JOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 9 1 MAY 21 Approximate description of the two-dimensiona director fied in a iquid crysta dispay G. Panasyuk, a) D. W. Aender, J. Key Liquid Crysta Institute Department of Physics, Kent State University, Kent, Ohio 44242 Received 19 October 2; accepted for pubication 2 February 21 A mode for describing properties of a iquid crysta dispay which combines the concepts of in-pane switching vertica aignment is proposed. There is good agreement between the resuts of this mode direct computer cacuation of the director the ight transmittance. The usefuness of the mode ies in the faster speed of cacuations compared to direct computer soution of the Euer Lagrange equations. 21 American Institute of Physics. DOI: 1.163/1.1359423 I. INTRODUCTION Recenty severa new types of iquid crysta dispays LCDs with wide viewing anges, good coor characteristics, fast response times were proposed. Among them are a device associated with a homeotropic to mutidomainike HMD transition, 1,2 a iquid crysta LC ce with patterned eectrodes, 3 an LCD ce which combines the concept of in-pane switching IPS with vertica aignment VA. 4 In each of these exampes there is a mutidimensiona director distribution, because the director depends on two or three space coordinates unike the case of the usua twisted nematic dispays. In our previous work 2 an approximate anaytica approach for describing the HMD ce at high votages was found. The purpose of this work is to deveop an appropriate anaytica mode to describe the director configuration for an LCD which combines the concepts of IPS with VA for votage differences, u, high enough that ight transmittance is observabe. The advantage of the mode we have deveoped is that its cacuationa time is much faster than for direct computer soution. In addition, it can be generaized to describe other mutidimensiona LCDs such as the threedimensiona HMD ce, where a direct computer soution has not been carried out. It can be used, for exampe, for fast estimation of the optima conditions for operation of these types of LCDs. II. GENERAL IDEAS OF THE DESCRIPTION In the ce studied in Ref. 4 vectors of the director n eectric fied E aways ie in a fixed pane xz pane. The schematic diagram of this two-dimensiona ce is shown in Fig. 1, where is haf the distance between adjacent eectrode fingers L equas 2 pus the width of an eectrode finger. The thickness of the ce is d. The system possesses the foowing symmetries: 1 2L periodicity aong the x coordinate, 2 mirror symmetry with respect to the vertica yz panes at xl/2, 3L/2,..., 3 the eectric potentia (x,z) is an odd function of x. Due to these symmetries it is sufficient to find the director ony inside the region a Eectronic mai: georgy@coumbo.kent.edu zd xl/2. The system possesses a series of wa defects. The centra pane of one of these was is the yz pane at x. Aong this pane the eectric fied E is perpendicuar to the initia homeotropic aignment, the torque due to the eectric fied is zero the director stays homeotropic aong the x ine even in the presence of E, as is iustrated in Fig. 1. Taking into account these symmetries, the free energy of the system per unit ength in the y direction can be written as dze2, 1 L/2 d F dx dzfd F f g 8 d where F d 1 2 K 1 x cos z sin 2 1 2 K 3 z cos x sin 2 2 F f 1 8 an"e 2 E 2, a. 3 In these formuas (x,z) is the ange between the director n ẑ we represent the director in the region xl/2 zd as nxˆ sin ẑ cos ). The dieectric constant for gass substrates is g, K 1 K 3 are the spay bend eastic constants of the nematic, are vaues of the dieectric tensor parae perpendicuar to the director, the dieectric anisotropy is positive ( a ). The main idea of the proposed mode is to numericay sove the dynamic equation 5 1 t F/ using the exact expression for the free energy 1 but an approximate expression for the eectric fied, which is found in this work. Here 1 is the rotationa viscosity fow is negected. In contrast, previousy used methods of direct computer cacuation e.g., the reaxation method 3,6,7 do not use an approximate form for the eectric fied but instead sove "D to get the eectric fied after each director update, based on Eq. 4. 4 21-8979/21/89(9)/4777/1/$18. 4777 21 American Institute of Physics Downoaded 11 Apr 26 to 131.123.234.122. Redistribution subject to AIP icense or copyright, see http://jap.aip.org/jap/copyright.jsp
