
where ρ is the LC density, σ is the stress tensor consisting of viscous and elastic
contributions and p is the hydrostatic pressure. The time derivative is the material
time derivative:
dv
dt=
∂v
∂t+ v · ∇v (5.55)
In the case of slow elasticity driven flow of LCs, it is possible to make two sim-
plifications to equations (5.54): The steadiness approximation and the low Reynolds
number approximation.
The steadiness approximation [12] is based on the assumption that changes in the
flow field are much more rapid than changes in the Q-tensor field. When this is true,
the partial time derivative in (5.55) can be ignored, and the flow is assumed to follow
the changes in the LC orientation. The validity of this assumption can be checked by
verifying that the characteristic times τQ and τv for the Q-tensor and the flow fields
respectively satisfy τv ¿ τQ, where [12, 9]:
τQ = µ1ξ2
L1
, (5.56)
τv = ρH2
α4
, (5.57)
where L1 is an elastic constant, α4, µ1 and ρ are LC viscosities and density respec-
tively, H is a characteristic length of the LC cell or container and ξ is the characteristic
92
length of the Q-tensor:
ξ =
√27CL1
B2. (5.58)
Typically ξ is in the order of a few nanometres resulting in τQ ≈ 10ns, whereas τv
may be a few orders of magnitude smaller. In cases when τQ ≈ τv the time derivative
cannot be ignored and time stepping for the flow field should be performed.
The Reynolds Number Re is a dimensionless parameter relating the inertial and
viscous forces of a flow (e.g. [110] p. 301):
Re =|v|H
ν, (5.59)
where H is again a measure or characteristic length of the container size and ν is
the kinematic fluid viscosity (dynamic viscosity divided by the density). When Re is
low, the nonlinear convective term in (5.55) is negligible, rendering the Navier-Stokes
equations linear.
The flow of the LC material can then be estimated at any instant in time (in prac-
tice, after each time step) by solving the steady state incompressible Stokes equations:
∇ · σ −∇p = 0
∇ · v = 0 (5.60)
5.7.1 Enforcement of Incompressibility
In the incompressible Stokes equations, the hydrostatic pressure acts as a Lagrange
multiplier to enforce the non-divergence of the flow field. However, it is a well known
problem in the field of computational fluid dynamics that a straightforward FE dis-
cretisation of the equations (5.60) results in numerical difficulties. These appear as
spurious pressure solutions, where the pressure field is oscillatory and the incompress-
93
ibility of the flow field is not satisfied [14].
Mixed Interpolation
Different approaches to overcome this problem exist. One possibility is to use the so-
called mixed formulation approach with higher order shape functions for interpolating
the flow solution than those used for the pressure. It is, for example, possible to use
second order functions for the flow and linear elements for the pressure. This is
a popular approach in two dimensional problems, where the number of degrees of
freedom is usually relatively small [10, 11, 14]. However, in three dimensions this
approach often results in prohibitively large systems due to the additional nodes
needed for the higher order elements. This was found to be especially true in this
work, bearing in mind that the flow solution is updated at the end of each time step.
Pressure Penalty
Alternatively, it is possible to enforce the incompressibility by the pressure penalty
formulation. In this approach, the continuity equation ∇ · v = 0 is replaced by [14]:
ε∇ · v = −p, (5.61)
where ε is a user defined large positive scalar coefficient. It is then possible to elim-
inate the pressure from the equations by substituting (5.61) into (5.60). Although
this approach reduces the number of degrees of freedom that need to be found, the
resulting system of equations becomes poorly conditioned due to the large value of
ε. This means that iterative Krylov sub-space solvers often do not converge to a
solution.
94
Pressure Stabilisation
An alternative approach, taken here, allowing for equal order interpolation functions
is to use the so-called Brezzi-Pitkaranta stabilisation technique [13]. This method
relies on introducing a perturbation to the continuity equation:
∇ · v = εh2e∇2p, (5.62)
where ε is a user defined small positive scalar coefficient and he is the local mesh size
of element e. The effect of the right hand side in (5.62) is to smooth the pressure
solution. A FE discretisation of the stabilised Stokes equations gives rise to the
following matrix system to be solved:
D C
CT T
v
p
=
f1
f2
, (5.63)
where the sub-matrices are D and C arise from the Stokes equations given in (5.60)
and T from the added stabilisation term.
An advantage of the stabilisation method is that the condition of the matrix is
improved due to the non-zero terms on the matrix diagonal due to the addition of T.
The condition of the matrix is further improved by introducing scaled shape functions
for the pressure, so that components of D and T are of comparable magnitude.
The stabilised formulation is tested on a container with a 90 bend, as shown
in figure 5.3, using different values for the stabilisation coefficient ε. In this test, σ
is taken to be that for an ordinary isotropic liquid (i.e only the viscous coefficient
α4 6= 0). The flow magnitude at the inflow is fixed to take a quadratic form, while
no-slip boundary conditions (|v| = 0)are applied to the side boundaries. The pressure
is fixed to zero at the outflow boundary.
Figure (5.4) shows the magnitudes of the flow and pressures on a plane through
95
Figure 5.3: Container with 90 bend for testing the stabilised Stokes flow.
ε = 10−4 ε = 10−6 ε = 10−9
Figure 5.4: Flow magnitude (top row) and pressure (bottom row) solutions obtainedusing three different values the stabilisation parameter ε = 10−4, 10−6 and 10−9).
96
the centre of the mesh for three different values of ε. The effect of over stabilisation
(ε = 10−4) can be seen in the first column where the flow field is not divergence
free. Similarly, in the last column, the effect of under stabilisation can be observed
as spurious pressure oscillations start to appear when the stabilisation parameter is
reduced to ε = 10−9. It was found that ε ≈ 10−7 − 10−8 typically result in non
divergent flow without introducing pressure oscillations.
97
Chapter 6
Mesh Adaptation
98
6.1 Introduction
The dimensions of some of the geometric features in an LC device may be very
small compared to the overall size of the device. For example, the bistable LC device
modelled in chapter 8, contains three dimensional posts with corners that are rounded
to correspond to arcs with radii in the order of tens of nanometres, whereas the
thickness of the cell is several microns. Similarly, spatial variations in the orientation
of the director field can be gradual throughout most of a device, but very high in
the vicinity of defects or aligning surfaces. In the Landau-de Gennes theory this
results in a Q-tensor field with a low gradient throughout most of a device and a
high gradient localised near regions of large distortions. Typically the diameters of
defect cores are, depending of the exact values of the material parameters an the
temperature, in the order of tens of nanometres or less. Often it is exactly these
small scale structural features and defects or disclinations in the director field that
are of interest. Consequently the spatial discretisation should be sufficiently accurate
in these regions in order to describe the geometry and to capture the variations of
the Q-tensor field.
Accurate representation of small scale geometrical features relies on the finite
element mesh provided by the user. The density of this mesh should be sufficiently
high in these regions in order to properly describe the geometry. In this work the
commercial GiD [111] program is used in the mesh generation. However, the regions
where the Q-tensor varies rapidly are not fixed and defect movement is possible during
the operation of a device. Instead of having a dense mesh throughout the whole
structure, it is often more efficient (especially in three dimensions) to increase the
mesh density locally in regions of rapid distortions of the Q-tensor and decrease it in
regions of slowly varying Q during the simulation. This is known as mesh adaptation.
A mesh adaptation algorithm for three dimensional tetrahedral meshes has been
implemented to be used in conjunction with the finite elements discretisation of the
99
Landau-de Gennes theory described in chapter 5. This algorithm performs mesh
refinement based on user specified criteria in regions where the accuracy of the inter-
polation is considered insufficient for describing the Q-tensor field. A brief overview
of finite element mesh adaptation is given in section 6.2. Then, in section 6.3, the al-
gorithm implemented as part of this work is described and example results are shown
in section 6.4. An alternative way of adapting the accuracy of the interpolation is
described and results for a reduced one dimensional problem are presented in section
6.5. Finally, possible future developments are suggested in section 6.6.
6.2 Mesh Adaptation
In general, a mesh adaptation algorithm consists of two stages: (1) Assessment of
the local error in a trial solution and (2) adaptation of the spatial discretisation to
improve the interpolation of the solution. After this, a new solution can obtained on
the improved mesh.
