Comprehensive threedimensional dynamic modeling of liquid crystal devices using finite element method
ABSTRACT In this paper, a comprehensive opensource threedimensional (3D) finiteelement method (FEM) is proposed to model the dynamic behavior of liquid crystal (LC) directors in complex structures. This dynamic model is based on interactively iterating the vector representation of director profile and potential distribution. The director update formulations are derived in detail from the Galerkin's approach of FEM, including the weak form approach to simplify the highly nonlinear iteration equation. The potential update formulations are derived from the Ritz's approach of FEM. A 2D inplane switching (IPS) structure is used as an example to compare our approach with the FEM based commercial software (2dimMOS). The results from both programs show an excellent agreement. Furthermore, our method also agrees well with the finitedifference method (FDM) in studying a 3D super IPS LC cell with zigzag electrodes.
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ABSTRACT: The influence of a bias voltage on the surfacedriven orientational transitions of liquid crystals (LCs) resulting from the weakening anchoring and anchoring transition was analysed theoretically and experimentally. The continuum theory of nematic LCs was used to model the LC deformation under different anchoring energies. A microelectroplated interdigitated Au structure was used in the nematic LCbased chemical and biological sensors. With a suitable bias electric field, the processes of the weakening anchoring energy and the uniform surfacedriven orientational transition due to targeted molecules binding to a functionalized surface were observed optically. These results can be used to improve the sensitivity, response speed and signal strength of LCbased chemical and biological sensors.Journal of Physics D Applied Physics 03/2011; 44(13):135103. · 2.53 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: A novel hightransmittance verticalalignment liquid crystal display using combined fringe and inplane electric fields is described. The strong horizontal electrical fields cause a greater number of liquid crystals to become tilted above and between the electrode pixels. As a result, transmittance is higher than that for cells driven by fringe field switching or by inplane switching, at the same time preserving a wide viewing angle and fast response. A high performance device of this type is particularly attractive in large liquid crystal displays.Liquid Crystals 01/2011; 38(4):469473. · 1.96 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We report electrically tunable liquidcrystalcore channel waveguides, in which the lightwave guiding can be tuned among cutoff, singlemode, and multimode. Ultrasonicassisted chemical etching is used to produce semicircular grooves on the optical glass substrate for encapsulating liquid crystal as the waveguide core. The liquidcrystalcore waveguide loss is reduced to the lowest value 1.3 dB/cm up to date, which is attributed to the smooth and uniform groove surface. The extinction ratio is >20 dB for optical switching application. The LC director distributions under various voltages are calculated by finite element method and the characteristics of guided modes are simulated by the fullvectorial mode solver considering the full anisotropy. The simulated waveguide characteristics are well consistent with the experimental results. The proposed liquidcrystalcore waveguide owns the features: simple and lowcost fabrication process, arbitrary device pattern, and integration with silicon platform.Journal of Lightwave Technology 01/2013; 31(22):35703574. · 2.56 Impact Factor
Page 1
194JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Comprehensive ThreeDimensional Dynamic
Modeling of Liquid Crystal Devices Using
Finite Element Method
Zhibing Ge, Student Member, IEEE, Thomas X. Wu, Senior Member, IEEE, Ruibo Lu, Xinyu Zhu, Qi Hong, and
ShinTson Wu, Fellow, IEEE
Abstract—In this paper, a comprehensive opensource threedi
mensional(3D)finiteelementmethod(FEM)isproposedtomodel
the dynamic behavior of liquid crystal (LC) directors in complex
structures. This dynamic model is based on interactively iterating
the vector representation of director profile and potential distribu
tion. The director update formulations are derived in detail from
the Galerkin’s approach of FEM, including the weak form ap
proach to simplify the highly nonlinear iteration equation. The po
tential update formulations are derived from the Ritz’s approach
of FEM. A 2D inplane switching (IPS) structure is used as an ex
ample to compare our approach with the FEM based commercial
software (2dimMOS).The results from both programs showan ex
cellent agreement. Furthermore, our method also agrees well with
thefinitedifferencemethod(FDM)instudyinga3DsuperIPSLC
cell with zigzag electrodes.
Index Terms—Liquid crystal devices, liquid crystal displays,
modeling, finite element method.
I. INTRODUCTION
L
able modeling of LC electrooptic behaviors is critical in both
developing novel devices and optimizing the current display
devices. Generally, the LC modeling includes two steps: eval
uation of the LC directors’ deformation under external electric
fields and calculation of the optical properties thereafter. There
fore, an accurate model of LC deformation is a prerequisite
for reliable device design and subsequent optical analysis. In
general, the dynamic modeling of the LC director calculations
can be implemented by finitedifference method (FDM) [1],
[2], finiteelement method (FEM) [3], [4], or a combination
of both. The success of these methods in accurate modeling
of LC devices has been demonstrated by the development of
LC modeling software, such as LC3D [2], DIMOS [5], LCD
Master [6], Techwitz LCD [7], LCDDESIGN [8], LCQuest
[9], and others.
SeveralfactorsmakeFEMastrongcontenderformodelingLC
devices.TheprimaryreasonisthatFEMisversatileinmodeling
IQUID CRYSTAL (LC) materials have been used widely
in directview and projection displays. Accurate and reli
Manuscript received June 6, 2005; revised July 24, 2005. This work is sup
ported by Toppoly Optoelectronics Corporation (Taiwan), Taiwan, R.O.C.
Z. Ge, T. X. Wu, and Q. Hong are with the Department of Electrical and
Computer Engineering, University of Central Florida, Orlando, FL 32816 USA
(email: zge@mail.ucf.edu; tomwu@mail.ucf.edu; qhong@creol.ucf.edu).
R. Lu, X. Zhu, and S.T. Wu are with the College of Optics and Pho
tonics, University of Central Florida, Orlando, FL 32816 USA (email:
rlu@creol.ucf.edu; xzhu@creol.ucf.edu; swu@creol.ucf.edu).
