Hydrodynamics of fringing-field induced defects in nematic liquid crystals

Consumer demand for high resolution and high refresh-rate displays has naturally led to the fabrication of liquid crystal displays with ever smaller pixels. As a consequence, fringing fields between adjacent pixels grow in magnitude, leading to abrupt changes in orientation. Electric field strengths above some threshold can lead to order melting and, in turn, disclinations. This paper presents accurate modeling of disclinations induced by fringing fields due to interdigitated electrodes in a nematic liquid crystal calculated by means of the Landau – de Gennes theory. Disclination paths are determined while taking into account the flow of the liquid crystal. Making use of interdigitated electrodes, precise electrical control over the creation and positioning of defects is demonstrated for homeotropic, planar, hybrid, and in-plane surface alignments.


I. INTRODUCTION
As liquid crystal displays increase in resolution and complexity, there is a tendency for defects to form in the liquid crystal layer, induced by strong electric fields or geometrical features.Pixels are typically designed so as to avoid the formation of defects, which can be detrimental to image quality.Similarly, high pixel densities are demanded by non-display applications of liquid crystals, including holography and beam-steering. 1 For example, Spatial Light Modulators (SLMs) for electrical hologram generation based on Liquid Crystal on Silicon (LCoS) panels are available 2 with pixel pitches of 3.74 μm.
As pixels shrink, large electric fields can arise in the inter-pixel gap leading to order melting and disclinations, in turn. 3isclinations can be of high density, playing an important and potentially advantageous role in device operation.Such fringing field induced disclinations are the product of pincement, an effect which is well studied in liquid crystal cells with planar alignment [4][5][6][7][8] or in multi-domain structures. 9For very high resolutions, pixels become narrow compared to the cell gap, which is fixed by the optical path length.Electrically driven reorientation is confined to the vicinity of the pixelated electrode since the electric field extends only partially into the liquid crystal bulk. Lindquist 11 demonstrated a phase grating formed by interdigitated electrodes with an electrode pitch of 8 μm and reported the successful fabrication of a 0.8 μm device.Predominantly, these studies focus on the disclination structures that result at steady-state; however, the impact of the electrode spacing on the response time has also been assessed experimentally 12 with electrode pitches as small as 2 μm.Proposed applications include reconfigurable diffraction gratings 11,17,18 and fast switching displays 13,19 with typical 13 electrode pitches of 8 μm or less.Additionally, such disclinations can be used to reconfigure liquid crystal colloids. 14,20olloidal particles are attracted to disclination lines since they essentially exclude a volume of high energy should they decorate a line. 21Small particles can be trapped within the saturn ring defect associated with a larger particle 21 or within disclinations arising in chiral-nematic filled wedge cells. 22Defect networks induced by topologically patterned substrates have been shown to trap particles and are reconfigurable by external fields, affording control over particle position. 23Recently, this trapping and positional control has been demonstrated by electrically induced defects due to crossed electrodes 24 and interdigitated electrodes (with hybrid alignment). 14Bacteria immersed in lyotropic liquid crystals can similarly be trapped and transported by defects. 25n this paper, the hydrodynamics of disclination formation by fringing fields due to interdigitated electrodes is studied theoretically.Conditions necessary to create disclinations are explored for homeotropic, planar, hybrid, and in-plane surface alignments with both positive and negative dielectric anisotropy liquid crystals.Both the anchoring strength and device proportions have a profound influence on threshold voltages required to induce disclinations, as demonstrated in Secs.IX and X.

II. MODEL
Both orientation and order of the liquid crystal may be described by the nematic order tensor 4 Q which is traceless so that it vanishes in the isotropic phase and symmetric.A tensorial representation, over a purely vectorial one, is necessary to fully describe the biaxial ordering adopted by molecules in the vicinity of defects.The Q-tensor may be defined as , where S 1 and S 2 represent the degree of order about the orthogonal unit vectors n and m, respectively.At equilibrium, Q minimizes the Landaude Gennes free energy functional, 4,26 where f (Q, @Q) is the free energy density given by and comprises elastic, f D (Q, @Q); bulk, f B (Q); and electrostatic, f E (Q) free energy densities defined as Summation over repeated indices is assumed.Here, A, B, and C are material bulk constants; L 1 , L 2 , L 3 , and L 6 are elastic constants; ε αβ is the permittivity tensor; u is the electric potential found by minimizing the electrical energy Ð Ω f E (Q)dV with respect to u; and P α is the flexoelectric polarization, 27 The surface free energy density, allows for different azimuthal and zenithal anchoring strengths. 28oefficients W 1 and W 2 are anchoring strengths, and the unit vectors ν, ζ, and the preferred alignment direction are orthogonal.
The order parameter at the surface is controlled by coefficient a. Dynamic evolution of the Q-tensor is governed by where ϵ αβγ is the Levi-Civita tensor and λ and λ γ are Lagrange multipliers that enforce the zero trace and symmetry requirements of the Q-tensor.0][31] The molecular field, h αβ , is given by the first variation in F (Q, @Q), The viscous molecular field, h 0 αβ , is defined as 29 where , and the material derivative d dt is @ @t þ v γ @ γ .Flow of the liquid crystal (assumed incompressible) is governed by the Navier-Stokes equation, written in terms of a generalized stress tensor, where ρ is the density, v α is the velocity, and p is the pressure.Three contributions to the overall stress tensor are considered: 29 the viscous stress, σ v αβ , the distortion stress, σ d αβ , and the Maxwell stress, σ e αβ , where E α ¼ À@ α u is the electric field and In the limit of a constant order parameter, the above formulation reduces to the Ericksen-Leslie theory.The more general variational theory proposed by Sonnet et al., 32 reduces to an equivalent set of equations when the Rayleigh dissipation functional is restricted to be a bilinear form in N αβ and A αβ and up to quadratic order in Q.