6.2.1 Assessment of the Error
In the so-called a posteriori error analysis a previously obtained solution is analysed
in order to find regions in the finite element mesh where the accuracy of the inter-
polation should be improved for increased accuracy and can be worsened for higher
computational efficiency. The error assessment stage can in general be classified either
as error estimation or error indication [112].
The error estimation method is based on defining an approximation of an error
measure within each element as the norm:
|| ei ||=|| ui − ui ||, (6.1)
where e is the error within element i, ui is the approximated solution and ui, the
100
exact solution. In most cases the exact solution u is not known, but in general it
is possible to provide a local estimate which is more accurate than u, so that an
approximation of the the error can be calculated. This estimate can for example be
formed by recovering a smoothed solution over a patch of elements using interpolation
functions of higher order than that used for u or on a finer mesh [100] .
Error indicators are based on heuristic considerations where a readily available
quantity, specific to the problem at hand, is chosen as an error indicator [112]. This
can for example be a gradient of the sought solution or some other physics-based
quantity.
In this work two different error indicators are considered. Firstly, the free energy
within each tetrahedron can be calculated, and elements where the total energy is
above some threshold value are chosen for refinement [10]. A second, simpler approach
considers only the value of the scalar order parameter. Elements that contain regions
where the order parameter is outside some user specified range of values are chosen
for refinement. In practise, the performance of the two error indicators is found to be
identical, provided the threshold values are chosen appropriately.
6.2.2 Adapting the Spatial Discretisation
Different methods for changing the interpolation exist. Three general schemes for
adapting the spatial discretisation can be classified: The h, p and r-methods (see e.g.
[100, 113, 114, 115] and references therein). In the h-method, the number of nodes
is locally changed. This can be achieved by splitting or recombination of existing
elements or by complete or partial re-meshing of the domain. In the p-method the
order of the interpolation polynomials is locally changed. Use of hierarchical elements
allows for addition or removal of higher order polynomials without changing the shape
functions of the lower order interpolants. In the r-method only the nodal positions
are relocated without changing the number of elements or the order of interpolation
101
functions. The advantage of this method is that the computational load remains
constant throughout the simulation, but often an additional system of equations needs
to be solved for determining the new node locations. Different combinations of these
three are also possible. In this work, the h-method is implemented in three dimensions
and a simple one dimensional test of the p-method is presented.
Local h-refinement can be achieved in various ways, but the resulting mesh must
be conforming (i.e. no hanging nodes may exist), and the mesh quality should not
degrade as a result of successive refinements. A tetrahedral mesh is said to be of
good quality when the elements are (nearly) equilateral. Low quality meshes may
interpolate poorly and the condition of the stiffness matrix tends to be worsened
[116, 117].
It is possible to obtain an improved mesh by complete remeshing the domain of
interest while ensuring that the new mesh density is appropriately changed from the
previous mesh. However, in three dimensions this process may be computationally
too expensive. Furthermore, programming a three dimensional mesh generator is
no easy task. Instead, the density of the existing mesh may be changed locally by
insertion of new nodes or removal of existing ones.
In [118, 119], refinement of tetrahedral meshes by bisection of a single element
edge has been described. Provided the edge to be bisected is chosen appropriately,
a degradation of the mesh quality is bounded below the initial mesh by a positive
constant. An alternative approach, taken here, is to subdivide elements selected for
refinement into eight sub tetrahedrons. This is a generalisation of a two dimensional
red−green refinement for triangular meshes (see e.g. [120]) into three dimensions
[121, 122]. The elements selected for refinement during the error estimation stage are
termed red, whereas transitional green elements need to be refined to ensure mesh
conformity. Figures 6.1 (a)−(f) show the possible ways an unrefined tetrahedron ,
fig. 6.1 (a), may be divided. For the red elements, fig. 6.1 (b), new nodes are added
102
at the mid sides of each edge resulting in a subdivision into eight smaller tetrahedra.
Additionally, if an element shares more than three edges with previously selected red
tetrahedra it is included in the list of red elements. However, if an element shares
three or fewer edges with the red tetrahedra (but at least one), different subdivision
possibilities exist: These are the various green elements, shown in figures 6.1 (c)-(f).
A
B
C
D
A
B
C
D
ab
ac
bc
ad
cdbd
(a) Unrefined tetrahedron. (b) Red
A
B
C
D
ab
A
B
C
D
ab
ac
(c) Green1 (d) Green2a
A
B
C
D
ab
cd
A
B
C
D
ab
bc
ac
(e) Green2b (f) Green3
Figure 6.1: Element refinement by the red-green method. Bisected edges are drawnin bold. Original nodes are labelled with capital letters whereas new nodes resultingfrom edge bisection are labelled using lower case letters.
103
6.3 Overview of the Mesh Adaption Algorithm
Mesh adaptation algorithms that include both mesh refinement and de-refinement
sometimes employ special tree-like data structures to represent the hierarchy of refined
and unrefined elements in a mesh and use recursive algorithms in the refinement of
neighbouring elements, e.g. [120, 123, 124]. However, the algorithm developed here is
required to work with the mesh represented by simple array data structures, as this
is the format used for the finite element program described in chapter 5. A benefit
of this is that the algorithm developed here is general and can be included in other
finite element programs with only small modifications to the code.
The mesh adaption algorithm works by starting the refinement process from a
copy of the initial user created mesh, making explicit de-refinement unnecessary and
thus reducing the complexity of the algorithm. Errors are estimated on a mesh from a
previous solution, which may or may not already be refined. Elements in the original
mesh that contain regions of high error are selected for refinement. This process is
repeated until no more refinable elements are found, or for a user defined number of
iterations. The steps taken in each refinement iteration are listed below, and explained
in more detail.
1. Choose refinable elements in mesh.
2. Expand region(s) of refinement if necessary.
3. Identify green elements.
4. Identify red and green surface elements.
5. Create new nodes and new elements.
6. Remove old elements chosen for refinement.
7. Interpolate Q-tensor field onto new mesh.
104
8. Repeat from step 1 or exit refinement algorithm.
1. A list of red tetrahedra is constructed. Two different criteria that can be
used either separately or in combination for finding these elements are implemented:
Firstly, the total free energy of the LC material is calculated within each element.
The free energy density is higher in regions with large distortions in the Q-tensor
field, such as in the vicinity of defects. If the integral of this energy density over the
volume of an element is above some user defined threshold, the element is marked as
refinable. Secondly, elements that contain nodes from the previous result where the
scalar order parameter is outside a user defined range can be marked for refinement.
2. It is often necessary to include elements that were not selected in step 1 to the
list of refinable elements. This may be due to two reasons: Sometimes an element
not previously marked red may be neighbouring several red elements (it shares four
or more of its edges with the elements already in the list of red elements). In this
case that element is also added to the list of red tetrahedra. When a structure with
periodic boundaries is simulated, it may be necessary to refine a region near one of the
boundaries. The periodicity must be maintained and it is then necessary to extend
the region of red elements to the opposite side of the mesh.
3. The tetrahedral elements of the mesh can be subdivided in different ways
depending on how many of its edges are bisected. In order to ensure conformity,
transitional green elements must also be created by subdivision into two or three
smaller tetrahedra. At this stage, lists of green tetrahedra are created depending on
the number of edges shared with any red elements selected in steps 1 and 2.
4. Two-dimensional triangular elements are used to represent alignment surfaces.
These must also to be refined when a red or green tetrahedron is located at the
surface. Red and green triangles are identified depending on the number of edges to
be bisected. Red triangles are divided into four and green triangles into two smaller
triangles.
105
5. Edge elements consisting of two nodes are created from the lists of red and green
tetrahedra. New node coordinates located at the centres of each edge are created.
Then, new two and three dimensional elements are created by dividing the red and
green triangles and tetrahedra.
6. All the red and green elements are removed from the mesh data structure and
are replaced by the newly formed smaller elements. This is necessary in order to
avoid overlapping elements.