4778 J. App. Phys., Vo. 89, No. 9, 1 May 21 Panasyuk, Aender, Key r u 2 on the eectrodes, 11 E r when z, It can be assumed that in most of the sab except near the eectrodes wa defect ayers of thicknesses of order of the correation ength the scae of spatia variations of n(r) is the same as for the eectric potentia. In this case, as can be seen from Eqs. 2 3, the characteristic vaue of the ratio F d /F f is determined by the parameter 4K m a u 2, FIG. 1. Schematic diagram of the 2D ce. where K m is the maximum vaue of the eastic constants K 3 in this case. To obtain an estimate of, we use the experimenta vaue 4 of 5.2 for a at 2 C 1 khz a typica vaue of 1.81 6 dyne for K m. Thus if u1/3 esu 1 V in Système Internationa,.5. Taking into account the condition n 2 1 negecting the eastic energy competey, the Euer Lagrange equations in the buk in a zero order approximation can be written in the simpe form: 5 where E. On the other h, n(r) changes rapidy with characteristic scae determined by the eectric correation ength (E x )(4K 3 / a ) /E x near the substrates the x pane to fit the homeotropic conditions there. Thus in a more reaistic case when is sma but not negigibe, we have to take into account, in the genera case, this sma scae variation in the director its infuence on the eectric fied. In Sec. II A we sove the boundary probem 1 11 to find the eectric fied in the zero order approximation, in Sec. II B this soution wi be modified to take into account changes of the director inside the substrate wa defect ayers. A. Soution of the boundary probem The genera form of the soution of the Lapace equation is LC p p exps op z p exps op zsin s p x where s op s p 2p1/L inside the LC sab, g1 r p 1p exps p zsin s p x 12 a 4 n"een, 6 13 E a n"en. 7 g2 r p 2p exps p zsin s p x Equation 6 can be soved to obtain the foowing reationship between n E: n E E, EE, ae 2 4, which means that the director is parae to E. Substituting Eq. 8 into Eq. 7 we can write the equation for the eectric potentia: 8 E, or 2 with E, 9 assuming that, do not depend on r. Thus negecting competey a the corrections produced by the presence of the ayers near the substrate the wa defect to the eectric fied which are proportiona to as was shown in Ref. 2 for the simiar situation in the HMD ce, the eectric potentia for the zero order approximation can be found by soving the Lapace equation 2 1 with the foowing conditions on the eectric fied: in the gass for zd z, respectivey. Using the continuity of (r) of the norma component of the eectric dispacement vector D n (r) at the LC gass interface at z d continuity of (r) at the z gass LC interface, one can express the p, 1p, 2p coefficients in terms of p : 1p 2ˆ p ˆ 1, pe p p, 2p p 1e p, 14 ˆ 1 e p ˆ 1 exp2s opd, where ˆ / g. To determine a the coefficients p, p, 1p, 2p competey, we introduce the eectrode pane potentia V(x) which is continuous a 2L periodic function of x has the vaues u/2 on the eectrodes, satisfies the symmetry reation V(x)V(Lx), is an odd function: V(x)V(x). Using these conditions, V(x) on the interva x may be written as Vx x 2 m1 f m sinmx/. 15 Downoaded 11 Apr 26 to 131.123.234.122. Redistribution subject to AIP icense or copyright, see http://jap.aip.org/jap/copyright.jsp
J. App. Phys., Vo. 89, No. 9, 1 May 21 Panasyuk, Aender, Key 4779 The number of f coefficients in the compete soution is of course infinite but an approximate soution may be found by truncating the series by some number m max N. Using the properties of V(x), one can rearrange V(x) in terms of sin s p x express a the coefficients 14 through f m.to obtain the atter coefficients, the variationa principe may be used to minimize the eectric free energy F f. Because LC vac satisfy the Lapace equation D n are chosen to be continuous at the upper interface at zd, itis possibe to rewrite F f as the eectrode pane functiona: F f 1 L/2 dx 8 LC z LC g2 z g2 at z. 16 Substituting here Eqs. 12 13 integrating the resuting expression expicity, we represent F f as a quadratic form with respect to the arbitrary coefficients f m. After that, differentiating F f with respect to these coefficients equating the resuts to zero, we wi get the foowing system of inear inhomogeneous agebraic equations with respect to f m : N n1 In this system where y mn f n r m, m1,2,...,n. 17 y mn p b mp b np B p, B p s op d ˆ 1e p 1e p 1, r m A p b mp B p, p 2 sin2p1 A p, 2 2p1 2 L, b np p1 np1 n 2 18 19 sin2p1, 2 where A p b np are the Fourier coefficients which arise after rearrangement of V(x) in terms of sin s p x. One important observation must be made about the system 17. Itbecomes inconvenient to sove Eq. 17 exacty for arge N. On the other h, because of nonanayticity of the eectric potentia near the sharp edges of the eectrodes, an accurate description of the eectric fied requires N5. Fortunatey, the matrix Ŷy mn in Eq. 17 can be represented as Ŷ Ŷ, where Ŷ is diagona Ŷ is sma with respect to Ŷ. Thus we can sove Eq. 17 by means of an iterative procedure f (n) N Ŷ 1 RŶ f (n1) N 21 for n1, which has rapid convergence for