Digital Object Identifier 10.1109/JDT.2005.858885
arbitraryLCdevicestructureswithadaptivemeshingtechnique.
Further, as a direct solver (seeking solutions via directly solving
linearsystemproblems)incontrasttoFDM(seekingsolutionsvia
iterations) [1]–[4], this method can generate accurate solutions
from solving large sparse matrix system, which is an inherent
property of FEM. Since efficient algorithms for large sparse
matrixsysteminotherareassuchasmathematics,mechanics,and
electromagnetics are well developed, they can be introduced to
theLCdevicemodelingwithoutmuchextraeffort.Inaddition,the
development of computer aided symbolic derivation techniques
nowadays makes automatic FEM formula generation in LC
modeling possible, which was previously the main obstacle to
introduce FEM into accurate LC simulations. Several papers
have already exploited the FEM in the study of advanced
LC devices [10]–[16]. However, their focus is more device
physics oriented. To the best of our knowledge, no literature
has ever reported an opensource comprehensive mathematical
derivation and formulation of the FEM in LCD modeling. The
lack of this part in LCD simulators has hindered the research
as well as applications of LCD technologies. Therefore, the
main objective of this paper is to propose a comprehensive
dynamicmodelingofLCdeformationsbyFEM,whichincludes
a detailed derivation of the FEM formulations. In addition, a
special technique, the weak form method [3], [4] to greatly
simplify formula derivations, is introduced as well.
In this paper, we first introduce the derivation of update
equations for LC director and potential profiles, as well as the
iterationschemefortheLCdynamics.Then,FEMisemployedto
solve the above derived highly nonlinear equations based on the
Galerkin’smethod[3],[4]fordirectorprofileandRitz’smethod
[3], [4] for potential profile. Derivations of the nonlinear update
equationsshowthatthehighestspatialderivativeisinthesecond
order (e.g.,
). Therefore, we introduce the weak
form method [3], [4] to simplify the director update equations.
This technique makes the firstorder 3D FEM interpolation
function (tetrahedron shaped) accurate enough. Furthermore,
some effects which influence the director distribution such as
surface anchoring and flexoelectric polarization [17], [18] are
discussed as well. To verify our derivations, a 2D inplane
switching(IPS)structure[19]isthenstudiedbytheabovederived
method and the FEMbased 2dimMOS [5]. Finally, a 3D super
IPS structure with zigzag electrodes [20] is simulated using
our FEM and results are compared to those derived from FDM
to validate our derivations as well.
1551319X/$20.00 © 2005 IEEE
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GE et al.: COMPREHENSIVE 3D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM195
Fig. 1.Vector representation of a LC director.
II. MATHEMATICAL FORMULATIONS
A. Dynamic Model of LC Devices
Modeling of the dynamics of LC directors in response to ex
ternal voltages generally starts off with minimizing the LC free
energy within the LC cell [21], [22]. However, owing to the
high nonlinearity of the free energy expressions and the cou
pling between elastic and electric energies, a direct solution of
the LC director deformation and potential distribution from the
free energy equation is almost impossible. Therefore, the dy
namic modeling usually involves an iteration process [2]: mini
mizing the LC free energy to update the LC director profile and
solving the Gauss law (or minimizing the electric energy) to up
date the potential profile interactively. In modeling the LC di
rectors, both vector representation [23], as shown in Fig. 1, and
tensor [24]–[26] methods can be employed. Detailed com
parison of these two methods is provided in [2]. Owing to its
mathematical simplicity, fast computing speed, and reliability,
the vector method is widely employed in LCD simulations. In
this paper, we refer the vector method in our derivations.
1) Director Profile Update Theory: Numerical simulation
of LC transient state at each time step under constant voltage is
based on minimizing Gibbs free energy [21], [22], which is the
difference between FrankOseen strain free energy [18], [27],
[28] and electric energy, by solving the Euler Lagrange equa
tion [2], [22]. The Gibbs free energy density in the LC cell can
be expressed as
(1)
where
(2)
(3)
and
in the given coordinates, as shown in Fig. 1;
wavenumber, which is a parameter specifying the equilibrium
twiststateoftheLC;
,and
of the LC materials; and
electric field, and potential function in the entire system region,
is the unit vector form of LC directors
is the chiral
aretheelasticconstants
are the electric displacement,
respectively;
correlates
free energy (a volume integral of the energy density) by solving
Euler–Lagrange equation with inclusion of a Rayleigh dissipa
tion function will result in [2], [22]
is the dielectric tensor of LC material which
and . It is known that to minimize the system
(4)
where
tiplier to constrain the unit length of the director
). Here the fluid flow effect is assumed to have
secondary importance and is thus ignored in (4). The Lagrange
multiplier
canbefurtherdropped,ifthecalculated
is normalized as
each iteration step. Therefore, (4) specifying the iterative rela
tion of LC directors under external applied voltages can be fur
ther simplified to the following form:
represents, and , andis a Lagrange mul
(i.e.,
,and
at
(5)
where
(6)
By taking a timedifference
can be further expressed as
, (5)
(7)
Since the dielectric tensor
, thepotential and the directorfunctions in
coupled with each other as indicated by (1)–(3). Computer as
sisted derivations show
has a highly nonlinear form con
taining the highest spatial derivative of
and
up to the second order (e.g.,
use the first order 3Dtetrahedral basis function to evaluate these
second order derivatives, weak form method is introduced, as
will be discussed in later sections.
2) Potential Profile Update Theory: At given LC director
distribution and electrode voltage, the potential profile
be determined by solving the Gauss law
includes the LC director
are
with respect to
). In order to
,
can
(8)
The displacement
tial
as indicated in (2) and (3). Here solving the above
equation is equivalent to minimizing the electric energy
in the bulk from the variational
aspect. The bulk region defined here includes the LC cell and
other regions, such as the substrates. Both direct solver (e.g.,
FEM with Ritz’s method or Galerkin’s method) and iterative
approach (e.g., FDM with iterations) can be employed to solve
the desired potential distribution. Due to the simplicity in
implementation and the inherited accuracy of direct solver, we
will take FEM with Ritz’s method in the following derivations.
is expressed in terms of the poten
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196JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Fig. 2. Iteration flowchart of the dynamic modeling.
