Polar flexoelectric deformations and second order elasticity in nematic liquid crystals

Transcription

1 Polar flexoelectric deformations and second order elasticity in nematic liquid crystals A.I. Derzhanski, H.P. Hinov To cite this version: A.I. Derzhanski, H.P. Hinov. Polar flexoelectric deformations and second order elasticity in nematic liquid crystals. Journal de Physique, 1977, 38 (8), pp < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1977 HAL is a multidisciplinary 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.

2 a On The The and the LE JOURNAL DE PHYSIQUE TOME 38, AOUT 1977, 1013 Classification Physics Abstracts POLAR FLEXOELECTRIC DEFORMATIONS AND SECOND ORDER ELASTICITY IN NEMATIC LIQUID CRYSTALS A. I. DERZHANSKI and H. P. HINOV Institut de Physique des Solides, Académie des Sciences de Bulgarie, Sofia 1113, Bulgarie (Reçu le 29 juin 1976, revise le 25 avril 1977, accepte le 2 mai 1977) 2014 Résumé. sur les déformations flexoélectriques polaires dans une couche homéotrope nématique (cas A) et dans une couche planaire nématique (cas B). Le champ électrique est perpendiculaire à la couche, l anisotropie diélectrique peut être positive ou négative. Les conditions de seuil exactes sont obtenues numériquement à partir des équations pour différentes valeurs des énergies de surface anisotropes WS1 et WS2, du coefficient flexoélectrique total e1z + e3x et de K13. Pour K13~ 0 et quelques valeurs critiques de WS1 et WS2 les déformations apparaissent soudainement; l énergie nécessaire à la création des déformations flexoélectriques (cas A et B) est considérablement modifiée. a étudié théoriquement l influence de l élasticité de second ordre (K13~ 0) 2014 Abstract. influence of second order elasticity (K13 ~ 0) on the flexoelectric polar deformations in a homeotropic (case A) and a homogeneous (case B) nematic layer with negative and positive dielectric anisotropy and an electric field normal to the layer is considered theoretically in this paper. The exact threshold condition is obtained, transcendental equations are solved numerically for different values of the anisotropic interfacial energies Ws1 and Ws2, the total flexoelectric coefficient e1z + e3x and K13. For K13 ~ 0 and some values of Ws1 or Ws2 (critical values), the deformations start with a jump. The required electric energy for the appearance of the flexoelectric deformations (case A and B) is corrected considerably. 1. Introduction. fundamentals of flexoelectric theory of liquid crystals were laid by R. Meyer [1]. He pointed out that nematic and cholesteric liquid crystals with asymmetric and polar molecules may deform under the influence of external electric fields linear flexoelectric effect vice versa in the case of deformation : they are polarized. The subtlety and diversity of this phenomenon was shown once again by the concept of a polar flexoelectric effect in the case of interaction of a homeotropic nematic layer with As 0 and asymmetric anisotropic interfacial energy (zero and infinity) and a constant electric field parallel to the initial orientation of the liquid crystal proposed by W. Helfrich [2]. In these conditions he pointed out that the flexoelectric deformation due to the interaction between the applied electric field and the flexoelectric polarization parallel to it, may exists for only one sign of the applied electric field. The critical threshold voltage obtained by Helfrich has the form : K33 is the Frank elastic coefficient of bend and e 1z + e3x is the total flexoelectric coefficient. W. Helfrich s concept for a polar flexoelectric effect (case A) is elaborated upon theoretically in this paper for different values of the anisotropic interfacial energy WS1 and WS2 and for different signs of the dielectric anisotropy. The elastic surface terms due to the second derivative of the director n (unit vector following the orientation of the liquid crystal molecules) with respect to the coordinates (elastic coefficient K13 0) are included in the surface energy. This is based on the theoretical conclusions drawn by J. Nehring and A. Saupe [3] that the secondorder elastic deformation energy is comensurate with the firstorder elastic deformation energy under the condition that all elastic coefficients are of the same order [4]. On the other hand the polar flexoelectric effects require small surface energy of interaction of the liquid crystal molecules with one of the surface walls which can be corrected to a great extent by including the surface elastic deformation energy due to the second derivatives as well. A new case B is also considered the polar flexoelectric effect interaction between a homogeneous nematic layer with negative and positive dielectric anisotropy and a constant electric field normal to the layer is demonstrated as well. (A unified Article published online by EDP Sciences and available at