any reasonabe N. In Eq. 21 f N R designate two sets f m r m of N eements each. FIG. 2. Graph showing the z dependence of E x at different vaues of the x coordinate. B. Modifications of the eectric fied caused by variation of the director on short ength scaes In this section we are going to estimate the infuence of the near-substrate wa defect ayers on the zero order soution E. In a the dispays mentioned in the Introduction, d is much ess than L usuay d/l.2. Thus if wa defect ayers were not present, a terms with x derivatives in the Euer Lagrange equations for the director eectric fied coud be negected. This gives us K b 2 zz 1 2 K sin2 z 2 a 4 1 2 E x 2 E z 2 sin 2E x E z cos 2, 22 z E z a n"ecos, 23 where K b ()K 1 sin 2 K 3 cos 2 KK 3 K 1. Negecting x derivatives in the Maxwe equation ÃE shows that one can aso negect the z derivative of E x in Eq. 23. It means that sma ength scae changes in the director fied near a substrate having homeotropic boundary conditions do not produce the same significant changes in E x.in such a situation E x is cose to its zero order approximation: E x E x. Figure 2 shows the z dependences of E x for different vaues of x throughout the most important region between nontransparent eectrodes, x. As iustrated in Fig. 2, z dependences are reay weak everywhere except in the vicinity of the eectrode edge at x. In the sma region x, however, the eectric fied is very strong, the correation ength is sma, the director is approximatey parae to the eectric fied as described by Eq. 8. Thus EE near x. Taking the constant of integration in Eq. 23 as the vaue of the expression in square brackets at a particuar point zz m (z m d) using the approximation E x E x, one can rewrite Eq. 23 after integration in the foowing form: E z 2 a cos 2 m E zm a sin 2E x sin 2 m E xm 2 a cos 2, 24 Downoaded 11 Apr 26 to 131.123.234.122. Redistribution subject to AIP icense or copyright, see http://jap.aip.org/jap/copyright.jsp
478 J. App. Phys., Vo. 89, No. 9, 1 May 21 Panasyuk, Aender, Key E z a E x, 25 which aows us to assume the foowing form for E z (x,z) in the region xx 1 : FIG. 3. Eectric fied components E z (x,zd/2) E x (x,zd/2). Equation 26 indicates E z is significanty different from E z, but the comparison E z E x for sma x is sti correct. where E zm E z (x,z m ), E xm E x (x,z m ), m (x,z m ), z m can be chosen from the foowing considerations. As was shown before, 2 for reativey arge votages more than u 1 V for the experimenta set of ce parameters 4 which is pointed out in Sec. III it is possibe to provide a reasonabe description of the sab by dividing it into two near-substrate ayers with thicknesses 1 2 2 a buk region between them where E is cose to E. It means that one can choose E zm E zm E z (x,z m )in Eq. 24 with z m cose to d/2. Indeed, our cacuations show that the director distribution for u1 V does not depend significanty on z m if it is taken from the interva.35d z m.6d. However, as u is owered, the buk region shrinks finay disappears when uu, where u can be estimated from the reation 4d with (E x ). The most important consequence of this is that E z deviates significanty from E z for these votages even in the centra part of the sab for x/2, there is no vaue of z m for which E zm E zm with a good accuracy in Eq. 24. It means that formua 24 fais to describe E z propery for uu for the important region of sma x. At first sight it is not important to know E z accuratey for the director cacuations because E z E x E x, which is iustrated in Fig. 3. But this is a wrong impression. The probem is that for sma x, when 2 cos 21 sin 22, E z E x is comparabe to.5(e x E 2 z )sin 2E 2 x in the eectric fied contribution to the Euer Lagrange equation for see the ast term in Eq. 22. To estimate E z propery, et us consider again a soution of Eq. 23 without assuming any particuar vaue for E zm for sma xx 1, where x 1 is the argest vaue of the x coordinate for which the foowing approximate reations a cos 2 sin 22 are sti satisfied within about 1% accuracy, which means that is ess than about 1.4 for a reaistic vaues of x 1 is a function of u the other ce parameters is found from the condition (x 1,d/2) 1 in the course of soving Eq. 4. Using these approximate reations, one can rewrite Eq. 23 as E z x,z a x,ze x x,ze, 26 where e does not depend on z or its z dependence is much weaker than in the first term in Eq. 26. Because x due to the presence of the wa defect E z x due to the symmetry of the probem when x, e must behave aso ike x at sma x. To determine e, et us notice, that for u, the ampitude a of the distribution aso goes to zero a u 2 when u, as foows from the Euer Lagrange equation for. Thus one can assume that ee z, where E,E z (E x ) is the ow