3) ModelingFlowchart: AsexplainedinSectionsIIA1and
IIA2, the coupling between director deformations and applied
electric fields makes it almostimpossible to obtainboth director
and potential profile solutions simultaneously. However, this
challengecanbeovercomebytheiterationapproach.Adetailed
iterative scheme is illustrated in Fig. 2. The initial LC director
profile can usually be preset by its boundary conditions. The
stability of this iterative updating scheme is an important issue
[29],whichreliesonaproperselectionofupdatingtimestep
in (7).
B. FEM Implementation
The solution for the abovementioned iteration scheme in
cludes two iterative steps: 1) solving the potential distribution
at a given director profile and 2) solving the director profile at
a given potential distribution. FEM is employed to solve these
iteration formulations.
1) Director Update Formulations via Galerkin’s Method:
The high nonlinearity of (1) indicates that direct solution of the
LC director distribution is difficult to obtain. However,
Galerkin’s method can bypass this difficulty. The Galerkin’s
method belongs to the family of weighted residual methods,
which seek the solution by weighting the residual of the
differential equations. Suppose we divide the study region
into element regions with
order interpolation function). The local element has a specific
shape, such as a tetrahedral block in the 3D FEM case. The
general solution of each director component
be approximated by interpolating from its values on the
global nodes using global interpolation functions as
nodes (considering the first
, orcan
(9)
where
th node value of and interpolation function in the entire region,
respectively. The specific form of
thispaper. After inserting
into(5), the residual
interpolation, can be written as
is the approximate solution of andare the
will be discussed later in
, error of the
(10)
The exact solution occurs only when
above approximation, the residual
value. However, with Galerkin’s method, the residual
minimized through weighting the residual
function
as
equals 0. Due to the
always leads to a nonzero
can be
by interpolation
and(11)
Here,
can be 1D to 3D integrals for different problems. Replacing
the
here by (9) yields
denotes the integral on the study domain, which
and
(12)
For each
variables
representation as
, we can obtainlinear equations with
. This equation group can be written in a matrix
(13)
where
denotes a matrix anddenotes a vector, with
(14)
(15)
Bytakingthesameforwardtimedifferencescheme
as (7), we can obtain the iteration relation via
Galerkin’s method in a matrix representation as
(16)
Equation (16) is the final iteration equation of the dynamic
model, which specifies the director profile correlation between
two consecutive time steps.
However, the abovementioned matrix
not easy to obtain because a good group of interpolation func
tion
defined on the entire domain is difficult to assign. Here
FEM can automatically generate an appropriate group of in
terpolation functions. The detailed derivation with FEM in ob
taining the matrix
and vector
The advantage of this method is evident that the global ma
trix
in (13) is uniquely determined by the mesh and the in
terpolation functions, as indicated in (A5) in the Appendix A.
As a result, in updating director profiles from (16), the matrix
and vectoris still
is provided in Appendix A.
Page 4
GE et al.: COMPREHENSIVE 3D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM 197
only needs to be calculated at the first iteration step and can
thenbestoredforsubsequentiterationsteps.Utilizingthisprop
erty greatly reduces the computing time in the director updating
procedure.
2) Potential Update Formulations via Ritz’s Method: From
the analysis of the potential update theory, the update of po
tential is equivalent to solve the Gaussian law of (8), or mini
mizing the system electric energy
a variational viewpoint. In our derivations, we use the Ritz’s
method to minimize the system electric energy, which inherits a
moreapparentphysicalandmathematicalmeaningthantoapply
Galerkin’s method in solving the Gaussian law.
The free electric energy
in the study domain is equal to
from
(17)
Here
and a scalar for isotropic media, such as the glass substrates.
Similarly, the approximate global potential solution
expressed by the global node value
function
as
is a tensor for anisotropic media, such as the LC layer,
can be
and global interpolation
(18)
Thus,
is a function of
, here
global variables
might be different with the total node
number in the director update. From mathematical theory, its
value is minimized where the relative partial derivative of
with respect to equals zero. This leads toequations as
(19)
This equation group can be further expressed as a matrix repre
sentation
(20)
Again, here
element in matrix
denotes a matrix and
has a form of
denotes a vector. Each
(21)
The detailed derivation of
FEM is described in Appendix B. Here for the nematic LC ma
terial, the tensor
is a symmetric matrix. This symmetric prop
erty can further simplify
to the following form:
and further implementation by
(22)
3) Weak Form Technique: As explained in above sections,
the matrix
for director and potential calculations have
definite forms and can be directly obtained from their cor
responding equations using FEM. Still, the difficulty lies in
the complicated derivations of global vector
though approximation of the derivations can be applied [12],
its accuracy in some complex device structures is not always
guaranteed. Therefore, the stateoftheart computer software
such as MAPLE [30] is used to generate FEMbased formula
tion derivations. We find the highest order of spatial derivative
of
with respect to , and
a result, the first order of interpolation/basis function [e.g.,
(A2)] will lose accuracy due to the fact that its second order
spatial derivative equals to zero. For example, let us assume
there is one term
shown in (A6). After substituting
with the first order interpolation function
derivative
diminishes to zero, which does not
always hold.