A. Numerical details
A finite element approach is taken to solve the aforementioned set of partial differential equations, 33,34 implemented in Matlab/C++.Weak forms and matrix equations that result are detailed in our previous work. 33An overview of the methodology follows.
In the cases considered herein, reorientation of the liquid crystal is driven by the applied voltage.Thus, for the sake of stability, it is advantageous to take a monolithic approach in computing the Q-tensor and electric potential whereby the solution vector takes the form {q 1 , q 2 , q 3 , q 4 , q 5 , u}, where Q ¼ q i T i .Basis tensors, T i , satisfy the tracelessness and symmetry constraints of Q. 35 Consequently, Lagrange multiplier terms in (8) may be dropped.Crank-Nicolson time stepping is employed with Newton-Raphson iterations within each time-step to deal with the nonlinearity of the governing equations.Velocity and pressure fields are solved for at the half time-step (consistently within each Newton iteration) so that the @Q=@t terms appearing in (12) can be evaluated by central differences, thus maintaining O(Δt 2 ) accuracy of the Crank-Nicolson scheme.The steadiness approximation is made, whereby the time derivative of the velocity appearing in ( 12) is dropped and the velocity is assumed pseudo-steady with respect to Q.
All variables, namely, q i , v α , and u, are discretized by second order triangular elements with the exception of the pressure, for which first order elements are used in a mixed order scheme for the flow as a means to inhibit oscillatory pressure solutions.Adaptive meshing (h-adaptation) driven by an empirical error estimate derived from the free energy bar the electrostatic contribution serves to reduce the number of degrees of freedom.The time-step is dynamically adjusted to maintain a constant time-integration error, 34 varying from 5 ns during nucleation to 0:2 ms near steady-state.
Elastic constants {L 1 , L 2 , L 3 , L 6 } can be obtained from the splay, twist, and bend elastic constants {K 11 , K 22 , K 33 } and K 24 using the expressions 41 wherein the equilibrium value of the order parameter, S 0 , is given by the stationary value of f B with respect to S, yielding The permittivity tensor is defined as In the limit of a constant order parameter, the flexoelectric polarization is equivalent to

C. Device description
A periodic structure with two interdigitated electrodes forms the modeling window with a period of 1:2 μm and height of 1:6 μm, as depicted in Fig. 1.The electrode gap is set to match the electrode width of 0:3 μm.An upper electrode is included which offers an additional degree of control over the liquid crystal bulk.Due to the stretched aspect ratio of the device, the upper electrode has negligible influence on the electric field due to the interdigitated electrodes when grounded (unbiased).
The fabrication of interdigitated electrodes at these length scales is difficult but can, for example, be achieved using deep UV, x-ray, or electron beam lithography techniques.In addition, similar switching mechanisms to those described herein are possible using the principle of fringe-field switching, 13,19 whereby the same potential is applied to in-plane electrodes positioned above a dielectric layer seated on a ground plane, which reduces the risk of electrical shorts.
Non-slip boundary conditions are applied for the velocity.Unless otherwise stated (T À T * ) ¼ À2 K is assumed.

III. HOMEOTROPIC ALIGNMENT
Reorientation of a liquid crystal due to fringing fields in a homeotropic cell is shown in Fig. 2. For simplicity of presentation, director streamlines are shown, whereby the director corresponds to the eigenvector associated with the largest eigenvalue of the Q-tensor.Prior to eigendecomposition, the Q-tensor is interpolated