votage asymptotic vaue of the eectric fied, which can be found as a soution of the boundary probem for the corresponding ow votage potentia (r): 2 x 2 z with the same conditions 11 on the eectric fied: r u 2 on the eectrodes, E r when z, 27 28 where E. Equation 27 is the Maxwe equation "D with everywhere inside the ce. The soution of this boundary probem 27 28 can be described in the same way as was done in Sec. II A with the same vaue of s p the foowing substitutions: LC LC, g1 g1, g2 g2, ˆ ˆ 1ˆ ( / ), s p s p ( / ). Thus one can approximate E z in the interva xx 1 by Eq. 26, where ee z. 29 Indeed, our cacuations show that the z dependence in E z is weaker than in (x,z); besides, for a votages important for ight transmittance a contribution of the first term in Eq. 26 for zd/2 is more than an order of magnitude arger than. This notice justifies the form of E z given in Eq. 26. For x 1 xx.8, sin 2 is comparabe to or even E z arger than cos 2,, because E z E x, accurate vaues for E z are not important for this region. For x cose to, the eectric fied is strong even for the owest votages of interest, EE, as was aready mentioned. For xl/2, E z is the argest component of E, is sma, one can aso assume that E z E z in that region. Comparison with the resuts of direct computer cacuation for E shows that E x E x throughout the ce with an accuracy about 1%, E z E z for x.8 with the same accuracy. These considerations aow us to use formua 24 for E z with z m d/2 for x 1 xl/2. Thus for the case of reativey ow votage u u 1 V for the experimenta ce, x 1 is reativey arge (x 1 /2), using E z from Eqs. 26 29 on the interva xx 1 instead of Eq. 24 is very important. Figure 4 shows a resut of cacuation of x dependence of E z for Downoaded 11 Apr 26 to 131.123.234.122. Redistribution subject to AIP icense or copyright, see http://jap.aip.org/jap/copyright.jsp
J. App. Phys., Vo. 89, No. 9, 1 May 21 Panasyuk, Aender, Key 4781 FIG. 4. Eectric fied component E z (x,zd/2) at u9 V obtained from a Eqs. 26 29, b direct computer cacuation, c Eq. 24 with z m d/2. some uu for xx 1 in accordance with Eqs. 26 29 curve a. This resut is in reasonabe agreement with the corresponding resuts of direct computer cacuation, but deviates significanty even has a different sign from a resut produced by the appication of Eq. 24 with z m d/2 curve c. As one finds from Fig. 4, E z is positive in the centra region of the ce, whereas both E z E z are negative. As indicated by Eq. 26, this interesting effect is due to the anisotropy of LCs disappears when a. When u increases, x 1 decreases quicky, formua 24 is appicabe for neary a x. On the other h, when votage increases, E z approaches E z E z E z even for sma x. To underst the infuence of the wa defect ayer on E x, et us consider the foowing ideas. Because E z is reativey sma for x2 w, it is reasonabe to use E z there when determining the infuence of the wa defect on E x. Again, the equation ÃE impies that we can negect the z dependence of E x for the sma x region. Thus one can rewrite the equation "D as x E x a sin 2 a E x z sin cos. 3 Considering Eq. 3 as an ordinary differentia equation with respect to E x E x (x), one can sove it aong the ine zd/2, where the ast term in Eq. 3 can be negected because z aong the ine zd/2, see the Appendix. Representing the integration constant of the resuting equation as a vaue of the expression in square brackets in Eq. 3 at the point x w, one can write the soution for E x as E x gxe x, gx a sin 2 a sin 2, 31 where ( w,d/2) E x E x ( w,d/2). On the other h, when x increases, the roe of x-derivatives decreases, the director distributions can be found from Eq. 22 where, however, a buk eectric fied distribution eectric fied far from the pane x, which is cose to E must be modified by the presence of the wa defect ayer. In particuar, one has to repace E for the part of the ce outside the wa defect ayer by a modified fied E (E x,e z ), which means that E x in Eq. 31 can be approximated by E x ( w,d/2). Note that for typica vaues of a see, for exampe, their experimenta vaues in Sec. III 6 9 for u12 V E x at x can be about twice E x E x ( w,d/2). To determine E x, et us consider a modified ce at the same votage which reproduces the eectric fied E at arge x by means of the zero order method as described in Sec. II A. As we know see, for exampe, Fig. 2, the zero order fied component E x varies sowy at sma x, which means that E x (,z)e x ( w,z) for any z. If the ce was not modified, the modified fied E woud describe the ce at a ower votage, because E x E x for x w see Eq. 31 EE for arger x. In order for the modified ce to have the same votage drop, it must have extended dimensions LL2x x the same d with some x. This ce can