7. After the new mesh is created, variables from the previous result are interpo-
lated onto the new mesh. The Q-tensor field is interpolated in terms of (θ, φ, ψ, λ1, λ2),
where θ, φ and ψ are Euler angles of the eigenvectors and λ1 and λ2 of the eigenval-
ues of the Q-tensor. The reason for performing the interpolation in terms of these
‘derived’ values instead of the actual Q-tensor components used in the calculations
is that the physical meaning of direction and degree of order are maintained. Figure
6.2 shows the effect of interpolation of the individual Q-tensor components within a
one dimensional linear element extending from x = 0 to x = 100. The eigenvalues
resulting from interpolation of the Q-tensor components are plotted between the two
nodes. The eigenvalues of the tensor are of equal magnitudes at both of the nodes,
but the orientation of the director changes by an arbitrary angle (B-A) through the
element. The interpolated Q-tensor at x = 50 then appears to be biaxial and with a
reduced order parameter.
7. Finally, the newly created mesh may be either further refined by repeating
steps 1 to 6, or the mesh adaptation algorithm may be exited and the simulation can
be continued using the new adapted mesh.
106
Figure 6.2: Example of error introduced by linear interpolation of the components ofa Q-tensor field representing a rotation of the director field of a constant order. Blackdots represent the original nodes and gray dots the new added node.
6.4 Example − Defect Movement in a Confined
Nematic Liquid Crystal Droplet
The switching dynamics of a spherical liquid crystal droplet was used in the testing
and development of the mesh adaptation algorithm. The simulated structure consists
of a spherical liquid crystal region of 1µm diameter immersed in a cube of solid
isotropic dielectric material. The anchoring of the LC material is assumed planar
degenerate, resulting in a pair of point defects located at opposing sides of the sphere.
A slight asymmetry is introduced by scaling two of the dimensions of the structure
by a few percent in order to ensure the existence of a unique LC configuration that
minimises the total free energy. Electrodes are placed at the top and bottom surfaces
of the cube containing the LC sphere. A part of the initial unrefined mesh for the
structure is shown in figure 6.3. The size of the initial mesh is 30748 tetrahedra and
5717 nodes.
The mesh adaptation algorithm is chosen to perform three refinement iterations
every seven time steps on the initial mesh based on the value of the scalar order
107
parameter. Refinement of a tetrahedron is performed when the value of the order
parameter within that element is below 75%, 30% and 1.5% of the equilibrium order
parameter S0. Typically, the resulting meshes consist of approximately 50000 tetra-
hedra and 9000 nodes. Results showing the director field during the switching process
are shown with the corresponding meshes in figures 6.4 (a) − (f). After the applied
potential is removed, the director field relaxes back to the inital configuration shown
in figure 6.4 (a) due to the slight asymmetry of the structure.
Figure 6.3: Partial 3-Dimensional view of initial unrefined mesh for LC droplet insidea cube of fixed isotropic dielectric material. Approximately a quarter of the dielectricregion (coloured white) and half of the liquid crystal (coloured grey) are shown.
6.5 Hierarchical p-Refinement
An alternative to the h-method implemented in three dimensions is the p-method,
where the order of the interpolation functions is increased locally to improve the
accuracy. One way of doing this is by use of hierarchical elements [100]. These are
higher order polynomials that can be added to elements without effecting the lower
order shape functions already present.
108
a b
c d
e f
Figure 6.4: (a), (c), (d) 2-Dimensional slices through the centre of a nematic dropletduring switching by an external electric field. Director colour indicates scalar orderparameter and background electric potential. (b), (d), (e) 3-Dimensional views ofcorresponding meshes.
109
Using the hierarchical approach, the discretised approximation of function u is
where ui are the discretised values of u and Ni the corresponding spatial interpolation
functions or shape functions. In a one dimensional system i = 1, 2 correspond to the
nodal degrees of freedom, whereas i > 2 correspond to higher order internal (bubble)
degrees of freedom.
Figure 6.5 (a) shows second, third and fourth order one dimensional hierarchical
shape functions plotted against the local element coordinate r. The superposition of
standard linear elements and a second order hierarchical element is shown in figure
6.5 (b).
0 0.2 0.4 0.6 0.8 10
0.5
1
N3
0 0.2 0.4 0.6 0.8 1−1
0
1
N4
0 0.2 0.4 0.6 0.8 1−1
0
1
N5
Local Coordinate, r0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Local Coordinate, r
Dis
plac
emen
t
u1
u3
u2
LinearLinear + O2 hierarchical
(a) (b)
Figure 6.5: (a) Second, third and fourth order hierarchical shape functions for a onedimensional finite elements implementation. (b) Example of superposition of firstand second order hierarchical element shape functions. Linear element (dashed line)is p-refined by the addition of a second order (solid line) shape function.
In a single dimension, the shape functions can be written as functions of the local
110
coordinate r ranging from 0 to 1 as:
N1 = (1− r)
N2 = r
N3 = 4(1− r)r
N4 =36√
3(1− r)(1/2− r)r
N5 = 81(1− r)(1/3− r)(2/3− r)r
In order to test the algorithm, a one dimensional finite elements discretisation of
the equations of the Landau-de Gennes theory using hierarchical elements has been
written. The convergence of this interpolation scheme was studied for a test case
where the director is fixed homeotropic on one surface of a thin cell and planar on
the other. No initial tilt bias is given, so that a region of ‘melting’ from horizontal to
vertical orientation of the director occurs at the centre of the cell. Although no real
defects can be considered to exist in a one dimensional geometry, the configuration
described here corresponds to the director and order parameter profile through the
centre of a −12
defect in two dimensions, as shown in figure 6.6 (a). The resulting
eigenvalues of the Q-tensor are plotted in figure 6.6 (b). The thickness of the cell is
taken as 0.1 µm., a single elastic constant approximation with K = 5pN/m2 is used
and the thermotropic coefficients are for the 5CB LC material at (T − T ∗) = −4K
(see appendix A).
The mesh density is uniform throughout the domain but the number of elements
is varied, i.e. no local h refinement is considered. The convergence of the scheme
using different orders of interpolation functions can be seen in figure 6.7 (a), where
the free energy of the system is plotted as a function of the element size. It can be
seen that adding the third order shape functions, O3, improves the accuracy of the
scheme considerably more than the second and fourth order functions, O2 and O4.
111
0 0.02 0.04 0.06 0.08 0.1−0.4
−0.2
0
0.2
0.4
0.6
0.8
Z / µ m
Eig
enva
lues λ
x
λy
λz
(a) (b)
Figure 6.6: (a) −12
defect in two dimensions (left) and the one dimensional direc-tor profile through the centre (right). (b) Eigenvalues of the Q-tensor in the onedimensional case plotted against the z dimension.
This is due to the fact that both the solution (see eigenvalues λx and λz in figure
6.6 (b)) and the third order functions are odd functions in the spatial coordinate z,
whereas the second and fourth order polynomials are even.
In this test the p-refinement is not local. That is, all the elements in the mesh
contain the same number of degrees of freedom. However, only a fraction of the
higher order degrees of freedom are in fact needed to describe the solution. This can
be seen in figure 6.7 (b), where the effective number of degrees of freedom (degrees
of freedom with displacement magnitudes larger than 10−6) are plotted against the
element size. The effective higher order degrees of freedom are concentrated at the
centre of the structure where the gradient of the solution is high (see figure 6.8).
This means that in a two or three dimensional implementation, where efficiency is
more important, higher order polynomials only need to be added locally to elements
containing defects.
6.6 Discussion
A three dimensional mesh adaptation algorithm has been developed and implemented.
The algorithm performs local mesh h-refinement in regions selected using an empirical
112
error indicator, making modelling of three dimensional defect dynamics feasible on a
standard PC workstation.
The performance of a hierarchical p-refinement scheme was tested in a simplified
one dimensional case. A three dimensional implementation of the p-refinement scheme
is more complicated than the simple one dimensional described here. This is because
higher order degrees of freedom must be assigned, in addition to the internal element
volumes, also to edges and faces separating neighbouring elements. Then, for exam-
ple in the case of a tetrahedron, addition of a higher order hierarchical polynomial
introduces a total of 11 new degrees of freedom (four faces, six edges and one volume).