3 description of flexoelectric effects for various experimental geometries different cases for K13 0 is given by A. Derzhanski and A. Petrov [5].) The differential equations and the boundary conditions are obtained after minimization of the functional of the liquid crystal free energy with respect to the deformation angle 8(z). The solutions demonstrated a deforming action of the linear flexoelectric surface torques as well as a deforming (case A, As 0 ; case B, As > 0) and an orienting (case A, As > 0 ; case B, As 0) action of the bulk dielectric torque. The character of the boundary conditions brings about a threshold requirement for these deformations. The threshold transcendental equations, obtained from the vanishing condition O(z) > 0 are solved numerically for the most interesting cases. Taking the secondorder elasticity for some values of the anisotropic interfacial energy, depending strongly on the value of the total flexoelectric coefficient e, + e3x (Critical Interfacial Energy) into account brings about an abrupt threshold with considerable deformations. Two examples the flexoelectric interfacial energy is exactly equal to the anisotropic interfacial energy (infinite anisotropic interfacial energy at the other wall) for ± K13 were solved. The sign of the elastic coefficient K13 is of great importance for the type of deformations in the layer. For example for case A, As 0, the negative sign of this coefficient for large or small values of the anisotropic interfacial energy (differing from the critical interfacial energy) ensures deformations changing smoothly with the electric field. The positive sign of K13 for small values of the anisotropic interfacial energy leads, above a certain value of the electric field, to smooth deformations up to the maximum possible. The influence of the sign of the elastic coefficient K13 may be replaced by a change of sign of the dielectric anisotropy of the liquid crystal. A characteristic feature of the case A, As > 0 and case B, As 0 is the absence of flexoelectric deformations for relatively small values of the total flexoelectric coefficient e lz. + e3x. The exact solution of the problem allows for the effective preliminary assignment of appropriate surface energies of interaction of the liquid crystal molecules with the walls attainable in the experiment. The inclusion of the dielectric anisotropy, as noted by Helfrich [2], on the one hand changes the polarity with mixedflexoelectric and dielectric deformations (case A, As 0 and case B, As > 0) and the existence of a threshold for both directions of the electric field, but on the other hand it lowers the voltage required for the effect. From here it follows that the operating voltage of this effect can change with the change of As. In the cases only flexoelectric deformations exist (case A, As > 0, case B, As 0), important relations between all constants of the liquid crystal (with a preliminary measurement of the values of the surface energies) may be obtained. 2. Case A, theory As 0, As sll 8ol8 The interaction between a thin homeotropic nematic monocrystal with As 0 and flexoelectric properties (e lz :f:. 0, e3x 0) and finite anisotropic elastic energy of interaction of the interface molecules with the cell walls WS1 and WS2, and a constant electric field parallel to the initial orientation is considered theoretically in this section. (The conductivity effects are neglected.) The free energy of this nematic layer written in an invariant vector form following Nehring and Saupe [3] and Meyer [ 1] reads : It is a sum of the elastic energy (with an accuracy to the second derivatives of the director n with respect to the coordinates), the flexoelectric energy and approximate dielectric energy, accurate to a large extent for nematic liquid crystals with small dielectric anisotropy ((Asls_L) 1 ). The following physical quantities in CGSE units are introduced in [1] : K11 K22 and K33 are elastic coefficients of splay, twist and bend respectively; e1z, and e3x are flexoelectric coefficients of splay and bend; sll and El are the permittivities of the liquid crystal along the liquid crystal optical axis (the long axis of the liquid crystal molecules) and along the normal ; Kl3 is the secondorder elastic coefficient ; WS1 and Ws2 are the anisotropic interfacial energies of the liquid crystal (theoretically between zero and infinity); V1.2 are the normals to the nematic layer surfaces. The introduction of a rectangular coordinate system OXYZ is most convenient for the problem thus set the X axis coincides with the lower interface surface, Z is the normal and Y brings the system to the right orientation. The constant electric field E applied along + Z is denoted by TE and along Z by JE. The anisotropic interfacial energies ensure that the preferred orientation of the liquid crystal is homeotropic (WS1 of the lower surface and Ws2 of the upper surface, Fig. 1). We suppose them to be substantially depending on z.

4 1015 Substituting the components of the vector field nx sin 0, nz cos 0 and Ez E in the expression for the free energy [1] transforms the latter into a functional of the deformation angle O(z). Varying this functional with respect to 8(z) (HEIO(z) is transformed into a functional of the inverse function z(o) and is varied as a functional with movable boundaries) gives the differential equation describing the deformations in the layer : FIG. 1. Polar flexoelectric deformations in nematic layers with a negative dielectric anisotropy : upper cells (case A), lower cells (case B). and the rather complicated boundary conditions for the values of O(z) at the two interfacial surfaces : do/dz 0, d is the thickness of the liquid crystal layer. The presence of the second derivative in the boundary conditions is a result of the elastic surface energy, due to secondorder elasticity (Kl 3 A 0). It is clear that this energy introduces essentially nonlinear terms in the boundary conditions and complicates the problem very much. Another characteristic feature of these boundary conditions is their manifest dependence on the electric field E. (Similar boundary conditions for K13 0 were first obtained by A. G. Petrov [6] in a theoretical investigation of low frequency flexoelectric oscillations and by Derzhanski and Petrov [7] when investigating low frequency flexoelectric oscillations and deformations for asymmetric interfacial energy without taking into account the secondorder elasticity.) A quadratic equation for d8/dz is obtained for the two surfaces (see Derzhanski and Hinov [8]) from (2) and (3) The following important conclusions can be drawn from (4) : the value of the derivative of the deformation angle O(z) at the surfaces depends not only on the flexoelectric energy (elz + e3,) E but on the dielectric energy as well (I As 1/4 n) E2. (This case was touched upon by Barratt and Jenkins [9].) The sign of the coefficient b is of great importance. It is evident that it is determined from the relation between the flexoelectric and the anisotropic interfacial energy. The sign of b at the two surfaces dictates the sign of d8/dz with respect to that of O(z) at the interfaces which determines the type of deformation in the liquid crystal layer under investigation. Two types of deformation may be realized in the problem under consideration with a special selection of the interfacial energies. The monotone deformation is determined by the expressions :