be produced from the region x L/2 zd of the ce under consideration by shifting the vaue of x where the potentia see Fig. 1 to the eft by x using the symmetry periodicity properties to extend the soution to a r. Cacuating the corresponding integra for the defect ayer using the expression 31 for E x, one finds an equation for x: E x w x. 32 Finay, x, E x, E z can be determined sefconsistenty in the foowing way. Choosing some arbitrary x for exampe, x.1) taking LL2x x, one cacuates E, as described in Sec. II A. Then, using this E the expression A9 for x from the Appendix which foows from Eq. 32 one cacuates a x. In genera the cacuated x wi differ from the initia choice of x, so the cacuationa procedure must be repeated severa times unti convergence is achieved. Detais of the cacuation are given in the Appendix. Let us discuss some important resuts concerning the eectric fied E. Our cacuations show that E z aways coincides with E z within 1% to 3% accuracy. Figure 5 iustrates characteristic resuts for the cacuation of E x using the mode scheme described above curve a, aso for comparison of the corresponding resut determined by direct computer cacuation curve b. As is shown in Fig. 5 curve a, E x increases significanty from E x when x decreases toward zero, which is at east in quaitative agreement with the resut of the direct computer cacuation. As votage u decreases, the peak in E x centered at x decreases practicay disappears when uu for the experimenta set of ce parameters, u 1 V. The atter resut can be understood easiy, using Eq. 31 taking into account that in Eq. 31 aso becomes sma, g(x) approaches 1 when u decreases. As foows from Eq. A9 of the Appendix, when u, x decreases, because cannot exceed /2 E x u. Therefore L approach the ce vaues L, E approaches E, the increase of E x towards the point x aso goes to zero when u. Downoaded 11 Apr 26 to 131.123.234.122. Redistribution subject to AIP icense or copyright, see http://jap.aip.org/jap/copyright.jsp
4782 J. App. Phys., Vo. 89, No. 9, 1 May 21 Panasyuk, Aender, Key FIG. 5. Eectric fied component E x (x,zd/2) at u16 V obtained from a Eqs. 31 32, b direct computer cacuation, c zero order soution E x. As one finds from Fig. 5, the vaues of E x are significanty ower than the cacuated E x vaues for both the mode direct computer cacuations when x w, sighty higher for the arge x part of the ce. However, we find that this difference between E x E x usuay produces ony sma changes in the corresponding director distributions. It means that the resut of the director fied cacuation from the dynamic Eq. 4 is reativey stabe with respect to sma variations of the eectric fied, used in Eq. 4. The most important consequence of this statement is that one can simpify significanty the expression for the eectric fied, used in the director cacuations. In particuar, one can approximate E x E x use expressions 24, 26, 29 for E z. It is worth mentioning that the use of the approximate fied expressions in the course of soving Eq. 4 speeds up the cacuation by a factor of about 3 with respect to the direct computer soution for the two-dimensiona probem being considered here. III. RESULTS AND DISCUSSION The resuts of the director cacuation the resuting optica properties are iustrated in Figs. 6 9. For each of these figures we used the foowing set of experimenta ce parameters: 4 K 1 13.2 pn, K 3 18.3 pn, 3.1, 8.3, L25 m, d5 m, 7.5 m. Figure 6 shows the x dependence of n x sin (x,d/2) for severa votages as cacuated in two different ways: a our mode based on Eqs. 4, 24, 26, 29, the approximation E x E x b direct computer cacuations reaxation method 6,8.Inthe atter case the Maxwe equation "D together with Eq. 4 are soved numericay. After discretizing "D, an equation inear in the discretized vaues of the votage can be soved to yied an update formua for the potentia (x i,z k ) at the current grid point (x i,z k ) after each iteration of Eq. 4 for the director. Because of the symmetries of the probem, namey mirror symmetry (L/2x, z)(x, z) the reation (x, z), we soved the equation "D ony on the interva xl/2x, where xz.25 m FIG. 6. Director component n x (x,zd/2) at 9, 12.5, 16 V as cacuated by the mode by direct computer cacuation. was the mesh step in both x z directions inside the LC sab, with the boundary condition u/2 on the eectrode. To take into account the boundary conditions at z, itis important to use the resuts of Sec. II A that the spatia scae for the decay rate of the eectric fied inside the gass substrates is of order L. In particuar, the decay rate of the owest harmonic is /L. Therefore we use a uniform mesh inside both gass substrates with boundary conditions z atz b 2L mesh steps x g x z g 2x inside the gass ayers. We