Journal of Applied Physics
from the computational mesh, shown in Fig. 3, onto a regular grid to facilitate plotting.Initially, when a voltage is applied, the liquid crystal in the inter-pixel gap reorients toward the horizontal, giving rise to a bend-splay wall.For low applied voltages, the core of the wall remains vertical and all changes in orientation are limited to a 0:3 μm thick layer above the lower alignment layer.
Increasing the applied voltage further compresses the wall, leading to a drop in order parameter and an increase in biaxiality within the wall.Above a specific value of applied voltage, +3:97 V, there is an exchange of eigenvalues of the Q-tensor, giving rise to a +1=2 disclination pair, as shown in Fig. 2(c).This transformation from walls to disclination lines is known as pincement. 4hroughout this paper, this value of voltage will be referred to as the threshold voltage, U th , and should not be confused with the Fréedericksz threshold.
The applied voltage need only exceed the threshold voltage for a short time (33μs for an applied voltage of +3:97 V) to induce disclinations, after which a much reduced voltage is capable of separating the disclinations.Figure 4 shows the variation in order parameter at the same time, 1 μs after disclination formation, where the drop in order associated with the +1=2 disclination lines is clearly visible.
Horizontal alignment between the disclination pairs is favored due to the applied voltage, encouraging their separation.The þ1=2 lines move downward until becoming stationary 14 nm above the lower alignment layer while the À1=2 lines rise.A parallel can be drawn with growth of horizontal and vertical domains seen in the π cell, which are topologically distinct states and are separated by disclination lines.In a π cell, an applied voltage gives rise to a uniform vertical electric field in the liquid crystal layer.Disclinations separating coexisting horizontal and vertical domains move with a constant velocity that depends on the applied voltage. 42In contrast, the use of interdigitated electrodes gives rise to an electric field that diminishes with distance within the liquid crystal layer, and, thus, disclinations separate until the force due to the electric field matches the elastic attractive force.shows the disclination trajectories when the applied voltage is removed.Confinement alters the behavior in comparison to isolated disclination pairs. 30,31hen the voltage is removed, velocities of þ1=2 and À1=2 disclinations differ markedly even if flow is neglected.Elastic distortion at the alignment layer expels the þ1=2 disclinations away rapidly.As seen in the annihilation of isolated disclination pairs, both theoretically 30,31 and experimentally, 43 here too the velocity field preferentially accelerates the þ1=2 lines.
In Fig. 6(a), the velocity field after applying +3:97 V for 0:1 ms is shown.At this time, the þ1=2 disclinations are already stationary above the lower alignment layer and have little influence on the velocity field.The À1=2 disclinations continue to rise and the velocity field near each disclination is in the positive z-direction and serves to compound the disclination velocity.Figure 6 (b) shows the velocity field 0:1 ms after removing the voltage.
By varying the potential difference across the lower electrodes, it is possible to control the vertical position of the +1=2 disclinations, as shown in Fig. 7.When the voltage falls below +1:03 V, the attractive elastic force overcomes the electric and the disclination lines annihilate to restore the starting vertical configuration.Disclinations annihilate at different locations than observed earlier in Fig. 5, where the applied voltage was removed abruptly.Gradually reducing the applied voltage serves to pin the þ1=2 disclinations above the lower alignment layers to layer, while the À1=2 disclinations descend.Relaxation may be accelerated by applying the same voltage to the interdigitated electrodes.
Varying the bias applied to the upper electrode provides a means to control the lateral separation of the À1=2 disclinations, as shown in Fig. 8(b).The bias voltage augments the electric field at the nucleation point and hence a lesser voltage of +3:89 V is able

IV. PLANAR ALIGNMENT
In this section, the disclination states induced with a planar alignment layer are studied.The surface director is chosen in the x-direction with a pre-tilt of θ 0 ¼ 2 , corresponding to antiparallel rubbing.A small pre-twist of f 0 ¼ 0:1 is chosen to allow in-plane reorientation of the liquid crystal to occur.When a voltage is applied, reorientation of the liquid crystal is most prominent above the center of each electrode, where a bend-splay wall arises.
Provided the applied voltage is above some threshold, U C , this bend-splay wall transforms into a twist wall.Thus, U C provides a measure of the elastic anisotropy.Typically, U C % 2U F , where U F is the Fréedericksz threshold, given by U The twist evolves at a rate determined by the director relaxation time 6 τ γ 1 ξ 2 E =ΔK.This depends on the electrical coherence length, , U is the applied voltage, d ¼ 0:2 μm is the effective height of the wall, and ΔK (K 11 þ K 33 )=2 À K 22 is the elastic anisotropy.Three factors are neglected in this approximation: flow of the liquid, which reduces τ; the influence of the pre-twist; and the change in electric potential due to reorientation (particularly, if Δε is large), which reduces U C .We find that, just below U th , transformation to a fully twisted wall takes 20 μs.
If the applied voltage is ramped sufficiently slowly, twisting of the wall is complete, but for stepped voltages, the transformation is merely partial when pincement occurs.Higher voltages are required to form disclinations from the more compact twist walls.Consequently, U th differs for ramped and stepped voltage waveforms, taking the values +5:41 V (for ramp rates of 55 mV/μs or less) and +5:19 V, respectively.If the pre-twist is increased, twist walls form more rapidly and U th for a stepped waveforms tends to the ramped value (for example, U th ¼ +5:36 V with f 0 ¼ 1 ).
Applying a +5:19 V voltage step gives rise to pairs of +1=2 disclinations after 18 μs that form 12 nm to the left of the electrode center (a displacement due to the pre-tilt).As the disclinations diverge, all twist that was once associated with the walls is lost.The director configuration shown in Fig. 9(a) results at steady-state, where planar orientation persists in the bulk.
When a bias voltage is applied to the upper electrode (that matches the potential of the positive lower electrode), it serves to augment or diminish the electric field above alternate fingers of Choosing splayed alignment characteristic of the π cell yields almost identical results both with and without a bias voltage.After disclination formation, despite the initial splay, the defects support above them an almost planar, splay free layer.The twisted nematic case with in-plane alignment at the upper alignment layer also behaves similarly.Disclinations support above them a twisted layer.With a bias, the twist in the bulk is lost.