Care must be taken with the ordering of the element node numbering in neighbouring
elements to ensure continuity of the higher order face and edge polynomials. A full
three dimensional hp-refinement scheme is left as future work.
In this work it is assumed that any solid surface-liquid crystal interfaces are static
and do not change during the simulation, so that the initial meshing should satisfy the
requirements of mesh density for proper description of the geometry of the LC device.
This is true in the case of most optoelectronic devices. However, nematic liquid
crystals find new applications as solvents for microemulsions and particle dispersions,
in e.g. biomolecular sensors [1] or in the self-assembly of crystal structures [2]. The
LC material interacts with the immersed nano- or micro-scale particles affecting their
position and orientation due to the elastic forces of the director field, so that the LC-
particle interfaces can no longer be considered static. Moving boundaries are possible
using the finite elements method, and are in fact extensively used e.g. in finite
elements models for structural mechanics [100] or Stefan problems [115] (a problem
where a phase boundary can move with time). However, as mentioned before, mesh
generation and remeshing is a cumbersome task, especially when a good quality mesh
without inverted or degenerate elements is needed. An alternative, more recently
developed method could be extending the finite element method with discontinuous
113
elements using the XFEM method [125]. In this method, moving discontinuities
are represented by additional discontinuous shape functions superpositioned on the
underlying standard finite element mesh eliminating the need for mesh adaption. The
XFEM method can be added on top of existing finite element code, and has been used
in e.g simulation of elastic fracture mechanics, multi-phase flow and representation of
microstructures [125, 126, 127].
10−4
10−3
10−2
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Element Size / µ m.
Fre
e E
nerg
y [A
rbitr
ary
Uni
ts]
O1O1+O2O1+O2+O3O1+O2+O3+O4
10−4
10−3
10−2
0
100
200
300
400
500
600
700
Element Size / µ m.
Effe
ctiv
e N
umbe
r of
Deg
rees
of F
reed
om
O1O1+O2O1+O2+O3O1+O2+O3+O4
(a) (b)
Figure 6.7: Comparison between results obtained using hierarchical elements of dif-ferent order. (a) Total free energy as a function of element size. (b) The effectivenumber of degrees of freedom as a function of element size
0 0.02 0.04 0.06 0.08 0.10
0.02
0.04
0.06
0.08
0.1
Z / µ m
Mag
nitu
de
u3
u4
u5
Figure 6.8: Magnitudes of higher order hierarchical degrees of freedom as a functionof the z-dimension. The number of 1-D elements is 50, resulting in an element size of2 nm.
114
Chapter 7
Validation and Examples
115
7.1 Introduction
In this chapter, examples of results obtained using the Q-tensor LC modelling software
developed for this work are presented. When possible, these are compared with
previously published work or with results obtained using other modelling methods.
First, defect-free cases where the elastic distortion energy dominates and the
Oseen-Frank theory is expected to give similar results to the Landau-de Gennes the-
ory are compared. Then, cases where order variations and defects play an important
role are considered. Finally, results for a zenithally bistable device, modelled in three
dimensions for the first time, are presented.
7.2 Three Elastic Constant Formulation
The dynamic three elastic coefficient formulation on the Landau-de Gennes energy is
validated by simulating the switching dynamics of a twisted nematic cell and com-
paring the results with predictions obtained using an established finite elements im-
plementation of the Oseen-Frank energy developed earlier at UCL [98]. Material
parameters for the 5CB LC material are used (see appendix A).
The cell thickness is chosen as 1 µm. The anchoring is strong on both surfaces
with 5 pre tilt and 90 twist through the cell. Cases with weak anchoring in the two
theories are compared in chapter 4. Starting from uniform director configurations
at time = 0, a 2V potential is applied across the cells for a duration of 3 ms, after
which the director fields are allowed to relax for a further 7 ms. The tilt angles at
z = 0.5µm are recorded and are plotted in figure (7.1).
The results show good agreement with only small observed differences. These can
be attributed to order parameter variations that occur near aligning surfaces where
the elastic distortion is high and to differences in the implementation of the two
algorithms.
116
0 2 4 6 8 100
20
40
60
80
100
Time [ms]
Tilt
[deg
rees
]
OF
LdG
Figure 7.1: Comparison of tilt angles at z = 0.5µm as a function of time using theOseen-Frank (dashed line) and the Landau-de Gennes (solid line) theories.
7.3 Switching Dynamics of a TN-Cell, with Back
flow
When a large holding voltage is removed from a twisted nematic cell, an optical bounce
in the transmitted light can be observed. The reason for this has been shown long
ago to be the director at the mid plane of the cell momentarily tipping over due to
shear flow, also known as back flow, of the LC material [128, 129].
The dynamics of a one micron twisted nematic cell with 5 degrees pre-tilt is
simulated with and without taking into account the effect of flow of the LC material.
The anchoring is assumed strong on both surfaces and the material parameters for
the 5CB liquid crystal material are used (see appendix A).
A 3V potential difference is applied across the cell for the duration of 2ms., after
which the potential is removed. The mid plane tilt angle is recorded and plotted
versus time in figure 7.2.
117
0 2 4 6 8 100
20
40
60
80
100
Time [ms]
Mid
plan
e T
ilt A
ngle
[Deg
rees
]
No FlowFlow
Figure 7.2: Switching dynamics of a twisted nematic cell, with and without flow ofthe LC material.
7.4 Defect Dynamics
The dynamics of defects are validated by studying the annihilation of a ±12
line de-
fect pair. This has previously been examined theoretically in [12, 63, 11], using two
dimensional discretisations. In [12] and [11], the process is modelled using finite differ-
ences and finite elements implementations of the Qian-Sheng equations respectively.
In [63], a finite differences implementation of the Berris-Edwards equations is used.
A mesh of 400×4×400 nm dimensions with periodic boundary conditions for the
(x, z)planes at y = 0 and y = 4nm is used. This is comparable to a 2D discretisation
where the defect lines are assumed to extend to infinity in the y-dimension. The
material parameters used are for the MBBA LC material (see appendix A).
Two distinct initial director configuration are considered: (a) Starting from a di-
rector configuration where the tilt angle is set to θa(x, z) = 12
(tan−1
(z
x−d
)− tan(
zx+d
)),
and (b) starting from the initial configuration θb(x, z) = θa(x, z)+ π2. In both (a) and
(b) the director is in the (x, z)−plane resulting in zero twist throughout the cell. The
coefficient d is the distance between the defects and the centre of the cell, d = 50nm
is used. Figures 7.3 (a) and (b) show the initial director fields for the two cases.
Dynamic simulations are performed both with and without the effect of flow of the
118
LC material. The positions of the defects are recorded at each time step by finding
the locations of the mesh nodes that correspond to minima in the order parameter.
The variation of defect positions with respect to time can be seen plotted in figure
7.4, while the resulting flow fields at time = 20µm are shown in 7.3 (c) and (d). The
−0.2 −0.1 0 0.1 0.2−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
x [µm]
z [µ
m]
−0.2 −0.1 0 0.1 0.2−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
x [µm]
z [µ
m]
(a) (b)
−0.2 −0.1 0 0.1 0.2−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
x [µm]
z [µ
m]
−0.2 −0.1 0 0.1 0.2−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
x [µm]
z [µ
m]
(c) (d)
Figure 7.3: The two initial director configurations for the defect annihilation cases(a) and (b), and the corresponding flow solutions (c) and (d) at time = 20 µs.
effect of including the flow is to speed up the defect movement, with the positive
defect accelerated more than the negative one. When the effect of flow is ignored the
two defects move at the same speeds, yielding in identical results in both cases (a)
and (b).
The flow field is found to be sensitive to both the defect separation as well as
the spatial discretisation. Nevertheless, good agreement is found between the results
119
0 0.01 0.02 0.03 0.04 0.05 0.06
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
Time [ms]
X [µ
m]
(a) and (b), no flow
(a)(b)
−
+
Figure 7.4: Defect positions with respect to time for the two initial configurations,with and without flow. In both cases when flow is ignored, identical results areobtained. The solid line represents the position of the positive defect and the dashedline the position of the negative defect.
obtained here and those published earlier.