aso used a ogarithmic mesh 9 inside the gass substrates for comparison set at z4l. We found, however, that the reative difference in the resuts of the director cacuations using these two meshes inside the gass ayers is very sma ess than.1%. On the other h, if one takes a finer mesh with a mesh steps decreased by a factor of 2, namey, xz.125 m in the LC ce z g 2z x g z inside the gass substrates in the case of a uniform mesh, there are somewhat arger changes in the director distribution. In particuar, we found a difference of about.2% to FIG. 7. Optica transmission vs votage. The soid curve is cacuated using the mode the dots are obtained from direct computer cacuation. Downoaded 11 Apr 26 to 131.123.234.122. Redistribution subject to AIP icense or copyright, see http://jap.aip.org/jap/copyright.jsp
J. App. Phys., Vo. 89, No. 9, 1 May 21 Panasyuk, Aender, Key 4783 FIG. 8. Isocontrast contour pots. The soid ines correspond to the direct computer cacuation the dashed ines to the mode..5% in the ampitude vaues a for the director ange for a votages considered here, about 2% to 3% for vaues of inside the wa defect substrate ayers where rises steepy from its zero vaue at the substrates the x pane for the case of arge votages. This is our estimate for the error of the direct computer cacuations for the.25 m mesh, used here in a figures for comparison with the mode. The program for the direct computer cacuation was checked carefuy by comparing with some specia cases, where the resuts are known. One of them is the case of interdigitated eectrode fingers surrounded by a uniform dieectric. The eectric fied for this case can be produced with very good accuracy with an error of ess than.1% in a way described in Sec. II A with the substitution g. The resut of the direct computer soution corresponds to this anaytica soution. Another case is the ow votage asymptote for the eectric fied E, which is found anayticay in Sec II B. This asymptote is vaid when u is so sma that E x sin E x E z E z for a space coordinates inside the ce. Our cacuations show that E x /E z.1 ony if u.7 V for the experimenta set of ce parameters. We find that our direct computer cacuation program produces the expected resuts EE for these ow votages. The fina check of the program is a comparison of the cacuated eectric fied director with the eectric fied E director soution n E /E, found anayticay in Sec. II vaid for high votages. The basis for this comparison is that when u, the near substrate ayers the wa defect ayer shrink, the director becomes parae to E, the eectric fied director approach their zero order anaytica soutions E n at the centra region of the LC ce outside the ayers. Our cacuations show that the resuts of the direct computer soution program aso agree with this imit. The program for the direct computer soution aso reproduces the director fied, cacuated by the mode after substituting E into the dynamic equation 4 with an expected accuracy O(E x /E z ) for ow votages, the director, cacuated after substituting E into Eq. 4 for the high votage imit. As is cear from Fig. 6 we have good agreement between the mode direct computer cacuations for the LC dispay appication votage interva 7u2 V, which the optica resuts of Figs. 7 9 show is the range of interest. Figures 7 9 iustrate resuts of cacuations of the ight transmittance based on the geometrica optics approach GOA, using a director pattern as input. The appication of GOA to one-dimensiona modeing of inhomogeneous LCs was originay done by Ong Meyer 1 appied extensivey by Ong. 11,12 A generaization of GOA to the mutidimensiona ces was described in Ref. 4. In particuar, Fig. 7 shows the votage-dependent optica transmittance for norma incidence using director patterns obtained by our mode by direct computer cacuation. Figures 8 9 dispay the same comparison for an isouminance contour pot waveength-dependent ight transmittance at u16 V. In Fig. 8 contours are abeed by percent transmission. The soid ines correspond to the direct computer cacuation the dashed ines to the mode. Agreement between the two director profies is good even for a poar ange higher than 4. Figures 7 8 are potted for norma incidence, the ight source is white ight. Figure 9 dispays the transmission versus waveength curves for the direct computer mode cacuations, data are shown for two anges of incidence: 6. The pane of incidence is the pane of the director. An interesting question is connected with the wa defect structure. An aternative to the wa defect is two discination ines with strengths m at some distance from the top bottom substrates which ie in the x pane are parae to the substrates. 2 The energy W w of the wa defect the energy W of the discination ines per unit ength in the horizonta y direction can be estimated 5,13,14 as FIG. 9. Optica transmission vs waveength. The curves are for two anges of incidence 6 are obtained by two different cacuationa methods. W w 2Kd 33 Downoaded 11 Apr 26 to 131.123.234.122. Redistribution subject to AIP icense or copyright, see http://jap.aip.org/jap/copyright.jsp
