# Anchoring energy and easy direction of non uniform surfaces

## Transcription

Anchoring energy and easy direction of non uniform surfaces

Anchoring energy and easy direction of non uniform surfaces G. Barbero, T. Beica, A. Alexe-Ionescu, R. Moldovan To cite this version: G. Barbero, T. Beica, A. Alexe-Ionescu, R. Moldovan. Anchoring energy and easy direction of non uniform surfaces. Journal de Physique II, EDP Sciences, 1992, 2 (11), pp.2011-2024. <10.1051/jp2:1992248>. <jpa-00247785> HAL Id: jpa-00247785 https://hal.archives-ouvertes.fr/jpa-00247785 Submitted on 1 Jan 1992 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Pllys. J. ii France 2 (1992) NOVEMBER1992, 2011-2024 2011 PAGE Classification Physics Abstracts 61.30 Anchoring (~), G. Barbero (~) Dipartimento Italy (~) (~) Institute Physics of (Received Abstract The of energy of the allows to Inti~oductiou. The orientation periodic a samples sample contains Materials, of an interference an (~) Moldovan degli Duca 24, Abruzzi Bucharest, MG 7 CP Department Rumania Physics,Splaiul of by Independentei, distribution of directions easy the on substrate on liquid crystal nematic a Alany aspects. Our papers devoted are to is this consider anchoring strength (the nematic distortion. very is of the literature nematic very the fs(n ar) ar is is the defined the important substrate understand to on the description phenomenological nematic. >vhich Of mechanism course large. very expressions used is the the llowever in is orientation the literature and ar orientation average n energy fs, in the absence of bulk of fs around its minimum. A curvature [iii, by Rapini and Papoular which ago as responsible number both surface proposed long one interaction fs(1) = researchers. nematic the This directions phenomenological w(n ar)~. The phenomenological approach + 2 [12-16], because it gives the possibility to describe, by means of few liquid crystal-substrate interaction. The microscopic approach is, on kind: microscopic the easy direction director) on the surface minimizing The anchoring strength is defined as popular expression of is easy surface axis. easy the The w. elective important both for subject [1-4]. The problem relevant to the nematic-substrate The from pl~enomenological [5-7] and microscopic [8-10] points of view. description of the surface is made by using the concepts of easy interaction of 313, macroscopic surface analysis shows that the interference coming from the two diferent term uniform direction, depending on the two easy term too, as suggested a few years ago by some solid a fundamental analyses Torino, 10129 August 1992) 4 considered. are elective an induced technological and of erect introduce 1 Technology Bucharest, Corso surfaces Rumania nematic containing of uniform non (~) and R. Torino, di 1992, accepted in final form June 9 properties published and of Alexe-Ionescu A-L- Politecnico Institute Bucharest, 77216 (2), Fisica, di direction easy Beica T. Politechnical and and energy of for parameters microscopic are the correct. parameters, the necessary also papers in the hand, imposed by complete a other orientation approach The widely used is for that confirms relevant to the JOURNAL 2012 PHYSIQUE DE II N°11 phenomenological approach try to connect the physical properties of the substrate induced on a given liquid crystal. The effect of the surface orientation geometry has taken into account with also the been [17-20]. published the surface is considered uniform and characterized by a well papers anchoring strength. is the But, well known, solid substrate easy energy as is never homogeneous over a large scale, even if the chemical composition of the surface does interesting to consider the effect of a spatial distribution of easy Hence it not change. seems