V. HYBRID ALIGNMENT
Hybrid Aligned Nematic (HAN) cells are constructed from a layer of liquid crystal sandwiched between two differently treated substrates: one yields homeotropic alignment and the other planar.Two possibilities exist with regards to the location of the interdigitated electrodes.If planar alignment is chosen near these electrodes switching proceeds much as it did for the planar cell, as shown in Fig. 9. Disclinations form above the center of each electrode and again the À1=2 lines move upward.There is some lateral movement associated with these disclinations due to the vertical change in the tilt throughout the structure, but this movement ceases when disclinations are positioned above the electrode edge.The bulk arrangement of molecules remains in the HAN state.
Choosing homeotropic alignment above the interdigitated electrodes yields more interesting results.The pre-tilt is fixed at 90 and 2 at the lower and upper alignment layers, respectively.If the applied voltage is below the threshold voltage of +3:96 V, switching is confined to a small region above the lower alignment layer, as was seen in the homeotropic case.Voltages in excess of this threshold induce disclinations at the same positions as in the homeotropic cell above the midpoint between the two electrodes.However, after forming, the À1=2 disclinations not only move upward but laterally too.
As the disclinations continue to rise, they also rotate until the situation shown in Fig. 10(a) is reached after 16 ms.It takes longer to reach steady-state than the homeotropic cell as this rotation is driven by the elastic force from the bulk.Disclinations have facilitated transformation of the bulk from a hybrid to a planar state.
The voltage can be increased or decreased and the thickness of the planar region can be controlled.If the applied voltage is reduced below +0:92 V the +1=2 lines move until they are almost at the same vertical position and then move laterally before finally annihilating.
A single voltage threshold exists that gives rise to two disclination pairs within the modeling window with a bias applied, +3:88 V. Figure 10(b) shows the resulting director field at steadstate, where the left-hand disclination forms sooner than the right.

VI. IN-PLANE ALIGNMENT
In-plane switching due to its wide viewing angle characteristics is commonly employed in displays.Consistent reorientation of

Journal of Applied Physics
the liquid crystal is ensured by the rubbing direction, which is typically a few degrees relative to the electrode edge.For the purposes of simulation, this pre-twist angle is chosen as f 0 ¼ 85 (relative to x) and the pre-tilt is 2 .Even low applied voltages (far below U th ) are sufficient to rapidly rotate the molecules by almost π=2 above the lower alignment layer, giving rise to an almost planar alignment (along x) in this region.Reorientation proceeds into the bulk, resulting in an almost linearly twisted bulk.By changing the applied voltage, the amount of twist can be controlled.This defectfree mode is attractive for display applications.
For larger applied voltages, the resulting disclination configuration depends on the exact voltage waveform applied, as sketched in Fig. 11.In case (a), as the voltage is ramped, a twist wall forms above the center of each electrode, but additionally a region of biaxiality arises above the left-hand electrode edge.As the voltage is ramped beyond +5:14 V, the biaxial region splits into two twist disclinations: a þ1=2 line that remains stationary at the electrode edge and a À1=2 line that moves rightward to the electrode center.Above +5:52 V, the central wall undergoes pincement, giving rise to a þ1=2 line that annihilates with the À1=2 line created previously.A À1=2 line from the wall remains, that rises upward.In case (b), a step voltage is applied.Similarly to (a), a biaxial region forms above the left-hand electrode edge, but now the twist wall forms further left, nearby the biaxial region (a shift that depends on the pre-twist, tending to electrode edge for f 0 ¼ 88:2 ).Between +5:14 and +5:44 V, the biaxial region splits into disclinations while the wall migrates to the electrode center.Above +5:44 V, the wall undergoes pincement rapidly, before it can move rightward significantly.The lower þ1=2 line settles above the left-FIG.6. Instantaneous velocity streamlines: (a) 0:1 ms after applying +3:97 V and (b) 0:1 ms after relaxation from +3:97 V. Streamline shade reveals the magnitude of the velocity field with white and black corresponding to slow and fast flows, respectively.During switching, the velocity in the neighborhood of the upper À1=2 lines has a maximum magnitude of 2:2 Â 10 À4 ms À1 .During relaxation, the velocity field peaks in magnitude at the lower þ1=2 defect cores with a maximum value of 3:5 Â 10 À4 ms À1 .FIG. 7. Steady-state vertical position of +1=2 disclinations as a function of voltage as the applied voltage is reduced from +3:97 V. Height is almost temperature invariant but for the weak dependence of the permittivity on the order parameter.

Journal of Applied Physics
hand electrode edge while the À1=2 line migrates diagonally up and rightward.
As the À1=2 lines rise due to the electric field, they lose a portion of their twist, while the layer beneath them twists toward x.Above the disclinations, the bulk becomes a linearly twisted layer but this process is a slow one.Figure 12(a) shows the steady-state liquid crystal orientation obtained 10 ms after applying a +5:44 V voltage step.By changing the applied voltage, the thickness of the π=2 twisted layer can be controlled.
When a bias voltage is applied to the upper electrode, there exists two threshold voltages for creating disclinations, as was observed for planar alignment.These thresholds are +3:95 and +6:04 V. Figure 12(b) shows the liquid crystal orientation that results at steady-state with a +6:04 V voltage step applied so as to induce disclinations above both lower electrodes.At 4 μs, the lefthand wall undergoes pincement, while the right-hand wall undergoes pincement at 36 μs after having drifted significantly toward the electrode center.