120
7.5 Defect Loops in the Zenithally Bistable Device
The work presented in this section shows a more substantial example of the mod-
elling capabilities of the tools presented in the earlier chapters: A periodic grating
structure known as the Zenithally Bistable Device (ZBD) is investigated. The device
has previously been modelled considering only two dimensions [130, 131], but here
results from three dimensional analysis are presented.
Conventional liquid crystal devices are usually monostable, that is, the liquid
crystal director field always relaxes to the same configuration after applied voltages
are removed. In contrast, bistable LC devices have two distinct stable configurations
or states to which the director field may relax, and in which they remain without
applied holding voltages. Advantages of bistability include lower power consumption
and the possibility of passive addressing of high resolution LC devices.
Stable states of the director field correspond to minima in the free energy of the
LC material. In monostable devices only one minimum is used in the operation of the
device, whereas in bi- and multistable devices two or more local minima separated by
barriers of higher free energy can be reached.
Bistability in nematic LC devices can be achieved in a number of ways including
the use of surface anchoring exhibiting bistability [134, 135, 136, 69], non-uniform
surface alignment patterns [137, 138] or surfaces shaped as two- or three-dimensional
micropatterns [19, 20, 130, 139], as is the case with the ZBD .
In the ZBD, the two states are knownas the continuous (C state) and the dis-
continuous or defect states (D state). In the C state the director field undergoes
continuous distortions whereas the D state is characterised by the presence of ±12
defect line pairs running along the peaks and troughs of the grating structure (the
positive defect along the troughs and the negative along the peaks).
Previously, simulations of the structure have been carried in two dimensions [130,
131]. The first of these [130] shows the switching process, back and forth, between the
121
two stable states under several simplifying assumptions; strong anchoring, a single
elastic constant approximation and a negligible dielectric anisotropy. The second
study [131] drops these assumptions, but concentrates on the defect movement during
the annihilation process from the D to the C state. Switching between the stable states
is controlled by the sign of the applied voltage, via the flexoelectric effect.
In two dimensional cases the grating is assumed to extend to infinity. However,
in reality the lengths of the gratings are finite and a 180 degree shift in the grating
structure has been found experimentally to stabilise the defect loops that form [132,
133]. The grating is now a three-dimensional structure that contains “slips”. The
effect of the slip region on the static defect structures has been modelled in three
dimensions and the results for this are presented next. A more comprehensive analysis
taking into account the time dynamics is necessary in order to study the role of the
slips on the switching dynamics and the stabilising effect they have on the defect
loops. This work is currently under way and results will be reported elsewhere.
The ZBD Grating Profile in Three Dimensions
In two dimensions, the surface profile has been previously represented in [131] using
the function:
Z(x) =H
2sin
(2πx
Px
+ α sin2πx
Px
), (7.1)
where Px is the grating pitch in the x−direction, H is the grating height and α is a
scalar coefficient used for determining the asymmetry of the grating profile. Figures
(7.5) (a) and (b) show the two dimensional director fields for the C and D states for
a grating structure described by expression (7.1), with Px = 1µm, H = 0.65µm and
α = 0.5. In three dimensions the surface profile is also a function of the y−coordinate,
122
(a) (b)
Figure 7.5: The continuous (a) and discontinuous (b) states found in the two dimen-sional representation of the ZBD grating structure.
In equations (7.2) and (7.3) Z1 and Z2 describe the the two grating profiles in the
x-direction. These are essentially the same as in (7.1), but the additional parameter
φ determines the phase difference between the two gratings. The functions Υ1, Υ2
and Υ3, based on sigmoid functions, are used to specify the surface profile in the y-
direction. The parameter s determines the steepness of the transition from a grating
to the slip region with larger value of s resulting in a more rapid transition, and w
is the width of the slip. Finally, h and H determine the relative height of the slip
region and the heights of the grating ridges respectively.
Three different grating profiles with different slip heights are considered. In all
cases H = 0.65µm, Px = 1µm, α = 0.5, φ = 180, s = 20 and w = 0.5µm are used,
but h is chosen as 0, 0.5 or 1. The resulting surface profiles are plotted in figures (7.6)
(a), (b) and (c). The cell gap is chosen as 2.5 µm in each case and periodic boundary
conditions are enforced along the (y, z)−faces and Neumann boundaries along the
(x, z)−faces. The anchoring on all alignment surfaces is homeotropic.
124
−1−0.5
00.5
1
0
0.5
1
0.20.40.6
Y [µ m]X [µ m]
Z [µ
m]
−1−0.5
00.5
1
0
0.5
1
0.20.40.6
Y [µ m]X [µ m]
Z [µ
m]
−1−0.5
00.5
1
0
0.5
1
0.20.40.6
Y [µ m]X [µ m]
Z [µ
m]
(a) (b) (c)
Figure 7.6: Three different surface profiles for the ZBD structure, with the height ofthe slip region set to 0, 0.5 and 1 times the ridge height in (a), (b) and (c) respectively.
Modelling and Results
For simplicity, a single elastic coefficient approximation with K = 15pN is assumed.
For reasons of computational efficiency the thermotropic coefficients are set to A =
0Nm−2, B = 0.64 × 106Nm−2 and C = 0.35 × 106Nm−2, resulting in slightly larger
defect core sizes than when using experimentally measured values. Since the emphasis
here is on the defect structures and not the switching between between the two states,
the effect of electric fields is not considered.
In order to model the defect state of the device, the initial director field is set
horizontal along the x−direction within the troughs. The LC is then allowed to relax
to the nearest stable state, which in this case is the D state. The C state can be
obtained in a similar fashion by starting from a vertical director profile within the
troughs.
At a distance from the slip region, the director field is contained in the (x, z)−plane
as shown in figures (7.5) (a) and (b). Closer to the slip, the director twists into the
y−direction in order to satisfy the homeotropic boundary condition imposed by the
vertical surfaces around the slip. Iso-surfaces of reduced order parameter due to the
±12
defect pairs are shown in figures (7.7)(a), (b) and (c) for the three surface profiles.
Closed defect loops are formed due to a continuous transition from positive to negative
defect through a twisting of the director parallel to the axis of the defect line (marked
with circles in the figures). The negative defect line is pinned to the convexly shaped
125
portions of the surface whereas the positive one follows the concave portions. The
strength of the LC anchoring to the grating affects the distance between the surface
and the defect line. Weaker anchoring allows the defect closer to the surface whereas
stronger anchoring expels the defect further into the bulk of the LC.
Further study of the ZBD geometry with an emphasis on the slip region is planned.
In particular, the role of the slip region as a possible defect nucleation site in the
switching process will be investigated.
7.6 Discussion and Conclusions
Results obtained using the modelling tools developed for this work have been pre-
sented. These were compared to previously published data or to results obtained using
other established methods in order to verify the correctness of the implementation.
The process of verification was started with simple cases that take into account only
a few LC characteristics at a time and then progressed to more complex situations.
First, defect free cases where the elastic distortions dominate were considered
in order to validate the response to electric fields and the three elastic coefficient
formulation. This was then taken further by additionally solving for the flow of the
LC material and observing the induced backflow after the release of a holding voltage.
Then, the dynamics of pair annihilating half integer defect lines were modelled.
This was done twice, first taking into account the flow of the LC and then without the
flow. As expected, including the effect of flow favoured the movement of the positive
defect, whereas when ignoring it the rate of movement of both defects was equal.
Finally, a larger problem was considered in order to demonstrate the scope of the
type of problems that can be tackled using the modelling software: The static defect
line configurations in the slip region separating the ends of grating structures in a
Zenithally Bistable Device was modelled in three dimensions. It is possible that this
126
region plays an important role in the switching dynamics of the device and further
investigation is currently under way.
127
Iso-surfaces of reduced order Iso-surfaces of reduced order
(a) (b)
Iso-surfaces of reduced order
(c)
Figure 7.7: Iso-surfaces of reduced order parameter showing the locations of the defectlines. Circles are drawn to indicate the regions of the ±1
2defect transitions.