A good solution of this issue is to introduce the weak form
method. Derivations of (6) based on commercial software show
that
can be summarized into a general expression
in (15). Al
is up to the second order. As
in , as
into
, the second order
(23)
where
to the first order,
derivativeterms(e.g.,
efficient. Neglecting the subscript index
in the following form as
stands for the terms with highest derivatives up
denotes these second order
)with
we now rewrite (15)
astheco
(24)
Substituting
from (23) into (24) yields
(25)
In (25), the first term with derivatives up to first order can be
accurately calculated by FEM with firstorder interpolation
function. Although there are many terms with secondorder
derivatives, they can always be divided into two categories: 1)
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198 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Fig. 3. 2D boundary planes from a 3D bulk.
either
illustration of taking the weak form to accurately evaluate these
termsin(25),wejustselecttwotypicaltermsbelongingtothese
two categories: 1)
or equals toand 2) neithernor equals to . As an
, and
2)
. Using the integra
tion by parts, we obtain
(26)
(27)
where
andlateral directions.Becausethedirector
is fixed under strong anchoring assumption in the vertical
rection, i.e.,
2D integral term in (26) equals zero. This property further sim
plifies (26) into
and denote the boundary coordinates in the vertical
di
at these boundaries, the
(28)
which makes
calcu
lable by the firstorder interpolation functions. On the other
hand,the2Dintegralterm
in (27) is not always equal to zero on the boundaries in
the
direction. However, as illustrated in Fig. 3, the term
Fig. 4.
interpolation function.
(a) 1D interpolation function. (b) 2D interpolation function. (c) 3D
in (27) can be further
expressed on
and planes by
(29)
The 3D interpolation function
terpolation function on the boundary planes (
andin this case), as shown in Fig. 4(b) and
(c). To evaluate this term, we can use 2D FEM on the lateral
boundary planes to calculate (29). This is equivalent to weight
by 2D FEM with 2D interpolation functions
generatedfrom
ontheboundary planes(  plane at
and in this case). The lateral derivative terms in
and directions (e. g.,
difference from the boundary point and its adjacent ones (e.g.,
decays to a 2D in
plane at
) can be calculated by a finite
). With the
assistance of FEM, these 2D integral terms are calculable and
can be added to the 3D integral terms evaluated by first order
interpolation functions. Other second order terms in (25) can be
calculated similarly in the same way as (26)–(29).
In other words, by taking the weak form method into deriva
tions of (25), these secondorder derivative terms will degrade
one order, which can be calculated by using the firstorder
Page 6
GE et al.: COMPREHENSIVE 3D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM199
interpolation functions. Therefore, introducing the weak form
greatly facilitates the derivation of vector
approximations to sacrifice simulation accuracy.
4) Boundary Conditions and Other Topics: In solving the
LC director and potential profiles, their boundary conditions
in the study cell need to be welldefined. As shown in Fig. 5,
typical potential boundary conditions on electrodes are the
Dirichlet boundary conditions, i.e., the potential values on
electrodes 1 and 2 are fixed to assigned voltages. In the far
regions along the
direction, such as the outer boundaries
of the supporting substrates where no electrodes are present,
the Neumann conditions apply to the potential, i.e., the spatial
normal derivative of potential (e.g.,
the implementation of these boundary conditions for the poten
tial, the fixed values on the electrodes can be forced by matrix
and vector manipulation in (20), and the Neumann condition
for potential in the far regions are naturally satisfied in FEM
with the assumption that the electric energy is totally confined
in the LC cells as well as the substrate regions.
For the LC director calculations, in the vertical
the director
on the top and bottom alignment
layer—LC interfaces (at
ified by initial pretilt and azimuthal angle values. With strong
anchoring assumption, the director values on these interfaces
are fixed. A more general case is to take the surface anchoring
energy into consideration. The surface anchoring energy den
sity from the RapiniPapoular model [31], [32] on the vertical
boundaries has the following forms:
without any further
) equals zero. In
direction,
and ) are usually spec
(30)
where
coefficients (J/m ),
twist angle, respectively. Equation (30) can be expanded and
expressed in terms of the director components
andare the polar and azimuthal anchoring energy
andare the pretilt polar angle and
andas
(31)
The surface anchoring energy from above density function can
be added to the total Gibbs free energy integrated from (1).
Here, one needs to beware the difference in unit between the
surface anchoring energy density (J/m ) and the bulk Gibbs
energy density (J/m ). In the lateral direction (
canbedeterminedbyDirichletboundarycon
ditions, i.e., the
mann conditions, i.e.,
boundaryplanes,whichisdeterminedbyspecificdeviceconfig
urations. Another type of lateral boundary condition frequently
employed in LCD simulation is the periodic boundary condi
tion. The director
equal values at two later periodic boundaries, such as planes
at
and in Fig. 5. Because the LC cell patterns
in the pixel are usually periodically repeated in some devices
direction),
values are fixed, or by Neu
at lateral
and its derivatives have
such as in the IPS mode, the calculation results from one period
are enough to represent the entire pixel. The periodic boundary
conditiongreatlyreducesthecomputingloadinsimulatinglarge
size pixels with periodic configurations.
Those abovementioned boundary conditions are most widely
employed in the LC device simulations. However, boundary
conditions for the potential and director update are not confined
to these categories. In stead, they need to be designated in ac
cordance to the specific problems studied.
In addition to the inclusion of surface anchoring energy,
flexoelectric polarization [17], [18] is another important effect
that needs to be considered under certain conditions. Flexo
electric effect describes the spontaneous polarization generated
by a deformation of the director in nematic phase composed of
molecules, which exhibit shape asymmetry and have permanent
dipole moments. The flexoelectricity induced polarization will
interact with the applied electric fields and deform the LC
directors. A phenomenological expression for this polarization
is given by
andare the splay and bend flexoelectric coefficients.
The coupling of this polarization with electric fields and LC
deformations will introduce an additional term to the Gibbs
free energy density in (1) as
by the FEM as described in above sections as well. Detailed
introduction and discussion of the flexoelectric effect in LC
modeling can be found in [17], [18], [33]–[38], and references
therein.
Other effects that might affect the deformations LC directors
and potential distributions in a cell under certain conditions can
be referred to detailed discussion in the [39].
[33]. Here
, thus it can be counted
III. RESULTS AND DISCUSSION
To illustrate and validate the FEMbased LC dynamic mod
eling with the above derived formulations, we use a 2D inter
digital IPS structure with rectangular electrodes as an example
to compare our own developed method with the FEMbased
commercialsoftware2dimMOS.Inaddition,weusea3Dsuper
IPS with zigzag electrodes to compare our method with the
FDM.