VII. NEGATIVE DIELECTRIC ANISOTROPY MATERIALS
A negative dielectric anisotropy material with properties identical to 5CB but with ε k and ε ?swapped has been simulated with flow neglected.For such materials, interdigitated electrodes promote in-plane reorientation of the liquid crystal.Regardless of the surface alignment, slowly ramped voltages yield in-plane orientational fields that are free of walls that might undergo pincement.
However, disclinations that appear fleetingly and annihilate inevitably can be induced by a step voltage, provided the alignment is not in-plane.Homeotropic and planar configurations have been simulated with (θ 0 , f 0 ) ¼ (89:9 , 89:9 ) and (θ 0 , f 0 ) ¼ (2 , 0:1 ), respectively.When a step voltage is applied, a twist wall arises above the electrode center with homeotropic alignment and between the electrodes with planar alignment.Below U th , the twisted region central to the wall expands laterally, giving rise to an in-plane orientational field at steady-state.Above the threshold, the wall is compressed giving rise to disclination pairs, as shown in Fig. 13.
The planar case is relatively insensitive to the surface alignment.Higher pre-twists serve to increase U th but disclinations paths remain similar.Conversely, in the homeotropic case, if the surface alignment is rotated toward x, it is possible to laterally shift the disclination paths.Annihilation can then be avoided by biasing the upper electrode.Ultimately, however, disclinations become pinned near the alignment layer at steady-state.
When a bias voltage is applied to the upper electrode with homeotropic alignment, the voltage threshold to create disclinations above both electrodes is +9:93 V.The defect trajectories are almost identical to the unbiased case but the left-hand disclinations diverge to a greater extent than the right before annihilating.With a bias applied for the planar case, the threshold to create both disclination pairs is +6:72 V. Disclination paths become curved due FIG. 8. Director streamlines at steady-state with homeotropic alignment.Black lines reveal trajectories followed by the disclinations and squares indicate nucleation points.In (a), +3:97 V is applied to the lower electrodes with no bias to the upper electrode, whereas in (b), +3:89 V is applied to the lower electrodes with a bias of þ3:89 V to the upper electrode.

Journal of Applied Physics
to the bias but both right-and left-hand disclinations reach the same height enroute to annihilation.
In reality, backflow can be severe for negative materials with homeotropic alignment. 44Dynamics are highly sensitive to the surface alignment, voltage signal, and Leslie coefficients (which are less extensively characterized in the case of negative materials).Consequently, detailed consideration of this special case is beyond the scope of this paper.Nevertheless, preliminary simulations suggest that, in this configuration, backflow can lead to very high defect populations particularly when the lower electrodes are grounded and a bias is applied to the upper electrode.

VIII. VOLTAGE THRESHOLDS
Table I shows the voltage thresholds required to induce defects for homeotropic, hybrid, planar, and in-plane alignments.For planar and in-plane alignments, U th differs for ramped (flow independent) and stepped (flow dependent) voltage waveforms.
In reality, several factors may cause the threshold voltage, U th , to differ from that presented here.First, the alignment layer and electrodes will be of finite thickness.A higher threshold voltage will be required to compensate for the falloff in electric potential within the alignment layer.Second, thresholds are highly sensitive to the material parameters.Third, the Landau-de Gennes theory is, strictly speaking, valid only in the vicinity of the nematic-isotropic phase transition.Voltage thresholds appear overestimated for temperatures further away. 3A possible reason is that higher order moments of the order parameter may be necessary to represent the biaxial phase prior to disclination formation adequately. 45inally, the assumption of invariance along y is not strictly true in practice.In reality, nucleation could occur at a specific point, for instance, in a region where an electrode is over-etched or where a spacer is located.Whether a wall or disclination state is favored depends on the free energies of the two states, which depend on the applied voltage.At the voltage U eq , the two states are equal in energy.A voltage above U eq favors the disclination state, whereas a voltage below favors the wall state.Above U eq , disclination lines would "unzip" along the line emanating from the nucleation point along y. 6 This mechanism is slower than the one considered in this paper but results in the same steady-state.
Threshold voltages are strongly dependent on the temperature.Order parameter variations occur on a length scale defined by the correlation length 30 When the width of a wall, given by ξ E , is similar to the correlation length, disclinations nucleate. 6This occurs for applied voltages of U th U 0 d=ξ.At a temperature above T * and below the clearing temperature T c (wherein the isotropic state is metastable and nematic state is stable), the correlation length is relatively large.Even low applied voltages are sufficient to reduce the order parameter and give rise to disclinations.As the temperature falls below T * , the isotropic state loses stability and the correlation length is reduced.A higher voltage is required to reduce the order FIG. 9. Director streamlines at steady-state with planar alignment.In (a) +5:19 V is applied to the lower electrodes and the upper electrode is grounded.Steady-state is reached after 2 ms.With a bias in (b), the approach to steady-state is slowed significantly, as the À1=2 lines migrate to the same height driven by their mutual repulsion (after 16 ms).

Journal of Applied Physics
and induce disclinations.For example, decreasing (T À T * ) from À2 to À4 K yields a 30% increase in U th for homeotropic alignment and also shifts the nucleation point downward.
An accurate expression for the temperature dependence of U eq in the case of planar alignment was given by de Lózar et al. 6 More approximately, we find that U eq / ln [k(T þ À T)] is a good fit for all four alignment configurations, where k is a constant and T þ is the temperature above which the nematic phase is unstable.Decreasing (T À T * ) from À2 to À4 K yields a more modest 4% increase in U eq for homeotropic alignment.Therefore, we expect defect formation via domain growth to be the dominant mechanism at low temperatures.
Splay or bend distortions of the liquid crystal induce a macroscopic (flexoelectric) polarization that can alter voltage thresholds as well as the dynamic behavior.
In the homeotropic alignment case, representative flexoelectric coefficients for 5CB 46,47 of (e 11 þ e 33 ) ¼ À8:4 pC/m and (e 11 À e 33 ) ¼ 7:1 pC/m yield defects with U th ¼ +3:94 V. Furthermore, walls and defect trajectories become inclined.The central wall (at x ¼ 0) inclines to θ % 85 and gives rise to a À1=2 defect line that follows a θ % 91:5 trajectory after nucleation.Walls and trajectories incline in the opposite fashion at the modeling-window edge.Lines of strength þ1=2 migrate laterally toward the positive electrode edge.In practice, liquid crystals are switched by alternating voltage signals that, coupled with the screening effect of ionic impurities, lessen the influence of the flexoelectric polarization.
An alternative means to induce disclinations is to ground the lower electrodes while applying a voltage to the upper electrode.Disclinations may only be formed in this manner for planar and in-plane cases.In these cases, the nucleation point tends to the electrode edge, where the electric field is maximum.Hence, reduced electric field thresholds [given now by E th ¼ U th =(1:6 μm)] of E th ¼ 8:57 V=μm (U th ¼ 13:71 V) for planar alignment and E th ¼ 10:78 V=μm (U th ¼ 17:25 V) for in-plane alignment are able to induce disclinations.
In the case of planar alignment, both stepped and ramped waveforms yield the same U th with the þ1=2 lines migrating to the same position shown in Fig. 9(a), whilst the À1=2 lines move FIG.10.Director streamlines at steady-state with hybrid alignment.Homeotropic anchoring is chosen at the lower alignment layer and planar at the upper.In (a), +3:96 V is applied to the lower electrodes without bias to the upper electrode, whereas in (b), +3:88 V is applied to the lower electrodes and þ3:88 V to the upper.