experimentally direction. This analysis could be important also directions the detected easy on In most of the defined the in direction which in case evaporated and a surface is by characterized than more axis, easy one as in the case of [21-23]. surfaces organized section 2 a semi-infinite periodic structure, with In analysis, which generalizes the one proposed long ago by Berreman [17] and de Gennes [18] and recently by Durand [19] et al., gives the opportunity of a periodic introduce the equivalent surface In section 3 the finite structure. to energy anchoring energy, in a semi-infinite sample, is taken into account. Our analysis, very to near performed by the results obtained section will show the Faetti generalizes in 2. We [20], one different in the surface there that the directions give rise to a kind of interference easy energy. characterized by an imposed deformation in the bulk is considered. In section 4 a finite sample results with the In this way >ve can define an equivalent easy axis. This is done comparing our relevant to a uniform sample. It gives, furthermore, the possibility to treat a well known one, homogeneous surface. In section 5 the main conclusions of surface with periodic structure as a given. our paper are Our strong paper is anchoring, sample Semi-infinite 2. Let us consider z-direction- a Les solid a follows. as considered. is The with fl be alichoi~ing by characterized surface and sti~ong the easy axes a energy, periodic for 0 < z < a of structure and < a z < wavelength ~ along (see Fig. I): ~ ~~ T a li O Fig. i. Semi-infinite ~ liquid crystal sample ~vith periodic ~ distribution x of easy directions. the N°il DIRECTION EASY NON OF UNIFORM SURFACES 2013 aforo<z<a, 8(z) (I) = fl for total The elastic of energy period, of the one < a ~. < z liquid crystal (unbounded along y) is nematic given by: F ~iv~)~d~dY lk it°' = 12) approximation, and by supposing that n is everywhere parallel to the Equation (2) holds in the strong anchoring case, I-e- in the event in which R, the angle made by the director n with y, at y = 0 is imposed by the surface In our treatment. the in constant one (z,y)-plane. case: ~(z, 0) 8(z) where is given by (I). By minimizing (2) [email protected] the function boundary have we be can minimizing (2) (I). condition determine to solved expand R(z, y) shown as Fourier in the equation ~ fiy2 ~~~ harmonic a known well £ ~> " ° = R(z, y) By putting series. R(z, y) for function. This function has to satisfy the problem, called Dirichlet's problem, in which function knowing its value on the boundary. Equation (4) Appendix I, but for further generalization it is better to is is the harmonic a directly, in This (3) obtain we [email protected] ~ 4 i-e- 8(z) " + ~ [D[cos(nqz) E(~sin(nqz)] + (5) e~~~Y n=1 imposing boundary and (3), condition ~l ~ 0 D[ ~'~ (2) q one = ~ obtains is the vector wave + ~" ) ~ /q found are to be ~)fl ~ ~ ~si~l'~q~) ~~~ ~°~~~~~~~' periodic of the ~ ~ ~ l~ coefficients ~ ~ " ~~ where expansion the ~ ~ ~~~ By substituting (5) and (6) into structure. j~ ~~~~~~~ ~~~ n= This is the equivalent is zero. total energy surface Furthermore per energy it is unit of the zero length along the sample [17, 18]. fl. These for a = z-directionNote results that are It for a can = obvious. 