4784 J. App. Phys., Vo. 89, No. 9, 1 May 21 Panasyuk, Aender, Key W 2m K 2 og R r c 2 2K 34 in a one eastic constant approximation 5 here K(K 1 K 3 )/2. In the ast formua m 2 K og(r/r c ) is the macroscopic energy of eastic deformation associated with one ine. A characteristic size R of the deformation can be taken as R, r c is the radius of the defect core r c is of the order of a moecuar size, 2 nm. 5 Further the core energy is m 2 K, where can vary between og R/r c. 13 15 The ast term in Eq. 34 is the energy associated with two sma wa defects which ie in the x pane connect the defect ines to the substrate panes. Because both of these two engths are inversey proportiona to the eectric fied, this ast term does not depend on votage, W increases with votage very sowy ogarithmicay. On the other h W w is proportiona to u. This means that when u becomes arger than some critica votage u c, which can be found from the equation W w W, 35 a first order phase transition to the aternative defect structure with two discination ines occurs. From Eq. 35 u c is estimated to be between 2 4 V depending on the particuar vaue of. On the other h, ony the wa defect structure has been observed in experiments 4 on this ce, at east for votages beow 6 V. Because the ine defect structure has not been seen, no attempt was made to deveop a fast cacuation mode for such a structure. The absence of the discination ines in the experiment needs further consideration, but may be due to energy barriers encountered in transforming from a wa defect to discination ines. IV. CONCLUSIONS A simpified mode was constructed to describe the director configuration in a two-dimensiona 2D ce with interdigitated eectrodes. The simpified mode provides an aternative to the traditiona method of direct computer soution of the Euer Lagrange equations for the director eectric fied in anayzing behavior of iquid crysta ces. Cacuations show good agreement between resuts of this mode direct computer cacuation of the director. Corresponding resuts for the ight transmittance, based on the director patterns provided by the mode direct computer cacuations, are in good agreement as we. The mode aso heps to underst some interesting features in the eectric fied distribution important for dispay appications, such as the behavior of the eectric fied component norma to the substrate in the region midway between the interdigitated eectrodes for reativey ow votages. Another advantage of the mode is that its cacuationa time is about 3 times faster than for the direct computer soution. A preiminary study indicates that it can be generaized to describe the three-dimensiona 3D HMD ce. 1,2 As foows from the geometry of the HMD ce, 2 the eectric fied component perpendicuar to the wa defect ayer is negigibe unike the 2D case one can negect the infuence of the wa defect FIG. 1. Graph showing z profies of for x1.5 1 m atu16 V. ayer on the eectric fied. Using the particuar symmetries of that ce 2 this observation, one can construct a 3D variant of the mode described in this work. ACKNOWLEDGMENTS This work was supported by the Nationa Science Foundation under the Science Technoogy Center ALCOM DMR89-2147. We woud ike aso to acknowedge Dr. Leo Homberg for his hep in optimization of the programming toos important for the present research. APPENDIX Substituting the expression E x g(x)e x into Eq. 32 one can rewrite it as w xe x wdx acos 2 acos 2 E x A1 with a2 / a 1. Because E x depends weaky on x at sma x, this simpifies to x wdx cos 2cos 2. A2 acos 2 To estimate this integra, et us consider the Euer Lagrange equation F/ aong the ine zd/2 for x w. We assume that the function a (x)ˆ (z/) gives an accurate representation for (x,z), the ange of deviation from the homeotropic state. Here ˆ is caed the z profie it is normaized to have a vaue of one at z, a (x) gives the maximum vaue of at a given x as z varies over the sampe thickness. The vaidity of this assumption is iustrated in Fig. 1. Both the z profie the parameter are sow functions of x inside the wa defect region, is cose to d/2 which means, in particuar, that z aong the ine zd/2 in the wa defect region. These curves were obtained by soving Eq. 4 using the approximate expression for E found in Sec. II see, in particuar, Eqs. 24, 26, 29, 31, 32. Anaogous direct computer cacuations give very simiar resuts. Negecting a terms with a first z de- Downoaded 11 Apr 26 to 131.123.234.122. Redistribution subject to AIP icense or copyright, see http://jap.aip.org/jap/copyright.jsp