128
Chapter 8
Modelling of the Post Aligned
Bistable Nematic Liquid Crystal
Structure
129
8.1 Introduction
In this chapter the operation of a bistable device, the post aligned bistable nematic
(PABN) [140] liquid crystal device is modelled. This is another example of bistable
devices whose operation relies on a structured solid surface in contact with the LC
material (see section 7.5 for more information on bistable technologies).
Due to the geometry of the PABN structure, its operation cannot be fully de-
scribed in two dimensions. In addition, defect dynamics of the director field is im-
portant in the switching process. For this reason, the 3D finite element discretisation
of the Landau-de Gennes free energy described earlier in this thesis is used to model
the dynamic behaviour of the device.
The device geometry is explained and previously published information is intro-
duced in section 8.2. The approach taken to modelling is explained in section 8.3 and
results are given starting from section 8.4.
8.2 Overwiew of the The PABN Device
The PABN device is a bistable liquid crystal device under development at the Hewlett-
Packard laboratories. The bistability of the PABN device is achieved by sandwiching
nematic LC material between two different surfaces. One of the bounding surfaces
contains an array or grating of microscopic posts, whereas the other surface is flat.
The anchoring on the flat surface (called from now on the top surface) is homeotropic.
The bottom surface, including the surfaces of the posts, is untreated and imposes
planar degenerate anchoring on the director. The result of this is that two distinct
stable director configurations exist. Between crossed polarisers, one of these appers
bright and the other dark [20].
The posts may be fabricated of photoresist on glass surfaces using photolitographic
techniques, or directly of the substrate itself, which may e.g. be a flexible plastic [20].
130
Various post shapes and sizes are reported to be possible, with dimensions ranging
from 0.1−3µm and cross sectional shapes including circles, ovals, squares and diamond
shapes. The distances between the posts and the cell gaps are also reported to be of
similar magnitudes [140].
Previous theoretical predictions of the two stable states obtained using the Oseen-
Frank theory in [19, 20, 140, 21] suggest that distinct director configurations with
different levels of tilt angle exist. These are known as the planar and the tilted states.
The predicted director fields in [19, 20, 140] suggest that the planar state is char-
acterized by a pair of −12
defect lines along the vertical edges of the posts. It is
suggested that a balance between the energies of the defects and the flat top surfaces
of the posts result in the stability of the planar state. The tilted state is argued to
be stable due to the absence of the defect lines of high energy. This argument is
supported by experimental evidence of the planar state becoming unstable when the
height of the post is increased sufficiently (resulting in longer line defects of higher
total energy) with respect to the cell gap.
More recently, in [21], four distinct stable configurations of the director field were
modelled. None of these states corresponds to the previously suggested planar states,
but the tilted states were topologically identical. The difference between the models
is that in [21], the director field is fixed along the edges of the posts, whereas in the
earlier publications this is not the case.
8.3 Modelling the PABN Device
Before the operation of the full device is modelled, a single corner of a single post is
considered. This is useful, since as will be shown later, the director configurations
found around the single corner are found to be essential for the operation of the
complete structure.
131
The material parameters used in the modelling are as follows: The single elastic
constant approximation K = K11 = K22 = K33 = 7pN was used, resulting in L1 ≈3.47 × 10−12. Thermotropic coefficients for the 5CB material at −4K below the
nematic-isotropic transition (see appendix A) and the modified coefficients (A =
0Nm−2K−1, B = 0.64 × 106Nm−2 and C = 0.35 × 106Nm−2) were used. Both sets
of coefficients were found to result in qualitatively identical results, but using the
modified parameters, the computational load is significantly reduced. A material
with negative dielectric anisotropy with values ε⊥ = 8 and ∆ε = −3 is chosen. The
flexoelectric polarisation is represented using an expression linear in the gradient of
the Q-tensor, which is the special case when the splay and bend coefficients are equal
e11 = e33 = 3S0
2e (see section (2.5)). The value of e is chosen as 10 × 10−12Cm−1,
which is comparable to both experimentally measured values [29] and theoretical
predictions [30]. The anchoring on the bottom surface, including the surfaces of the
post, is assumed planar degenerate and the anchoring energy density is written as:
fs = asTr(Q2) + W (viQij vj), (8.1)
where v is the local surface normal unit vector and the coefficients as and W have the
same meaning as described in section (4.4.1). The value of the anchoring coefficient W
is chosen large enugh to prevent topological changes through breaking of anchoring.
It was found that W ≥ 5×10−4J/m−2 is sufficient. The value of as is set to as = W6S0
.
The anchoring of the top surface is assumed strong homeotropic.
8.3.1 The Geometry of the Modelling Window
Two differerent modelling windows are needed, and can be seen in Fig. 8.1. Fig. 8.1a
shows a cell containing a full post in the periodic structure while the calculation cell
in Fig. 8.1b contains anly one corner (a quarter of the cross-section) of a post and a
132
fraction of the total height of the structure.
The grating structure consisting of microscopic posts is assumed periodic (al-
though it does not need to be [140]) allowing the modelling of a structure extending
to infinity by considering a single cell with periodic boundary conditions. The external
dimensions of the modelling window are 1.2× 1.2× 3.05µm, and the boundary condi-
tions on the (x, z) and (y, z) planes are periodic. For the separated corner structure
shown in figure 8.1 (b), the external boundary conditions are left free, corresponding
to Neumann boundaries in the single elastic coefficient approximation. The base of
the structure, including the post consists of isotropic dielectric material. The corners
and edges of the post are rounded to a radius of 20 nm. The top surface of the base
(including the post and its sides) are LC-solid-surface-interfaces, where planar degen-
erate anchoring conditions are applied. Strong homeotropic anchoring is applied at
the top surface at z = 3.05µm. Planar electrodes are placed at z = 0 and z = 3.05µm.
In the actual device, some degree of asymmetry is introduced to the geometry in
order to ensure that a preferred alignment in the azimuthal direction exists [19, 20].
Here, this is achieved by the choice of the initial director field orientation, so that
a main diagonal (x = y) can be identified along which on average the director is
aligned.
8.4 Modelling Results
8.4.1 A Topological Study of a Single Corner
First, the stable configurations for the single corner are modelled by minimising the
total free energy in the absence of electric fields. Three topologically distinct config-
urations are found by starting the minimisation process from different initial director
configurations. These will be referred to as the horizontal, continuous vertical and
discontinuous vertical states, after the orientations and distortions of the director
133
(a) (b)
Figure 8.1: The geometries of the 3-D modelling windows (a) for the full device, and(b) for the isolated corner.
fields found in the states. Furthermore, a defect state is modelled by minimising the
free energy in the presence of an externally applied electric field.
The horizontal state is characterised by the director field lying nearly parallel to
the bottom surface (in the (x, y) plane) of the modelling window. The director field
bends in a continuous fashion aroud the corner. Figure 8.2a shows the director field
on a slice in the (x, y) plane at z = 0.3µm through the isolated corner structure.
In the continuous and discontinuous vertical configurations, the bulk xy-components
of the director field is approximately perpendicular to those of the the horizontal state
(see Fig. 8.2b). However, the director field can be described as flowing over the corner,
rather than bending around it, so the z-components of the director field is significant
near the vertical surfaces of the corner. The difference between the continuous and
discontinuous vertical states can be seen in figure 8.3a and b , where the director field
is displayed on a vertical slice through the diagonal of the isolated corner structure,
the (x = −y, z) plane. Defects in the director field can be seen near the top and
bottom corners of the structure in the discontinuous vertical configuration, whereas
Figure 8.2: Director profiles for the horizontal (a) and continuous vertical (b) stateson a regular grid along the (x, y) plane through the centre of the isolated cornerstructure at z = 0.3µm. The discontinuous vertical state is not shown, as it appearsnearly identical to the continuous vertical state from this point of view.
in the continuous vertical configuration these are not present.