A. TwoDimensional Conventional IPS Structure
Fig. 5 depicts a conventional 2D IPS structure. The LC ma
terial employed is MLC6692 (from Merck) with its parameters
listed as follows:
pN,
tional viscosity
Pa s. The LC cell gap is 4
LC pretilt angles on both substrates are 2 , and the rubbing an
gles are 80 away from the
axis in the
the electrodes 1 and 2 are located in the same bottom planes at
, with an equal length of 4 m and a separation distance
of 8
m.
In the simulation, the electrode 1 is applied with 6 V, and
the electrode 2 is grounded. The LC directors on the vertical
boundaries are assumed to be strongly anchored in our simu
lations. The
direction periodic boundary conditions in Fig. 5
are applied to both potential and LC director calculations. In the
2dimMOS, since no periodicboundary conditionsare available,
pN,pN, and rota
m, the
plane. In Fig. 5,
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200JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Fig. 5. A conventional 2D inplane switching structures.
we just expand two more periods in both left and right
tions to approximate the periodicity in the 2dimMOS (a total of
five periods). Here we take the centertruncated region (one pe
riod) from the 2dimMOS to compare with our simulation based
on the derived method in Section II.
Fig. 6(a) and (b) plot the director distribution at 4 and 16 ms
with 6V applied voltage, respectively. Apparently, the effect
of director tilt deformation near the interdigital electrode side
regions becomes more and more evident as the time changes
from 4 to 16 ms. The director twist deformation effect in
the interelectrode regions also shows the same tendency.
The transmittance dependence on different
8, 12, and 16 ms are plotted in Fig. 7. At
the transmittance in the curve almost reaches the maximum
value which is 35% under real crossed polarizers. Due to
the structure symmetry, the directors in the regions above
each electrode’s surface have little twist in response to the
electric fields at different times, which results in dark regions
above the electrodes. From Fig. 7, our simulation results agree
well with those obtained from 2dimMOS, which indicates
our derivations are dependable.
direc
positions at 4,
ms,
B. ThreeDimensional Super IPS Structure
The super IPSstructure with zigzag electrodesis then studied
by both derived FEM and FDM. The electrode shape and di
mension in the bottom  plane are depicted in Fig. 8. And the
crosssection view of the cell in  plane is same as Fig. 5. In
our simulation, same LC material and cell gap as the 2D sim
ulation is taken in this configuration. The pretilt angles of LC
directors are 2 from the  plane, and the rubbing angles are
90 away from the
axis in the  plane. Six volts are applied
on the left electrode, while the right one is grounded to 0 V. The
LC directors on the vertical boundaries (in
sumed to be strongly anchored in our simulations. Fig. 8 only
shows a single period of the electrode structure within a pixel.
The periodic boundary conditions in the lateral directions (both
in
anddirections) are employed in both potential and LC
director calculations.
Fig. 9(a)and (b) showsthecrosssection view ofthepotential
and director distributions at
sampling the crosssection view of  plane, the
m, and m is taken for the sampling of
direction) are as
andms, respectively. In
is taken at

(a)
(b)
Fig. 6.
distribution at 16 ms with 6 V applied.
(a) LC director distribution at 4 ms with 6V applied. (b) LC director
crosssection view. As can be seen from these figures, the LC
directors in the regions between the electrodes have been driven
to the parallel direction of electric fields in these regions at
ms, and in these regions not much difference is observed for
ms and ms. However, it takes a long time for
the directors in the regions above the electrode surfaces to be
fully rotated, as seen from
ms [see Fig. 10(a) and (b)]. This is because the electric
fieldlines abovetheelectrodesaremoreverticallydistributed(
direction), while the electric fields in the interelectrode regions
aremostlyalongthehorizontaldirection.Similarlyasinthe2D
case, the less rotation in these regions generates dark regions as
illustrated in Fig. 11(a)–(c).
Fig. 11(a) to (c) shows the transmittance patterns at different
time steps of
and
(top) and FDM (bottom). It can be seen that these two methods
generate almost the same transmittance patterns. The main dis
crepancy of these two method comes from the center positions
of electrodes (at
m), where the electrode angle is very
ms [see Fig. 9(a) and (b)] and
ms from the derived FEM
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GE et al.: COMPREHENSIVE 3D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM 201
Fig. 7.Transmittance versus position at time equal to 4, 8, 12, and 16 ms from the derived FEM based method and the 2dimMOS.
Fig. 8.Super IPS structure with zigzag electrodes in simulation.
sharp. The FEM and FDM might generate different LC director
orientations, i.e., at these positions the LC director might rotate
either in the
direction or in the
other regions, the results of these two methods match well with
each other. Both methods show that the vertically distributed
electric fields above the electrodes make these regions have rel
ative low transmittances. Here the asymmetric dark region pat
terns in the
direction originate from the 2 pretilt angle in the
 plane.
In addition to IPS structure described above, the derived
method is also applicable to other complex LCD structures,
such as multidomain vertical alignment (MVA) [40] and pat
terned vertical alignment (PVA) [41] modes.
direction. However, in
IV. CONCLUSION
We have derived a comprehensive dynamic LC director cal
culation model based on FEM. The vector represented LC
director is calculated by iteration between the calculations of
potential and director distributions. Weak form is introduced
to simplify the derivation of director calculation formulations.
Excellent agreement is obtained between this method and the
commercial 2dimMOS software in calculating a 2D IPS struc
ture. Our method is also validated by the comparison between
FEM and FDM in simulating a 3D super IPS structure, in
which good agreement is obtained. The derived method is also
applicable to other complex LC device designs and optimiza
tions thereafter.