J. App. Phys., Vo. 89, No. 9, 1 May 21 Panasyuk, Aender, Key 4785 rivative such as ( z ) 2, 2 xz, x z negecting E z as was discussed preceding Eq. 3, one can rewrite the equation F/ as K a 2 xx K b 2 zz 1 2 K sin 2 x 2 a 8 E x 2 sin 2, A3 where K a ()K 3 sin 2 K 1 cos 2, K b ()K 1 sin 2 K 3 cos 2, KK 3 K 1. The term with 2 zz in this equation can be determined by recaing that the z dependence of at sma x is neary the same as the z dependence for x cose to w therefore using the arge x form of Eq. A3, where one can negect a x derivatives put E x E x : K b 2 zz a 8 E x 2 sin 2. A4 Equation A4 is appicabe in the region z near d/2 x near w. Substituting 2 zz from Eq. A4 into Eq. A3 gives an equation K a 2 xx 1 2 K sin 2 x 2 a 8 E 2 x sin 2 K b E K b x 2 sin 2 A5 for (x, d/2) on the interva x w, where ( w,d/2). After substituting here E x from Eq. 31 mutipying Eq. A5 by x this equation can be rewritten as 1 d 2 dx K a x 2 ae x 2 d 16 dx f where A6 f acos 2 2 sin 2 2. A7 acos 2 We have used K b ()/K b ( )1 for simpicity because this ratio is a sow function of the second term with this ratio in Eq. A5 is much smaer than the first term in the square brackets for. Taking into account that for x cose to w x derivatives can be negected, Eq. A6 can be integrated the constant of integration determined, giving d dx E x a 8K a f f. A8 Here f () f ( ) for a in the interva /2 f () f ( ) ony if. Substituting dx from this equation into Eq. A2, one finds that x 1 2E x 8K a 2 dy 1K cos y f y/2 f cos ycos 2, A9 acos y where K(K 1 K 3 )/2 K(K 3 K 1 )/(2K). Finay x E can be found sef-consistenty as was described after Eq. 32. As our cacuations show, ony three or four iterations are needed to find x E with an error of ess than.1%. In this way, however, one has to sove the dynamic equation 4 after each iteration, because in Eq. A9 is unknown from the beginning. To avoid the necessity of soving Eq. 4 repeatedy, we propose another way to cacuate. The main idea is to sove Eq. 22 at x w to estimate as the maximum vaue of ( w,z) aong the z coordinate. In the course of soving Eq. 22, one can take E x E x use E z ( w,z) as was discussed after Eq. 29. To sove Eq. 22, et us divide the gap between substrates at x w into two intervas 1 2, where 1 w 2 w d z at z 1 w. A1 Again, because E x is a weak function of z, it is possibe convenient to substitute E x by its vaue Ē x averaged over the z coordinate for each interva. After mutipying Eq. 22 by z soving the resuting equation using the boundary conditions ( w,)( w,d) z w,z 1 ( w ) for each interva, one finds the foowing soutions: K dw b w A11 f 1,2 w f 1,2 a 8 1,2 t. Indices 1 2 correspond to a particuar interva, t is the normaized z coordinate in each ayer: t1, where t1 corresponds to the border between the intervas. Functions f 1,2 (e 1,2 ) 2 cos 2w arise after taking the first integra of the resuting equation, where e 1,2 Ē x1,2 for each interva. Evauating the two Eqs. A11 at the point t1 where, summing them, an equation of sef-consistency for dw a K b w f 1 w f 1 8 d dw K b w f 2 w f 2 A12 is obtained. Equation A12 is soved with respect to after each iteration in the course of finding x E sefconsistenty. 1 S. H. Lee, H. Y. Kim, Y. H. Lee, I. C. Park, B. G. Rho, H. G. Gaabova, D. W. Aender, App. Phys. Lett. 73, 471997. 2 G. Panasyuk D. W. Aender, J. App. Phys. 87, 649 2. 3 H. Mori, E. C. Gar, Jr., J. Key, P. Bos, Jpn. J. App. Phys., Part 1 38, 135 1999. 4 W. Liu, J. Key, J. Chen, Jpn. J. App. Phys., Part 1 38, 27791999. 5 P. G. de Gennes J. Prost, The Physics of Liquid Crystas Oxford Science, New York, 1993, see Eqs. 3.89 5.32. 6 S. Dickmann, Ph.D. dissertation, University Karsruhe, Karsruhe, Germany, 1995. 7 J. E. Anderson, P. J. Bos, C. Cai, A. Lien, SID Symp. Dig. 3, 628 1999. 8 W. H. Press, B. P. Fannery, S. A. Teukosky, W. T. Vettering, Numerica Recipes in C Cambridge University Press, Cambridge, Eng, 1988. Downoaded 11 Apr 26 to 131.123.234.122. Redistribution subject to AIP icense or copyright, see http://jap.aip.org/jap/copyright.jsp
4786 J. App. Phys., Vo. 89, No. 9, 1 May 21 Panasyuk, Aender, Key 9 D. Haas, H. Woher, M. W. Fritsch, D. A. Mynski, SID Symp. Dig. 21, 31199. 1 H. L. Ong R. B. Meyer, J. Opt. Soc. Am. A 2, 198 1985. 11 H. L. Ong, J. App. Phys. 64, 6141988. 12 H. L. Ong, J. App. Phys. 64, 48671988. 13 M. V. Kurik O. D. Lavrentovich, Sov. Phys. Usp. 31, 1961988. 14 M. Keman, Points, Lines Was Wiey, New York, 1983. 15 I. F. Lyuksyutov, Sov. Phys. JETP 48, 1781978. Downoaded 11 Apr 26 to 131.123.234.122. Redistribution subject to AIP icense or copyright, see http://jap.aip.org/jap/copyright.jsp