Journal of Applied Physics
laterally to the midpoint between the electrodes and to a height of 51 nm.However, for the in-plane case, disclinations only form if a stepped waveform is applied, forcing the reorienting liquid crystal to clash with the in-plane alignment.Furthermore, these disclinations annihilate, unless inhibited by applying a potential difference across the lower electrodes within 180 μs.

IX. WEAK ANCHORING
In this section, the effect of altering the anchoring strength, W, is considered.For the sake of simplicity, anchoring strengths in the zenithal and azimuthal directions are assumed equal, where (7).In order to maintain a surface order parameter of S 0 at equilibrium, 28 the coefficient a is set to During switching, the wall that arises prior to disclination formation shifts downward as the anchoring strength is reduced.Consequently, the electric field is larger at the nucleation point and a lower applied voltage will give rise to disclinations.
Figure 14 shows the voltage required to induce disclinations as a function of anchoring strength for both homeotropic and planar alignments.Additionally, the vertical position of the nucleation point, z th , is given for the pertinent value of threshold voltage (z th rises slightly for voltages in the excess of U th for both homeotropic and planar anchoring).With homeotropic alignment, the nucleation point remains in the bulk except for very weak anchoring strengths, whereas with planar alignment, the nucleation point tends to the alignment layer for comparatively modest anchoring strengths.
A notable feature of Fig. 14(b) is the turning point in the curves for weak planar alignment with ramped applied voltages.Initially, when the nucleation point tends to the surface, nucleation still converts a region of bend at the alignment layer to a region of splay.As the anchoring strength is reduced further the liquid crystal increasingly prefers to twist at the alignment layer rather than bend.A higher voltage is required to cause nucleation of the more compact twist wall, resulting in a turning point.When the applied voltage is stepped, bend persists at the alignment layer, even when the nucleation point tends to the surface, and turning points are absent.
In order to investigate the effect of K 24 on the nucleation process, a value was chosen based on a Cauchy relation for K 24 ¼ (K 11 À K 22 )=2, which follows from the Maier-Saupe molecular approach. 48With homeotropic anchoring, K 24 causes a marginal upward shift in the nucleation point and voltage threshold.This marginal effect is anticipated because the nucleation point remains in the bulk except for very weak anchoring strengths.For planar alignment, the effect is more pronounced since the nucleation point tends to the surface when the anchoring strength is relatively high.Once again, a larger K 24 reduces the tendency of the liquid crystal to twist at the alignment layer.An increased value of K 24 ¼ K 22 (the upper limit for positive definiteness of the free FIG.12. Director field at steady-state with in-plane alignment (a) without and (b) with a bias voltage applied to the upper electrode.In (a), a linearly π=2 twisted bulk sits atop the disclinations, whereas in (b), the twist is lost.

Journal of Applied Physics
energy 48 ) is found sufficient to inhibit twist entirely at the alignment layer leading to a voltage threshold that is monotonically decreasing with W.
After disclinations are induced, they separate.With strong homeotropic anchoring and +3:97 V applied, the þ1=2 lines become stationary 14 nm above the alignment layer.As the anchoring strength is reduced, the stationary point shifts downward.An anchoring strength of W ¼ 7:4 Â 10 À4 Jm À2 with K 24 ¼ 0 is sufficiently weak to ensure that the þ1=2 disclinations descend all the way to the lower alignment layer.If the anchoring is weakened yet still the þ1=2 lines smear along the alignment layer giving rise to a region of biaxial order along the alignment layer between the electrodes.
Weak anchoring affects more significantly the relaxation process.First, the rate at which the þ1=2 lines rise depends on the anchoring strength.Second, for weaker anchoring strengths, the ascent of the þ1=2 lines may be delayed or inhibited entirely.However, for this to occur the applied voltage prior to relaxation must be sufficient to seat the þ1=2 lines at the alignment layer.
With homeotropic alignment, the À1=2 lines are immediately expelled from the lower alignment layer on relaxation if W ! 4:7 Â 10 À4 Jm À2 .For weaker anchoring, the þ1=2 lines remain stationary at first but rise eventually when they enter the range of attraction of the descending À1=2 lines.For W 4:0 Â 10 À4 Jm À2 , the þ1=2 lines remain stationary until annihilation.
Expulsion upward of the þ1=2 lines converts a region of bend to splay above the alignment layer with homeotropic alignment.Conversely, with planar alignment, expulsion converts splay to bend.The latter is less energetically favorable, hence slightly stronger anchoring (W !5:2 Â 10 À4 Jm À2 ) is required to assure immediate expulsion.
Contrary to what one might expect, higher temperatures enlarge the biaxial regions of the alignment layer associated with the þ1=2 lines.A high energy is associated with this biaxiality.