0 considered be or a = Keeping as this in the energy mind the JOURNAL 2014 definition due energy of q, it is easy to show that F/~ is distortion imposed to the periodic D[ I is the of the average axis, easy y Be angle imposed by tilt considered be can as Appendix 2). The results [24] for the particular case Let value easy this In w. F total )k /~ = previous the torded for second term, for small 0 < z y - In the ~ < from and 0 < y < condition boundary the c<, I-e- the to elastic that: (8) Be, silicon in which + surface. under follows It analysis presented the that (see consideration in reference surfaces. energy~ on 0 < z < energy of the ~(VR)~dzdy the structure generalise anchoi~ing situation from far very periodic the vicinal on finite the easy and a a nematic still the ones of equation ~, and they have the < z < same sample is given by axes are 8(z)]~dz. jw /~[S(z, 0) (9) o o considered the contribution. surface deviation c<, section weak the sample is equation (9) case c<. - elastic o As in ~~~ = to this in with the event i-e- Note treatment. ~~ ( ~ ~" due anchoring general now a more anchoring energies are the but fl)2/~, (a to = at y axis consider us (I), surface obtained sample Semi-infinite 3~ a)fl (~ N°11 c< the of proportional by the surface ~ R(z, y) - the + au II where lim is the PHYSIQUE DE easy R(z, y) first The along unbounded the represents term surface is term y-direction and undiscontribution, and the in the parabolic form, valid the bulk written calculations By minimizing (9) trivial still given by equation (4), but it has now that for satisfy the axis. show is to ()) 81~)1" )lsl~> 0) + (lo) °> y=o at y = 0. In (10) L = Neuman-Dirichlet's knowing a relation k/w is the extrapolation length. The problem (mixed), in which we have betwen Appendix I this problem putting expansion (5) into derivative normal its equation (10) Do,, " 2 ~~ l aa + we fl(~ +~qL ~n l instead substrate of (6). S(z, y) Note tends that to also iii its obtain now determine value on the means of a for the border. As called now function shown expansion. Fourier expansion is harmonic a in By coefficents a) ~ ~iqL (ii ~~ and easily only by solved be can problem mathematical to this case ~° ~~~~~~~~~~ ~° ~~~~ ~°~~'~~~~~ equation (8) holds, Be (See Appendix 2). By assuming 8~ ~~~~ I-eas the very easy far from direction the solid imposed N°ii by EASY the uniform surface sample defined non z-uniform DIRECTION 8~ treatment, UNIFORM NON be can SURFACES 2015 by minimizing deduced a surface of energy a as: Is a)~ wo(S = ~wp(R + fl)~ a)/~ and assumed z-independent. w(~ is By putting [aa a)]/~, previously reported. that if for fl(~ Note + one as results have to be 0 < z < a and a < z < ~ the anchoring energies are different, previous modified. in which the chemical of the surface is everywhere the However in our nature case w(0 a) w(a ~) be justified. to to < < < z < same, assume z seems condition (10), equation (9) becomes: Using the boundary where wa fifs/fiR = w(a/~) OF = and wp obtains 0 = = 8~ = = ( jk1/~ ~(VR)~dzdy + L = o By substituting into o ~ 2 and II) ~ L Equation (13) shows the elastic energy is that no>v the localised total energy shown L. As over f ~~~~~ in this in Appendix (~f~ ~~ ~=i l ~ )2 (1+ n2q2 n=1 8 ~) proportional is case ~( ~ 1(~~) ~~f~ ~"~~ (12) y=o obtain we (°~fl)~f ~ ~ Y o (12) equations (5) ~ /~ ())~ dz). (a to fl)~/L, I-e- 1. ~) (~ ~) Consequently ~~~~ ~ i and (13) be can rewritten (nq)2 (1~ ~~ l~~ interference 4~ Sample Finally let surfaces works between of us of the finite consider kind ~) ~ ~~~ ~~~~ l' ~~~~ 8~~~ as ~' Equation (16) ~~~ well the if qL two thickness a real described » I, I-e- surfaces and problem above. ~ ~ of Let for L a us ~~ ~ Equation (16) ~. -~ characterized weak ~~L by different anchoi~ing sample suppose of first can be interpreted as an directions. easy energy, thickness that d the = 2e surfaces limited are by two arranged solid in an JOURNAL 2016 PHYSIQUE II DE N°11 y e -fl -a 0 ~ a x fl ~ Fig. 2.