Figure 8.3: The director field on a regular grid along the diagonal (x = −y, z) planethrough the separated corner structure. (a) Stable continuous vertical configuration,(b) stable discontinuous vertical configuration
In the defect configuration the director field lies in the (x, y) plane, with a −12
defect line extending along the edge of the post (see Figs. 8.4a and b). This state
135
is found to be unstable unless an externally applied electric field is present and the
flexoelectric coefficient e is zero, and is a transitional configuration separating the two
topologically nonequivalent stable vertical states. The defect configuration can be
achieved by applying an electric field which due to the negative dielectric anisotropy
aligns the director field along the horizontal (x, y) plane (to demonstrate the topology
of the defect structure, e has been set to zero in figure 8.4).
The effect of a non-zero flexoelectric coefficient e is to counteract the horizon-
tally aligning dielectric response and to cause a director field deviation into the z
direction. The degree and direction of this deviation depends on the magnitudes and
signs/directions of e and the electric field. This vertical deviation is necessary for
switching between the two stable states. After removal of the electric field, the direc-
tor field relaxes to the vertical state that is closer to the angular deviation caused by
the flexoelectric effect. If e is chosen as zero, the director field always relaxes to the
continuous vertical state due to the geometry of the corner and the aligning effect it
has on the director field.
(a) (b)
Figure 8.4: Defect line along a post edge during switching. (a) a magnified viewof (x, y) plane at z = 0.3 µm cutting through the post. Darker background colourindicates a reduction in the order parameter near the defect core. (b) 3-D view ofsame post edge with a dark iso-surface for the order parameter showing the extent ofthe line defect.
136
The sum of the elastic, thermotropic and surface energies are calculated and plot-
ted for comparison in figure 8.5. Both the modified thermotropic coefficients and the
experimentally obtained values for the 5CB material were used. Qualitatively the
results are similar, with the horizontal state being the energetically most favourable
followed by the continuous and discontinuous vertical states, while the defect config-
uration results in the highest energy. However, using the more realistic coefficients
for the 5CB material, the difference in the total energy between the four director
configurations is larger than when using the modified coefficients.
1 2 3 40.95
1
1.05
1.1
1.15
1.2
1.25
1.3
Rel
ativ
e E
nerg
y
Horizontal
Vertical Continuous
Vertical Discontinuous
Defect
modified5CB
Figure 8.5: Sums of elastic, thermotropic and surface energies for the four directorconfigurations using the modified thermotropic coefficients (black) and for the 5CBmaterial (white). The energies are normalised with respect to the respective horizontalstates.
8.4.2 Modelling the Full Structure − The Two Stable States
Director configurations for two stable states were obtained by minimizing the free
energy starting from two distinct initial director profiles. Fig. 8.6a and 8.6b show
the results for the planar and the tilted states in the (x, y) plane cutting through the
post at z = 0.3µm. Although the Q-tensor maintains the head-tail symmetry of the
director, vectors are used here for clarity to represent the x and y components while
the background color represent the z component of the director field. Fig 8.6c and
8.6d show the same states in the (x = y, z) diagonal plane.
The states of the complete structure can be seen as combinations of those described
137
for the isolated corner, occurring at opposing corners of the post. The director field
can adopt two distinct configurations at the edges A and B of the main diagonal of
the post (see Fig. 8.6); the two vertical configurations introduced in the previous
section. In both cases the director is nearly parallel to the z-axis along the edge,
but the difference is observed at the top and bottom corners of the edges, where
the director field can adopt either continuous or discontinuous configurations. In
Fig. 8.6c, the director field bends in a continuous fashion around the post at both
edges A and B, resulting in the stable planar state. In Fig. 8.6d, the director field
is continuous at edge A and discontinuous at edge B, resulting in the stable tilted
state. Near the edges C and D, the director bends in a continuous fashion around
the post while remaining close to the (x, y) plane, corresponding to the horizontal
configuration described earlier.
Fig. (8.7) shows a comparison of the tilt angles through the cell in the z-direction
for the planar and the tilted states at a corner of the modelling window (e.g. x =
y = 0). The difference in tilt angle between the two states can be seen concentrated
near the bottom of the structure, where the direction of alignment around the posts
dominates.
8.4.3 Modelling the Full Structure − The Switching
Dynamics
A 50 ms period was modelled, during which switching back and forth between both
stable states was considered (from planar to tilted and back to planar switching).
The dynamic simulation was started without applied electric fields using the steady-
state Q-tensor configuration previously obtained for the planar state. Keeping the
top electrode grounded the bottom electrode voltage is set to +20 V at T = 1ms and
maintained for a total of 3ms, after which the director field relaxes to the tilted state
when the voltage is removed. Then, at T=30 ms, the bottom electrode voltage is
138
0 0.6 1.20
0.6
1.2
X − [µm]
Y −
[µm
]
B
A
C
D
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
(a) Stable planar state in the (x, y) plane atz = 0.3 µm.
0 0.6 1.20
0.6
1.2
X − [µm]Y
− [µ
m]
B
A
C
D
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
(b) Stable tilted state in the (x, y) plane at z= 0.3 µm.
(c) Stable planar state -diagonal
(d) Stable tilted state -diagonal
Figure 8.6: The director field (x, y) plane at z = 0.3µm for the (a) planar and (b)tilted states. The planar (c) and tilted (d) states in the (x = y, z) plane runningdiagonally through the modelling window. In (a) and (b), the background colorcorresponds to the z-component of the director, where positive z direction is out ofthe page.
139
0 0.5 1 1.5 2 2.5 30
20
40
60
80
Z [µm]T
ilt A
ngle
[deg
rees
]
Planar StateTilted State
Figure 8.7: The tilt angles of the stable planar and tilted states along a corner of themodelling window as a function of z.
set to −20 V for 3 ms, causing the director field to relax to the planar configuration
after removal of the voltage. The free energy of the LC material was also calculated
at each time step during the simulation.
Fig. 8.8(a−n) show a series of sketches of the director field on the (x = y, z)
diagonal (vertical) plane during the planar − tilted − planar switching sequence.
Triangles indicate −12
defect lines and circles show point defects at the top and bottom
corners of the edges of the post on the (x = y, z) plane. The total energy variation
through the entire switching cycle is shown in Fig. 8.9a. The two peaks are due to the
added electric energy at the times when voltages are applied. The switching process
can be described in 10 steps related to the director profiles shown in Figs. 8.8(a−n)
and the sum of the total thermotropic, elastic and anchoring energies shown in Figs.
8.9a and 8.9b:
1. T=0−1ms, Fig. 8.8a. The device is in the stable planar starting configuration
without applied electric fields.
2. T=1ms, Fig. 8.8b. A positive potential is applied to the bottom electrode which
starts to re-orient the director of the LC material with a negative dielectric
anisotropy towards the (x, y) plane.
3. T=1−4 ms, Figs. 8.8(c−e). Planar degenerate anchoring keeps the director
140
parallel to the post surface while the electric field forces it to the (x, y) plane
due to the negative dielectric anisotropy. The combination of these two torques
causes −12
defect lines to be formed along the edges of the main diagonal section
of the post (circles in Figs. 8.4(a−b) and 8.8(c−d)). The flexoelectric effect
further re-orients the director field to the discontinuous configuration at the
edges, breaking the line defect into two point defects. The transition occurs at
the right hand side edge before the left hand side edge due to the direction of
the tilt in the LC above the post.
4. T=4−12 ms, Fig. 8.8f. After the electric field is removed the relaxation starts
from the top of the device due to the aligning effect of the homeotropic anchor-
ing. The director field remains stable around the post.
5. T≈12ms, Fig. 8.8g. The aligning effect of the homeotropic anchoring reaches
the top of the post forcing the point defect down along the left hand edge
resulting in the recombination of the two defects at the lower corner of the edge
and leaving the left hand side edge in a continuous configuration. This process
can be seen as a reduction in free energy in Figs. 8.9a and 8.9b at T≈12ms.
6. T=12−30 ms. Figs. 8.8h and 8.8i. The director field further relaxes into the
stable tilted configuration of minimum free energy, marked ‘T’ in Fig. 8.9b.
7. T = 30 ms, Fig. 8.8j. The device is in the stable tilted state when a negative
potential is applied to the bottom electrode re-orienting the director field to the
(x, y) plane.