TABLE I.
Voltage threshold (with no bias) for disclination formation, U th , and voltage at which the free energies of wall and disclination states are equal, U eq , at (T − T*) = −2 K.With hybrid alignment U eq is low because the bend distortion of the bulk in the wall state is lost in the defect state.Electric fields thresholds are calculated as E th = 2 U th /(0.3 μm) and E eq = 2 U eq /(0.3 μm).

Journal of Applied Physics
Therefore, expulsion of the defects is energetically favorable on relaxation, unless the anchoring strength is significantly weakened.

X. SCALABILITY AND ELECTRODE PROPORTIONS
Figures 15(a) and 15(b) show the effect of altering the fill factor, the proportion of the device horizontal occupied by electrode on the threshold voltage.Recall that with homeotropic alignment, the nucleation point lies in the inter-pixel gap.For large fill factors, the inter-pixel gap is narrow and even modest voltages can give rise to an electric field at the nucleation point sufficient to induce disclinations.Hence, U th in (a) exhibits a downward trend with fill factor.However, closely spaced electrodes are more prone to short-circuits.Another factor to consider is the risk of causing dielectric breakdown of the alignment layer or liquid crystal (which occurs at 100 V/μm 49,50 ).To assess this risk, the electric field threshold, E th , is a more useful measure than U th .For (a) and (b), it is defined as the ratio of the potential difference between the lower electrodes and the inter-pixel gap.Conversely, E th begins to increase for high fill factors due to the descent of the nucleation point toward the alignment layer.Consequently, the alignment layer becomes a stronger influence on the wall and resists pincement.For planar alignment, (b), nucleation occurs above the electrodes and is driven by the falloff in the electric potential.Therefore, low fill factors (narrow electrodes) are conducive to disclination formation.
Figures 15(c) and 15(d) show the effect of scaling the horizontal size of the modeling window on the threshold voltage.For scaling factors between 1=2 and 3, U th varies linearly with scaling factor for both homeotropic and planar alignments since the effective wall height, d, is proportional to the scaling factor.Had the vertical dimension been simultaneously scaled with the horizontal, this linearity would have been maintained without upper limit.However, when the cell gap is fixed, the influence of the upper electrode (which is grounded) becomes apparent for scaling factors above 3.For homeotropic alignment, the upper electrode serves to

Journal of Applied Physics
reduce the electric field at the nucleation point between the electrodes, leading to an increase in U th for large scaling factors.For planar alignment, when the scaling factor is larger than 32=3, the electric field at the nucleation point (above the lower electrodes) is determined by the potential difference between the lower and upper electrodes and the cell gap.Consequently, U th becomes independent of the scaling factor.

XI. CONCLUSIONS
Fringing fields beyond a critical field strength have been shown to induce disclinations irrespective of the surface alignment for liquid crystals with positive dielectric anisotropy.Disclinations nucleate in the inter-pixel gap when alignment is normal to the substrates and above the electrodes when alignment is tangential.Precise positional control of the disclination height is possible by adjusting the voltage across the interdigitated electrodes, while the lateral position may be controlled by applying a voltage bias to the upper electrode.
Weak anchoring shifts the nucleation point toward the alignment layer, a shift more evident with planar and in-plane alignments, yielding disclinations with lower voltages.Narrow electrode widths are conducive to disclination formation for planar and in-planar alignments, whereas narrow inter-pixel gaps give rise more readily to disclinations for homeotropic and hybrid alignments.
Flow of the liquid crystal serves to accelerate defect motion and perturbs their trajectories in the case of positive dielectric anisotropy materials.For negative materials, backflow can arise leading to richer transient behavior that warrants further study.
Essentially, the devices studied herein act as programable defect arrays and could find application as reconfigurable diffraction gratings, using defects to realize high resolution.For holographic applications, the pixel pitch determines both the viewing

Journal of Applied Physics
angle and image resolution.In order to obtain a viewing angle in excess of 30 , a pixel pitch of 1 μm is required but, at these scales, breakdown of the driving transistor is the limiting factor when using Liquid Crystal on Silicon (LCoS) technology due to the short channel length. 2 If sub-millisecond switching is required, electrode arrays of period , 1 μm are recommended since, when defects are present, the response time (which depends on the correlation length) increases for large defect separations. 30Furthermore, fringing field induced defects could prove an invaluable tool to reconfigure liquid crystal colloids.Pixel pitches of 2-10 μm should provide adequate control since particle trapping within defect lines has been demonstrated 21 for microspheres of diameter 1:5 μm.