- liquid Finite distribution of antisymmetric crystal directions easy and way, sample arranged the hence thickness of in angle tilt is odd an by limited 2e antisymmetric an solid two surfaces periodic with way. with function respect the to sample, as shown in figure 2. middle frame having the origin of the y-axis in the By using the reference half thickness is given by total period and (see Fig. 2), the energy of one middle of the F = k 2 ~(VR)~dzdy /~ + 0 _e 8(z) where is odd function still w-r-t- given by (I). Simple -R(z, -y). y : R(z, y) = ~° S(z, y) The y = expansion Simple -e. = coefficients y + 2 now are calculations £ determined 2 ~" ~ ~ ~ " = ch(nqe) and Sn 8(z)]~dz /~[S(z, -e) (17) 0 considerations Its sample Fourier + by the show R(z,y) that expansion is is now a harmonic written Ensin(nqz)]sh(nqy). boundary condition (18) (10) rewritten for give e Cn w [Dncos(nqz) ~° where 2 of the = sh(iiqe). aa + fl(~ ~ + L a) ' sin(nqa) (flqLcn + Sn flq cos(flqa) i fl (flqLcn + Sn) flq fl a ~ a ~ ~jg~ EASY N°ii By operating length along the in as previous the is found z-axis I DIRECTION lk Ii = NON OF UNIFORM relevant case, anchoring, weak to 2017 total the energy for unit be to ~~[l~)lT'~j I) l~ +1 1°1~)~ + SURFACES (2°) nq where Tn Sn/Cn = and °~ os (~ + j ~~ = is the average surface (Six, -e)1, 121) angle. tilt y = ~ ~ fl a 0 ~ a a x fl e Fig. 3. Finite distribution of In is an the even event liquid in crystal directions easy which with function the sample arranged surfaces respect to thickness of in arranged are the limited 2e symmetric a middle by solid two surfaces with periodic way. in of the symmetric sample, as in the tilt angle figure 3, R(z,y) is Elsin(nqz)I ch(flqy), (22) way, hence and shown given by R(z, y) ~~ ~ « + = lDlcos(nqz) + n=i where the expansion coefficients are ~* given by now ~"~ + 0 ~"~ ~ ~~ fl(~ ~ ~ a ~ ' fl sin(iiqa) flq(flqLsn+Cn)~ ~ q(nq~~~~~~~nl' ~ ~~ ~) ~~~~ JOURNAL 2018 By substituting the total expansion (22) sample of half energy PHYSIQUE II DE expression (17) length along into unit per N°11 and taking the z-axis into equations (23), account is section, it is supposed that the two regions with easy axes a and fl have to be in exact practical sample, one can expect a random arrangement of as in figures 2 and 3. Iii a calculations the two regions on the two plates. In this case the above reported be easily can generalised. Of course the results depend on the actual arrangement of the considered surfaces. (19-21) different considered This that equations cliange for from the arrangement means an one above. However the aim of our analysis is to show the expected results for two very simple, and completely different, surface arrangements. direction of a uniform surface we choose as criterion In order to deduce the "equivalent" easy the equality of measurable quantities. From the experimental point of view we can only measure birefringence or other physical anisotropies of the sample. These quantities are proportional, in the limit of small angles to the average tilt angle, (R~). Consequently it is possible, square characterized according to our point of view, to define two samples as equivalent if they are bj under the defining, for the periodic consideration, by of By (R~). structure (R~) same means In this register (R~) ~ /~ /~ S~(z, y)dzdy, = e and for the z-uniform 0 by characterized structure )/ of (Sll (25) -e ~ = a R(y) tilt angle the same quantity by means (26) Sl(y)dy> -e "equivalent" the easy axis is by putting obtained (S() In the sion event (18) which in and the sample is in into (25) the equations (19) one (S~). = arrangement, by substituting obtains, in the limit of e » ~, antisymmetric where m j~(~ ' To the contrary for an uniform ~~~~2 ~) ~ (27) (iiq)3(1 ' antisymmetric expan- [email protected]) (~g) 2 + sample, nqL)2' for which Su(y) = -(yle)Ssu, [25] equation (26) gives ~~u) It follows that in this case the condition ~~~ = 3 (27) gives: su. (~~) EASY N°11 Consequently DIRECTION "equivalent" the (~ ~~ ~ Since in this case equivalent the axis is easy both In ~ ~ a, (32) L). and R(z, y) hence is given by (~ ~ ~)~ R(q, a, L). + homogeneous sample, (RI in the present ~j (~ (°~ equivalent the cases now = R(q, (33) = R]. Consequently given by ~ S( ~ arrangement, e is distortion no ~j (~ (°~ = @) give ~ (R~) 2019 e symmetric in the SURFACES [25] ~ ~ (~ e homogeneous sample is non expansion (22), similar calculations to be ~ + = If the UNIFORM axis is found easy ~ °~ R) NON OF ~ ~ + e axis easy contains R(q, a, proportional term a )~ (° ~ (34) L). to ~ R(q, 1° ~ ~) vanishing the for a fl = interference or a 0 = or reduces term a ~. = L) in the that Note a, (35) anchoring of strong case I-e- for L - 0, to L m Rlq, a, 0) 2 @) ~)~~~] = Rlq, > L). a, n=1 origin from a geometrical c<), as expected. An proposed a few years ago by Yokoyama interference effective anchoring energy term in the was variation of the surface tilt angle experimentally et al. temperature [23] to interprete the observed in nematic samples. It is important to underline that the analysis presented in section holds only for a fl « I, I-e- a fl. In fact the parabolic expression 3 and in the present one deviations of the surface tilt angle for the surface used in the text is valid only for small energy ai « I and ii pi « I, and consequently a fl. In from the easy axis. This implies that ii ai pi the general case of ii I the analysis has to be performed by using for I and ii anchoring energy the Rapini-Papoular or more general expressions. In this frame the surface condition. complicated, due to the non linear character of the boundary the analysis is more interference surface However term is still expected in the energy. an This that means effect, and the interference only vanishes it term for is always present. takes It anchoring (I.e. weak very for L - -~ -~ -~ -~ Conclusions. 5 We have of a considered periodic the effective the two optical easy the effect distribution easy axes measurements axis on with is plus a an are the macroscopic surface different two weigthed In term. fact More axes. (with respect average interference involved. easy properties of in this to interference This case a nematic a liquid crystal precisely we have shown that the geometrical extension) of term sample is is usually negligible considered uniform if if the JOURNAL 2020 PHYSIQUE II DE N°11 Otherwise the sample is macroscopically than the light wave-length. goal of our analysis has to be modified. Our results confirm the model characterized by two coworkers [23] relevant to surfaces ago by Yokoyama and also be applied explain recently presented in this directions. The analysis to paper can easy induced by the transitions observed surface temperature in nematic samples limited by SiO the observed evaporated surfaces covered by lecithin [26]. It can be applied also to connect evaporated liquid crystal limited by surfaces containing of samples surface orientation a average doped with an additive giving homeotropic alignment [27]. spatial regions inhomogeneous proposed long thinner are the and Acknowledgements~ This work has Strigazzi (Torino) for and (Orsay) to G. Durand Appendix In section In this relevant the 2 f(z) TENIPUS of the Italian-Romanian and of programme Many thanks are due to S. Faetti (Pisa) and A. direction discussions the meaning of the equivalent easy on critical reading of the manuscript. useful many for a solution the Dirichlet problem is given by the problem will be solved in a general sample is solved by the function [28] of same semi-infinite a ii where frame I. appendix to in the realized been partially supported by MURST. collaboration, R(z,0) = Equation (Al holds in our z, y) ? f" coincide case the with strong anchoring in the (j)~ = case. way. Fourier a The Dirichlet expansion. problem jAi.1) dj ~ direction easy