APPENDIX A
FEM IMPLEMENTATION FOR DIRECTOR UPDATE VIA
GALERKIN’s METHOD
FEM is employed to get the solutions for the matrix
vector discussed in Sections IIB1. In FEM theory, a study do
main
can be divided into element domains with
In each element domain, there are
tion functions (considering firstorder interpolation functions)
as well. In any element domain (e.g., th element) the local di
rector value
can be interpolated by its
using
local basis functions as
and
nodes.
nodes withinterpola
local node values
(A1)
where
tion in the th local element, respectively. The form of
andarethe thnodevalueandinterpolationfunc
is
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202JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Fig. 9.
in ?? plane at ? ? ? ?m at ? ? ?? ms with 6 V. (b) LC potential (top figure)
and director (bottom figure) distributions in ?? plane at ? ? ? ?m at ? ? ??
ms with 6 V.
(a) LC potential (top figure) and director (bottom figure) distributions
welldefined.Forexample,thefirstorderbasisfunctionina3D
tetrahedral element would be
(A2)
Thecoefficients
coordinate values and the volume of the tetrahedron [3], [4].
Detailed expressions of these coefficients are provided in Ap
pendix C. Similarly, the residual
can be weighted by the element basis function as
,and canbedeterminedbythenode
in each element domain
and(A3)
Fig. 10.
in ?? plane at ? ? ? ?m at ? ? ?? ms with 6 V. (b) LC potential (top figure)
and director (bottom figure) distributions in ?? plane at ? ? ? ?m at ? ? ??
ms with 6 V.
(a) LC potential (top figure) and director (bottom figure) distributions
Substituting the
timedifference scheme leads to
and expanding the above equation with a
(A4)
where
(A5)
and
(A6)
By applying the assembling process of typical FEM procedure,
the global matrix
and global vector
from the element matrix
and element vector
in (13) can be obtained
.
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GE et al.: COMPREHENSIVE 3D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM203
Fig. 11.
FDM (bottom figure) at ? ? ?? ms with 6 V. (b) The transmittance patterns
in the ?? plane by FEM (top figure) and FDM (bottom figure) at ? ? ?? ms
with 6 V. (c) The transmittance pattern in the ?? plane by FEM (top figure) and
FDM (bottom figure) at ? ? ?? ms with 6 V.
(a) Transmittance pattern in the ?? plane by FEM (top figure) and
APPENDIX B
FEM IMPLEMENTATION FOR POTENTIAL UPDATE VIA RITZ’s
METHOD
To derivethedefinite form of the matrix
we first express the electric field
in Sections IIB.2,
into a vector form as
(B1)
Therefore, the free electric energy
panded to
in (17) can be further ex
(B2)
Adetailedcheckof
of the
’s spatial derivative. Therefore once the
substituted in (19), the derivative of
be achieved by differentiation by parts
formrevealsthatitisaquadraticfunction
in (18) is
with respect tocan
(B3)
Fromaboveequation,eachelement
be obtained by differentiating the (B3) with respect to
ofmatrix in(20) can
(B4)
This will lead to the expression of (21), which can be uniquely
defined by FEM as below.
By applying the similar mesh and derivation as discussed in
the FEM implementation of director profile section, we can ob
taintheelementmatrix
foreachelement,whichhas
and
basis functions. The
nodes
can be written as
(B5)
From the above equation, one can get a definite
and the coordinate values of the small local domain
value once
are
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204JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
specified, as illustrated in Appendix C. By applying the assem
blingprocessofFEM,theglobalmatrix
the element matrix
.
canbeobtainedfrom
APPENDIX C
FEM INTERPOLATION FUNCTIONS
The local element shape for 1D to 3D cases is shown in
Fig. 4(a) to (c). For simplicity in the expression, we just ne
glect the superscript element index
nate values
and the local interpolation functions
compared to
and
For the 1D case in Fig. 4(a), the firstorder interpolation (or
basis) function can be expressed as
for both the local coordi
, as
.
(C1)
(C2)
These two interpolation functions equal zero out of the local
region. As we can see that the
and
on the other one.
For the 2D triangle shaped first order interpolation (or basis)
functions as shown in Fig. 4(b), they have the expression as
is equal to 1 on the th node,
(C3)
(C4)
(C5)
where
is the area of the triangle, which is equal to
(C6)
here
denotes the determinant of the inside matrix. And
, andare equal to
(C7)
(C8)
(C9)
respectively. Here the interpolation function is equal to zero out
of the triangular region.
For the 3D tetrahedron shaped firstorder interpolation (or
basis) functions as shown in Fig. 4(c), they can be expressed as
(C10)
(C11)
(C12)
(C13)
where
is the volume of the tetrahedron, which is equal to
(C14)
and
, andare equal to
(C15)
(C16)
(C17)
(C18)
respectively. The interpolation function is equal to zero out of
the tetrahedral region.
REFERENCES
[1] A. Taflove, Computational Electrodynamics: The FinteDifference
TimeDomain Method.Reading, MA: Artech House, 1995.
[2] J. E. Anderson, P. Watson, and P. J. Bos, LC3D: Liquid Crystal Display
3D Directory Simulator, Software and Technology Guide.
MA: Artech House, 1999.
[3] J. Jin, The Finite Element Method in Electromagnetics, 2nd ed.
away, NJ: WileyIEEE Press, 2002.
[4] Y. W. Kwon and H. Bang, The Finite Element Method Using
MATLAB. BocaRaon, FL: CRC Press, 2000.
[5] AutronicMELCHERS GmbH
http://www.autronicmelchers.com/index.htm
[6] [Online]. Available: http://www.shintech.jp/eng/index_e.html
[7] [Online]. Available: http://www.sanayisystem.com/introduction.html
[8] V. G. Chigrinov, H. S. Kwok, D. A. Yakovlev, G. V. Simonenko, and V.
I. Tsoy, “Invited paper: LCD optimization and modeling,” in SID Symp.
Dig., vol. 28.1, May 2004, pp. 982–985.
[9] LCQuest [Online]. Available: http://www.eng.ox.ac.uk/lcquest/
[10] J. B. Davies, S. E. Day, F. Di Pasquale, and F. A. Fernandez, “Finite
element modeling in 2D of nematic liquid crystal structure,” Electron.