Figure 5 (
Figure5(a) shows the path disclinations follow when a voltage of +3:97 V is applied both with and without flow.Flow causes disclinations to form 14:6 μs earlier and accelerates their separation.Disclinations reach the same steady-state height of 488 nm after 2 ms both with and without flow.Figure5(b) shows the disclination trajectories when the applied voltage is removed.Confinement alters the behavior in comparison to isolated disclination pairs.30,31When the voltage is removed, velocities of þ1=2 and À1=2 disclinations differ markedly even if flow is neglected.Elastic distortion at the alignment layer expels the þ1=2 disclinations away rapidly.As seen in the annihilation of isolated disclination pairs, both theoretically30,31 and experimentally,43 here too the velocity field preferentially accelerates the þ1=2 lines.In Fig.6(a), the velocity field after applying +3:97 V for 0:1 ms is shown.At this time, the þ1=2 disclinations are already stationary above the lower alignment layer and have little influence on the velocity field.The À1=2 disclinations continue to rise and the velocity field near each disclination is in the positive z-direction and serves to compound the disclination velocity.Figure6(b) shows the velocity field 0:1 ms after removing the voltage.By varying the potential difference across the lower electrodes, it is possible to control the vertical position of the +1=2 disclinations, as shown in Fig.7.When the voltage falls below +1:03 V, the attractive elastic force overcomes the electric and the disclination lines annihilate to restore the starting vertical configuration.Disclinations annihilate at different locations than observed earlier in Fig.5, where the applied voltage was removed abruptly.Gradually reducing the applied voltage serves to pin the þ1=2 disclinations above the lower alignment layers to layer, while the À1=2 disclinations descend.Relaxation may be accelerated by applying the same voltage to the interdigitated electrodes.Varying the bias applied to the upper electrode provides a means to control the lateral separation of the À1=2 disclinations, as shown in Fig.8(b).The bias voltage augments the electric field at the nucleation point and hence a lesser voltage of +3:89 V is able

Figure 5 (
Figure5(a) shows the path disclinations follow when a voltage of +3:97 V is applied both with and without flow.Flow causes disclinations to form 14:6 μs earlier and accelerates their separation.Disclinations reach the same steady-state height of 488 nm after 2 ms both with and without flow.Figure5(b) shows the disclination trajectories when the applied voltage is removed.Confinement alters the behavior in comparison to isolated disclination pairs.30,31When the voltage is removed, velocities of þ1=2 and À1=2 disclinations differ markedly even if flow is neglected.Elastic distortion at the alignment layer expels the þ1=2 disclinations away rapidly.As seen in the annihilation of isolated disclination pairs, both theoretically30,31 and experimentally,43 here too the velocity field preferentially accelerates the þ1=2 lines.In Fig.6(a), the velocity field after applying +3:97 V for 0:1 ms is shown.At this time, the þ1=2 disclinations are already stationary above the lower alignment layer and have little influence on the velocity field.The À1=2 disclinations continue to rise and the velocity field near each disclination is in the positive z-direction and serves to compound the disclination velocity.Figure6(b) shows the velocity field 0:1 ms after removing the voltage.By varying the potential difference across the lower electrodes, it is possible to control the vertical position of the +1=2 disclinations, as shown in Fig.7.When the voltage falls below +1:03 V, the attractive elastic force overcomes the electric and the disclination lines annihilate to restore the starting vertical configuration.Disclinations annihilate at different locations than observed earlier in Fig.5, where the applied voltage was removed abruptly.Gradually reducing the applied voltage serves to pin the þ1=2 disclinations above the lower alignment layers to layer, while the À1=2 disclinations descend.Relaxation may be accelerated by applying the same voltage to the interdigitated electrodes.Varying the bias applied to the upper electrode provides a means to control the lateral separation of the À1=2 disclinations, as shown in Fig.8(b).The bias voltage augments the electric field at the nucleation point and hence a lesser voltage of +3:89 V is able

FIG. 1 .
FIG. 1. Illustration of the surface alignment configurations analyzed in Secs.III-VI, where gray line segments represent the starting director configurations (at time ¼ 0) chosen for the simulations.Pre-tilt, θ, and pre-twist, f, at the upper (u) and lower (l) alignment layer are indicated, where n ¼ ( cos θ cos f, cos θ sin f, sin θ) T .
Journal of Applied Physicsto induce disclinations.Due to the lateral motion, flow now perturbs the disclination paths.

FIG. 3 .FIG. 4 .
FIG.3.Mesh in the vicinity of the electrode edges at four different times.Throughout the modeling window, the element edge-length varies from 0:25 to 400 nm.Element count peaks at 34 μs with 8400 triangles.

FIG. 5 .
FIG. 5. Vertical disclination position as a function of time when (a) +3:97 V is applied and (b) the relaxation process that follows when the voltage is removed.Squares indicate nucleation points and diamonds annihilation points.If flow is neglected, disclination creation is delayed by 14:6 μs.

FIG. 11 .
FIG. 11.Disclination and wall arrangements observed above one electrode with in-plane alignment.The horizontal axis represents the voltage with threshold voltages labeled.In (a), the applied voltage is ramped, and in (b), a step voltage is applied.

FIG. 13 .
FIG.13.Director fields obtained after 100 μs with a negative dielectric material (with properties identical to 5CB but with ε k and ε ?swapped and flow neglected).(a) Homeotropic alignment: an applied voltage of U th ¼ +9:21 V induces disclinations that annihilate at 1:1 ms, leaving a HAN like orientation at steady-state.(b) Planar alignment: an applied voltage of U th ¼ +6:61 V induces disclinations that annihilate at 0:5 ms to leave a twisted nematic like state.

FIG. 14 .
FIG. 14. Disclination formation threshold, U th , as a function of anchoring strength for (a) homeotropic and (b) planar alignment.Dark lines correspond to K 24 ¼ 0 and gray lines to K 24 ¼ (K 11 À K 22 )=2.Nucleation point height, z th , as a function of anchoring strength for (c) homeotropic and (d) planar alignment with K 24 ¼ 0.