Let of means assume us the (imposed by the surface). f(() is given by, function f(z)=aforAn<z<On E n f(z) = By putting the origin of the z-axis this event equation (Al.I) gives ~~~ ~~ Simple £Y j fl _ calculations fl for On in Ao < we z (Al.2) Bn. < have An ~j arctg [~~ ~l IO, c<) n~, On = = n~ + a, Bn ~~~ ~~ ~) ~~~~~ ~ ~ = (n + 1)~. In ~~~ ~~ that sho>v (S)> = /~ S(z, y)dz ~ ~~ = ~ ~~ ~~ IA lA) o S(~, y) along x is independent of y. possible to determine the total energy, as done in section 2. We means complicated than the one given in section 2. underline however that equation (Al.3) is more direction imposed on a nematic In section 3 the effect of the weak anchoring on the easy liquid crystal has been analy2ed by means of Fourier expansion. In this appendix we show that I-e- the By average of value (Al.3) of it is DIRECTION EASY N°11 OF NON SURFACES UNIFORM 2021 solve the problem by means of general formula is very difficult. Let us consider the same has to satisfy the boundary condition (10). problem of the section 3. The harmonic function By indicating with f(z) the actual value of R(z, 0) and by using equation (Al.I) and equation (10) we obtain for f(z) the integral equation to -) /~ /~~j~~di )lf(z) + 8(z)1 (A15) ° = because the kernel has a strong singularity. This equation is not which is a singular equation, integrable in the ordinary sense [29]. The appropriate solution can be expressed in term of constructed Cauchy's principal value. A complete theory ofsingular integral equations has been for our aim, Fourier information but it is difficult However expansion gives all the to apply. about the physical mechanism, and it is simpler than to solve the singular integral equation. used this technique in the text. For this reason we relation (14) let us consider the periodic function In order to show Afor0<z<a, g(Z) " z < ~. (Al.6) [dncos(nqz) + ensin(nqz)], (Al.7) B for By expanding g(z) in Fourier < a series g(z) £ + = obtain we ~° The value ~ Aa + B(~ a) ~ of sin(nqa) + cos(nqa) I ' nq over ~~ ~~~ ~~ ~ g~(z) integrated A~a ~~ ~~~ ~ ' ~ ~'~~ nq ~ is B~(~ a) (( ~ ~ + = £ (d( + e() ~. (Al.9) ~ By substituting Appendix (Al.8) into (Al.9) we obtain easily (14). 2. appendix the calculations reported in section 4, relevant to a sample of finite characterized by the generalized. More precisely, we consider now a sample given by equation (I) on a surface, and by a strong anchoring imposing a tilt angle In this will other be one. The figure 4. The the boundary the other one. thickness of the sample will be by denoted R(z, y) minimising the total condition (10) on the lower surface, function energy and the e. reference The is still harmonic boundary frame and condition thickness axis easy Rb on the is shown in it has R(z,e) satisfy to = Rb on JOURNAL 2022 PHYSIQUE DE N°11 II y e a fl 0 Fig. Finite 4. liquid crystal periodic distribution angle is fib- Routine I a of thickness are limited e and present by surfaces solid a another at one y = e on at y 0 = which the which on anchoring give: calculations R(z, y) sample of directions easy x ~° = ~ f Cy + [Dncos(nqz) + + Ensin(nqz)] sh[nq(e (A2.I) y)] n=1 where ~ " = the total energy nq ~~ ~ that If we the last whose consider, in F is term role is as done in independent can derive ~ ~"~ section ~~~~ of Sb. Hence ~j~)~ it can analysis. z-uniform sample, the ~(j be ~~~'~~ considered as 4, a jk~~[ total energy is found ~~~ expressions (A2.3) and (A2.4) we deduce that 8~ plays the role of the conclusion by applying the criterion proposed in section 4, same the additive an in the = We nq~flq~n ~ ) By comparing j~~~" ~ fundamental not ~~~)~~~ In qL jA2_2) ~ constant + 2CL is: ~~~~L~~~ Note ~~ ~jj)~ ~ that fl(~ ~ ~~~ En follows + 2(a Dn It aa ~ ~ be: to (A2.4) easy axis. connected N°11 (R~). By to of means EASY DIRECTION (A2.I) and ~~ (R~) (A2.2) ~~/ = + NON we obtain: ~~~~ ~ UNIFORM OF ~~ ~) + + e SURFACES 2023 ~ e~ + Ri (q, a, (A2.5) L) e where ~~ The ~~ i 1~ ~l ~~ ~ quantity same for ~~ll))1)t5Sii~ ~ z-uniform sample ~~ (S() ~)~ = (A2.6) series The interference + eRb(3L converges Ri term + ~~~ ~ ~)~ ~ ~~~~ + e)8e ~ ~~' ~~~~~ is: (~) = e By comparing (A2.5) with (A2.7), the equivalent e~8( i 1~ ~l ~ easy (A2.7) e~ + e axis is given by: eR~Rb(3L + e) + IL + e)Ri e~R( rapidly. By approximating it by means of its " (A2.8) 0. first addendum, the becomes: ~~ ~ e p)2 (~ ~~~'~~ 2~r3 sample considered as uniform this term can be important or not according to the exa the perimental technique used to appreciate this fact. In the case of optical measurements sample will be considered as uniform if ~ is smaller than the wave-length of the visible light, Consequently for samples studied by optical means, the interference I-e0.5 ~m. term is could be important for samples negligible and Se Be. On the contrary, the interference term studied e-g- by capacitance in which the uniformity of the sample is not tested. measurements independently interference Of if ~le is small the importance of the term is also small, course of the technique employed. conclusions hold, of course, for the antisymmetric and symmetric cases considered The same For -~ = in section 4. References [ii Cryst. Liq. Cryst. 179 (1990) 217. Physics of Liquid Crystalline Materials, I-C- Khoo and F. Simoni 1982) p.301. H., &Iol. Cryst. Liq. Cryst. io8 (1988) 317. Jerome B., Rep. Frog. Pliys. 54 (1991) 391. Nehring J.,1<met2 A-R-, Shefer T-J-, J. Appl. Pliys. 47 (1976) 850. Yang K-H-, Rosenblatt C., Appl. Pliys. Lent. 43 (1983) 62. Yokoyama H., Van Sprang H-A-, J. Appl. Pliys. 57 (1985) 4520. Bernasconi J., Strassler S., Zeller H-R-, Pliys. Rev. A 22 (1980) 276. Parson J-D-, J. Pllys. France 37 (1976) 1187. S., S., Breach, [3] Yokoyama Faetti Mol. [2] Faetti [4] [5] [6] [7] [8] [9] [lo] Sluckin T-J-, Texeira P., to be published. Eds. (Gordon and JOURNAL 2024 PIIYSIQUE DE II N°11 [1ii Rapini A., Papoular M., [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] J. Pllys. Colloq. France 30 (1969) C4-54. P., Faetti S., Fronzoni L., J. Pllys. France 44 (1983) 1061. Chiarelli P., Faetti S., Fronzoni L., Pliys. Lent. ioiA (1984) 31. Nobili M., Lazzeri C., Schirone A., Faetti S., Mol. Cryst. Liq. Cryst. 212 (1992) 97. Rosenblatt C., J. Pliys. France 45 (1984) 1087. Rosenblatt C., Griffin A-C-, Uma Hari, Liq. Cryst. 7 (1990) 359. Di Lisi G., D-W-, Pliys. Rev. Lent. 28 (1972) 1683. Berreman de Gennes P-G-, The Physics of Liquid Crystals (Oxford University Press, 1974). Barbero G., Durand G., J. Pllys. II France 1 (1991) 651. Faetti S., Pbys. Rev. A 36 (1987) 408. Monkade M., Boix hi., Durand G., Europliys. Lent. 5 (1988) 697. Cognard J., )[al. Cryst. Liq. Cryst. Suppl. 1 78 (1982)1. Yokoyama H., I<obayashi S., I<amei H., J. Appl. Pllys. 56 (1984) 2645. Jerome B., Pbys. Rev. A 42 (1990) 6032. The total unit length along the z-axis, of a z-uniform sample caracterized by energy per ~~ ~k~~~ + @(z,0) uniform axis @e, VT E (0,1) and having. 0 is given by easy a Chiarelli = ~wu 2 (@su z-uniform w-r-t- @su is fe)~, sample. reached where Of @su " course for fsu @u(-e) in this Barbero Illeman Petrovski case ~~ = 1+ (Lule) is wu @u depends where , the Lu = anchoring linearly on energy y. The of the minimum 2 e considered of (22) k/wu. Cryst. Liq. Cryst. (to be published). G., Beica T., Alexe-Ionescu A-L-, Nloldovan R., Liq. Cryst. (to be published). AI., Points, lignes, parois, (Les Edition de Physique, Les Ulis, 1977). Moscow, 1975). I-G-, Lectures on the Theory of Integral Equations (MIR Publishers [26]1<omitov L., Sparavigna A., Striga22i A., [27] [28] [29] and = 1 Mol.