8. T≈ 31−33 ms, Fig. 8.8(k−l). The transition now starts at the right edge of
the post where the discontinuous configuration changes into the continuous one
through the creation of a line defect.
141
(a) Stableplanar state
(b) (c) (d) (e) (f) (g)
(h) (i) Stabletilted state
(j) (k) (l) (m) (n) Stableplanar state
Figure 8.8: Simulation results of planar to tilted to planar switching.
9. T = 34 ms, Fig. 8.8m The voltage is removed and the aligning effect of the
homeotropic surface anchoring starts to reorient the director field from the top
of the device.
10. T = 34−50 ms, Fig. 8.8n. The director field relaxes to the stable planar state.
The total energy reaches the stable value marked ‘P’ in Fig 8.9b.
8.5 Discussion and Conclusions
The stable director configurations have been modelled for both a reduced represen-
tation of the PABN geometry, consisting of an isolated corner, and a more complete
device, consisting of a periodic array of posts. The stable director profiles of the
isolated corner are found to be present in the complete post structure, and being an
essential feature that enables the bistability of the device. This means that it may be
possible to investigate the effect changes in the post shape or other parameters have
142
1 4 10 20 30 33 40 500
2
4
6
8
10
12
Time [ms]
Ene
rgy
[Arb
itrar
y U
nits
]
(a)
1 4 10 20 30 33 40 50
0
0.2
0.4
0.6
0.8
1
1.2
T
P
Time [ms]
Ene
rgy
[Arb
itrar
y U
nits
]
(b)
Figure 8.9: The sum of the total thermotropic, elastic and surface anchoring energiesduring the planar-tilted-planar switching sequence.
on the overall device performance by considering the case of the separated corner.
The advantage of this is the reduced computational cost as compared to modelling
the full structure.
The results presented here show that for the given geometry the free energy of
the stable tilted state is lower than that for the stable planar state. The planar state
corresponds to a local minimum in free energy and remains stable as long as the
energy barrier between the two states is greater than the energy associated with the
torque exerted by the top surface which favours a high tilt angle.
When the dimensions of the structure are kept constant the choice of material
parameters becomes critical for bistability. While the values quoted earlier in section
8.3 allow for modelling of bistable operation of the current geometry with reasonable
computational cost (related inversely to the defect core size), other values were also
considered. In general: Increasing the value of the elastic constant increases the torque
exerted by the top surface, destabilizing the planar state. Conversely, increasing the
defect energy by choosing different values of the thermotropic energy coefficients
stabilizes the planar state. Furthermore, the planar degenerate anchoring must be
strong enough to keep the director on the plane of the surface of the post at all times
(or sufficiently close to it), in order to prevent topological changes through breaking
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of surface anchoring.
Some of the previous theoretical predictions of the two stable states obtained using
the Oseen-Frank theory in [19, 20] suggest that the planar state is characterized by
a pair of −12
defect lines in the director field. However, the results presented here
predict a different, defect free planar state. Furthermore, the previously identified
defect lines are shown to correspond to an intermediate stage that acts as an energy
barrier separating the planar and the tilted states. The director configurations of the
tilted state in both simulations are qualitatively the same.
Switching between the stable states was found to be a two stage process: First
the director field was forced into the defect configuration, mainly by the effect of
the negative dielectric anisotropy of the LC material. Then, due to the flexoelectric
polarization, the director field would adopt either the continuous or the discontinuous
configuration at the post edges, depending on the direction of the electric field E and
the sign of the flexoelectric coefficient e. The defect configuration cannot be achieved
when the dielectric anisotropy is below some threshold value irrespective of the value
of the flexoelectric parameter. Similarly, if the value of e is too small, transitions from
the defect configuration to either the continuous or the discontinuous configurations
do not occur.
The results presented here explain the full process of switching between the two
states using monopolar electric fields in the PABN structure. The modelling methods
presented here can be used to explore the effect of altering the post shape and varying
material parameters on the operation of the device.
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Chapter 9
Summary and Future Work
145
9.1 Summary or Achievements
The work presented in this work has concentrated on the static and dynamic three
dimensional computer modelling of nematic liquid crystal materials. Three main
areas can be identified: Implementation of a 3D coputer model for calculating the LC
Q-tensor field, advances in phenomenological description of the solid surface-liquid
crystal interface and the application of the developed tools in the modelling of bistable
LC devices.
A 3D finite element formulation of the Landau-de Gennes continuum theory and
its dynamic extension taking into account the flow of the LC material, the Qian-
Sheng formalism, has been implemented into a computer program. The Q-tensor
representation with variable order and biaxiality is used to describe the LC material.
This combined with an automatic mesh refinement algorithm and a stable non-linear
Crank-Nicholson time integrator makes modelling of defect dynamics feasible on a
standard personal computer.
The aligning effect solid surfaces have on liquid crystals, known as anchoring, is of
fundamental importance to the operation of most LC devices. Usually the anchoring
is anisotropic, the polar (away from the surface plane) and azimuthal (in the surface
plane) anchoring strengths being unequal. In this work, a power expansion on the
Q-tensor and two mutually orthogonal unit vectors is used as a surface energy density
to represent the effects of anisotropic anchoring in the Landau-de Gennes theory. The
expression is shown to simplify in the limit of constant uniaxial order to a well known
anisotropic anchoring expression in the Oseen-Frank theory, making it possible to
assign experimentally measurable values with a physical meaning to the coefficients
of the tensor order parameter expansion.
The modelling software was used to investigate the operation of a bistable technol-
ogy, the post aligned bistable nematic devce (PABN). The two stable configurations
of the device, the tilted and the planar state, were identified and the dynamics of
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switching between these were modelled. The two stable states were found to be sep-
arated by barriers of higher free energy corresponding to line defects extending along
the edges of microscopic posts present on one of the surfaces of the device. Traversal
of the energy barrier is necessary for the switching between the two stable states.
This was found to be possible due to a combination of the dielectric anisotropy and
the flexoelectric effect in presence of externally applied monopolar electric fields.
9.2 Future Work
In general, two main areas of future work can be identified: Applications of the
modelling software and further developments of the methods.
Traditionally the geometries involved in LC applications can be rather simple, as
is the case e.g with many display devices where some liquid crystal material is sand-
wiched between two flat glass surfaces inducing a well defined direction of alignment.
However, the drive drive for devices with higher resolution, faster switching and the
possibility of bistability imply smaller and more complicated geometries. Computer
modelling of such novel devices often allows for faster and cheaper design and optimi-
sation than would be possible by manufacturing actual prototype devices. Using the
computer program developed for this work, it is possible to explore the effects of e.g.
chages in material parameters, device geometry and applied voltage waveforms on the
operation of the device without the need of expensive laboratory and manufacturing
equipment. An example of proposed future work in this direction is further investiga-
tion of the ZBD device described in section 7.5. Other applications for the modelling
includes investigating controlled formation and manipulation of defects through the
use of sub micron sized electrodes on LCOS substrates.
The other direction for future work involves extending the current modelling capa-
bilities of the program. This may e.g be in the form of taking into account additional
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physical effects, making better use of computing resources through code parallelisa-
tion or by making use of more efficient interpolation schemes.
As mentioned earlier in section 3.7, some LC mixtures contain finite concentrations
of positive and negative ions. These ions are not fixed in space and time, and changes
in their distributions can be estimated by solving the drift-diffusion equations, coupled
with the Poisson equations governing the electric potential which in turn affects the
LC orientation through the dielectric response and the flexoelectric effect.
The possibility of adding higher order terms to the interpolation functions (shape
functions) in regions where variations in the Q-tensor are rapid were explored in
section 6.5. This scheme, in combination with the currently implemented mesh re-
finement could add significant saving in calculation time. Alternatively, the possibility
of using some other kind of functions, specifically chosen for the problem, could be
used for even greater efficiency. It could for example be possible to represent defects
using only a few degrees of freedom (say, for position, defect strength and orientation)
compared to the hundreds or thousands that are currently needed using linear shape
functions.
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Appendix A
Values of Material Parameters
Used in this Work
149
Material Parameters for 5CB Material Parameters for MBBA