Lett., vol. 32, pp. 582–583, Mar. 1996.
[11] F. Di Pasquale et al., “Twodimensional finiteelement modeling of ne
matic liquid crystal devices for optical communications and displays,”
IEEE J. Sel. Topics Quantum Electron., vol. 2, pp. 128–134, Apr./May
1996.
[12] F.A.Fernandez,S.E.Day,P.Trwoga,H.F.Deng,andR.James,“Three
dimensional modeling of liquid crystal display cells using finite ele
ments,” Mol. Cryst. Liq. Cryst., vol. 375, pp. 291–299, 2002.
[13] F.A.Fernandez,H.F.Deng,andS.E.Day,“Dynamicmodelingofliquid
crystal display cells using a constant charge approach,” IEEE Trans.
Magn., vol. 38, no. 2, pp. 821–824, Mar. 2002.
[14] H. J. Yoon, J. H. Lee, M. W. Choi, J. W. Kim, O. K. Kwon, and T.
Won, “Comparison of numerical methods for analysis of liquid crystal
cell: Inplane switching,” in SID Symp. Dig., vol. 50.1, May 2003, pp.
1378–1381.
Reading,
Piscat
[Online].Available:
Page 12
GE et al.: COMPREHENSIVE 3D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM 205
[15] S. H. Yoon, C. S. Lee, S. I. Yoon, J. H. Lee, H. J. Yoon, M. W. Choi,
J. W. Kim, and T. Won, “Threedimensional numerical simulation for
understanding the fringe field effect on the dynamic behavior of liquid
crystal,” Mol. Cryst. Liq. Cryst., vol. 413, pp. 333/[2469]–343/[2479],
2004.
[16] I. A. Yao, J. J. Wu, and S. H. Chen, “Threedimensional simulation of
the homeotropic to planar transition in cholesteric liquid crystals using
the finite elements method,” Jpn. J. Appl. Phys., vol. 43, pp. 705–708,
Feb. 2004.
[17] R. B. Meyer, “Piezoelectric effects in liquid crystals,” Phys. Rev. Lett.,
vol. 22, pp. 918–921, 1969.
[18] P. G. De Gennes and J. Prost, The Physics of Liquid Crystals, 2nd
ed.Oxford, U.K.: Oxford Science, 1993.
[19] M. Ohe and K. Kondo, “Electrooptical characteristics and switching
behavior of the inplane switching mode,” Appl. Phys. Lett., vol. 67, pp.
3895–3897, Oct. 1995.
[20] S. Endoh, M. Ohta, N. Konishi, and K. Kondo, “Advanced 18.1inch
diagonal superTFTLCD’s with mega wide viewing angle and fast re
sponse speed of 20 ms,” IDW’99, pp. 187–190, Dec. 1999.
[21] R.N.ThurstonandD.W.Berreman,“Equilibriumandstabilityofliquid
crystal configurations in an electric fied,” J. Appl. Phys., vol. 52, pp.
508–509, Jan. 1981.
[22] D. W. Berreman, “Numerical modeling of twist nematic devices,” Phil.
Trans. R. Soc. Lond., vol. A 309, pp. 203–216, 1983.
[23] G. Haas, S. Siebert, and D. A. Mlynski, “Simulation of inhomogeneous
electric field effects in liquid crystal displays,” in Proc. 9th Int. Display
Conference, Society for Information Display and Institute of Television
Engineers of Japan, vol. 89, 1989, p. 524.
[24] D. W. Berreman and S. Meiboom, “Tensor representation of Oseen
Frank strain energy in uniaxial cholesterics,” Phys. Rev. A, vol. 30, pp.
1955–1959, Oct. 1984.
[25] S. Dickmann, J. Eschler, O. Cossalter, and D. A. Mlynski, “Simulation
of LCD’s including elastic anisotropy and inhomogeneous field,” SID
Dig. Tech. Pap., vol. 24, pp. 638–641, May 1993.
[26] H. Mori, E. C. Gartland Jr., J. R. Kelly, and P. J. Bos, “Multidimensional
director modeling using the Q tensor representation in a liquid crystal
cell and its application to the ? cell with patterned electrodes,” Jpn. J.
Appl. Phys., vol. 38, pp. 135–146, Oct. 1999.
[27] C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc., vol.
29, pp. 883–???, 1933.
[28] F. C. Frank, “On the theory of liquid crystals,” Discuss. Faraday Soc.,
vol. 25, pp. 19–???, 1958.
[29] J. E. Anderson, C. Titus, P. Watson, and P. J. Bos, “Significant speed and
stability increases in multidimensional director simulations,” SID Tech.
Digest, vol. 31, pp. 906–909, 2000.
[30] Maplesoft, a divison of Waterloo Maple, Inc. [Online]. Available:
http://www.maplesoft.com
[31] A. Rapini and M. J. Papoular, “Distortion d’une lamelle nématique sous
champ magnétique conditions d’ancrage aus parois,” J. Phys. Colloq.,
vol. 30, p. C4, 1969.
[32] D.Demus,J.Goddby,G.W.Gray,andHW.Spiess,HandbookofLiquid
Crystal, V. Vill, Ed: WileyVCH, 1998, vol. 1.
[33] P.RudquistandS.T.Lagerwall,“Ontheflexoelectriceffectinnematics,”
Liq. Cryst., vol. 23, pp. 503–510, May 1997.
[34] A. Mazzulla and F. Ciuchi, “Optical determination of flexoelectric coef
ficients and surface polarization in a hybrid aligned nematic cell,” Phys.
Rev. E, vol. 64, p. 021708, July 2001.
[35] A. J. Davidson and N. J. Mottram, “Flexoelectric switching in bistable
nematic device,” Phys. Rev. E, vol. 65, p. 051710, May 2002.
[36] C. V. Brown and N. J. Mottram, “Influence of flexoelectricity above the
nematic Fréedericksz transition,” Phys. Rev. E, vol. 68, p. 031702, Sept.
2003.
[37] D. L. Cheung, S. J. Clark, and M. R. Wilson, “Calculation of flexoelec
tric coefficients for a nematic liquid crystal by atomistic simulation,” J.
Chem. Phys., vol. 121, pp. 9131–9139, Nov. 2004.
[38] L. A. ParryJonesand S.J. Elston,“Flexoelectricswitching in zenithally
bistable nematic device,” J. Appl. Phys., vol. 97, p. 093515, Apr. 2005.
[39] L. M. Blinov and V. G. Chigrinov, Electrooptic Effects in Liquid Crystal
Materials.New York: SpringerVerlag, 1994.
[40] A. Takeda, S. Kataoka, T. Sasaki, H. Chida, H. Tsuda, K. Ohmuro, T.
Sasabayashi, Y. Koike, and K. Okamoto, “A superhigh image quality
multidomain vertical alignment LCD by new rubbingless technology,”
SID Tech. Dig., vol. 29, pp. 1077–1080, May 1998.
[41] K. H. Kim, K. H. Lee, S. B. Park, J. K. Song, S. N. Kim, and J. H.
Souk, “Domain divided vertical alignment mode with optimized fringe
fieldeffect,”inProc.18thInt.DisplayResearchConf.(AsiaDisplay’98),
1998, pp. 383–386.
Zhibing Ge received the B.S. and M.S. degrees
in electrical engineering from Zhejiang University,
Hangzhou, China, and University of Central Florida,
Orlando, in 2002 and 2004, respectively, and is
currently working toward the Ph.D. degree at De
partment of Electrical and Computer Engineering,
University of Central Florida, Orlando. His Ph.D.
study concentration is in liquid crystal display
modeling, transflective liquid crystal displays, and
numerical analyses and optimization of liquid crystal
devices.
Thomas X. Wu received the B.S.E.E. and M.S.E.E.
degrees from the University of Science and Tech
nology of China (USTC), Anhui, China, in 1988
and 1991, respectively, and the M.S. and Ph.D.
degrees in electrical engineering from the University
of Pennsylvania, Philadelphia, in 1997 and 1999,
respectively.
From 1991 to 1995, he was with the faculty of the
Department of Electrical Engineering and Informa
tion Science, USTC, as an Assistant and Lecturer. In
Fall 1999, he joined the Department of Electrical and
Computer Engineering, University of Central Florida (UCF), Orlando, as an
Assistant Professor. His current research interests and projects include complex
media, liquid crystal devices, electronic packaging of RF SAW devices, elec
trical machinery, magnetics and EMC/EMI in power electronics, chaotic elec
tromagnetics, millimeterwave circuits, and CMOS/BiCMOS RFICs.
Dr. Wu was awarded the Distinguished Researcher Award from the College
of Engineering and Computer Science, University of Central Florida, in April
2004. Recently, he was listed in Who’s Who in Science and Engineering, Who’s
Who in America, and Who’s Who in the World.
Ruibo Lu received the M.S. degree in applied
physics from Department of Physics, East China
University of Science and Technology, Shanghai,
China, in 1995, and the Ph.D. degree in optics from
Department of Physics, Fudan University, Shanghai,
China, in 1998. His research work for the Ph.D.
degree focused on liquid crystal alignment and
ferroelectric liquid crystal devices for display and
advanced optical applications.
He was part of the faculty in Department of
Physics, and later in Department of Optical Science
and Engineering, Fudan University, Shanghai, China, from 1998 to 2001. He
was an optical engineer in Lightwaves 2020 Inc., San Jose, CA, from 2001 to
2002. Since then, he joined the School of Optics/CREOL (now as College of
Optics and Photonics), University of Central Florida, Orlando, as a research
scientist. His research interests include liquid crystal display technology, wide
viewing angle for liquid crystal TVs, liquid crystal components for optical
communications and optical imaging using liquid crystal medium.
Xinyu Zhu received the B.S. degree from Jilin
University, China, in 1996, and the Ph.D. degree
from Changchun Institute of Optics, Fine Mechanics
and Physics, Chinese Academy of Sciences, China in
2001. His Ph.D. research work involved mainly the
reflective liquid crystal display with single polarizer.
He is currently a research scientist at the College
of Optics and Photonics, University of Central
Florida, Orlando. His current research interests
include reflective and transflective liquid crystal
displays, liquidcrystalonsilicon projection display,
wide view liquid crystal displays, and adaptive optics application with nematic
liquid crystals.
Page 13
206JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Qi Hong received B.S. degree from the Nanjing
University of Aeronautics and Astronautics, Nan
jing, China, in 1992, and the M.S.E.E. degree from
the University of Central Florida, Orlando, in 2002,
where he is currently working toward the Ph.D.
degree in the electrical engineering. His doctoral re
search topics include liquid crystal device modeling,
wide viewing angle and fast response liquid crystal
display.
He was design engineer at the Xiaxin Electronics
Company Ltd., Xiamen, China, from 1992 to 2000.
ShinTson Wu (M’98–SM’99–F’04) received the
B.S. degree in physics from National Taiwan Uni
versity, and the Ph.D. degree from the University of
Southern California, Los Angeles.
He is currently a PREP professor at College of
Optics and Photonics, University of Central Florida
(UCF), Orlando. Prior to joining UCF in 2001, he
worked at Hughes Research Laboratories, Malibu,
CA, for 18 years. His studies at UCF concentrate
in foveated imaging, biophotonics, optical commu
nications, liquid crystal displays, and liquid crystal
materials. He has coauthored two books: Reflective Liquid Crystal Displays
(Wiley, 2001) and Optics and Nonlinear Optics of Liquid Crystals (World
Scientific, 1993), four book chapters, and over 220 journal papers.
Dr. Wu is a Fellow of the IEEE, Society for Infformation Display (SID), and
Optical Society of America (OSA).
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