5 weak 1016 The interfacial angles are smaller than Tr/2 because of the unidirectional action of the volume and surface torques (see Barratt and Jenkins [9]). This solution requires little anisotropic interfacial energy at the upper wall anchoring of the liquid crystal molecules with this wall ((ei + e3x) Er > WS2 (el + e3x) > 0, Fig. 1). The deformation with a maximum in the liquid crystal layer is determined by the expressions : In the cases WS2 > (ei + e3x) Et and WSl (el. + e3x) Et, dor,,,idz change signs which leads to the monotone deformation changing into a deformation with a maximum in the liquid crystal layer and the deformation with a maximum in the cell turns into a monotone one. The differential eq. (2) and the boundary conditions (3) allow, for WS1 :f:. WS the elementary solutions 00 and 0 n/2 as well. In order to find which is the real physical solution in this case, a dynamic stability investigation is required. This is impossible at the moment. For this reason we shall follow Dafermos [10], Leslie [11] and Barratt and Jenkins [9] and shall assume that the only solution over the threshold is the one with lower energy. (At the same time we shall keep in mind that other solutions with lower energy than the solutions

6 1017 considered may exist.) The criterion leads to the following conclusions : the deformations (5) and (6) are of lower energy than the homeotropic orientation if the following inequalities are valid (see Barratt and Jenkins [9]) : For a deformation with a maximum inside the liquid crystal layer : The homogeneous orientation (8 n/2) will be of lower energy than the homeotropic (0 0) when the following inequality is valid : The solution (5) and (6) are exact solutions and fully exhaust the problem of polar flexoelectric deformation in the problem thus posed for case A. We shall however in this paper touch upon the problem of threshold characteristics alone, giving the cases most advantageous for practical applications when realizing polar flexoelectric deformations. The complete tabulation of (5) and (6) will be carried out in another paper. The following very important threshold transcendental equations for the nondimensional parameter U / Uo (Uo is the threshold for fixed vanishing boundary conditions for the homeotropic nematic layer) were obtained following Barratt and Jenkins [9] and Leslie [11] from (5) and (6) with Z d after changing the variables respectively to sin A sin 0/k and sin A sin 0/sin (Jm and vanishing conditions 0 (z 0, d) and Om > 0 : Monotone deformation : Deformation with a maximum in the liquid crysta 1 1 yer :

7 The 1018 The threshold conditions (7) and (8) are solved numerically for the cases : 1. Infinite anisotropic interfacial energy (practically Ws1 d 104 erg/cm) and finite interfacial energy WS2 for MBBA at room temperature (T 21 OC) for the following values of K13 0 (see Nehring and Saupe [4]) : 0; 1.6; 3.2; and 4.8 dyn and the following values of the total flexoelectric coefficient (elz + e3x) 10 4; 3 x 10 4 ; and 6 x 10 4 in cgs units (see W. Helfrich [12]). 2. For K13 0, WS2 d 10 8 ; 107; 10 6 ; 10 5 and 10 4 erg/cm, finite anisotropic interfacial energy WS1 and the same values of the total flexoelectric coefficient (e1z + e3x) 3. Case B, theory As 0. interaction between a thin homogeneously oriented nematic monocrystal with a negative dielectric anisotropy and flexoelectric properties (e lz :f:. 0, e 3x :f:. 0) and finite anisotropic elastic energy of interaction of the liquid crystal molecules with the cell walls WS1 and WS2 and a constant electric field E normal to the layer is considered theoretically in this section. After substituting the components of the vector field nx cos 0, nz sin 0 and E, E in the expression for the free energy (1) and minimizing the latter with respect to 0(z) we obtain the following differential equation, describing the deformations in the liquid crystal layer : and the following boundary conditions for d0/dz at the two surfaces : Only a monotone deformation, defined with the expressions from [9], is realized in the problem considered : The differential eq. (9) and the boundary condition (10) allow for WS1 WS2 the elementary solution 00 as well. Similarly to case A this solution will be with an energy lower than (11) if the inequalities : are valid.

8 1019 The solution (11) is the exact solution and fully covers the problem of flexoelectric deformation thus posed for case B. We shall however only consider the threshold characteristics showing the most favourable cases from a practical point of view for realizing polar flexoelectric deformations for this configuration. The complete tabulation of (11) will be given in a separate paper. In analogy to case A, after changing the variables sh A,,2 sin O/hsl,2 and the vanishing conditions OS1 and as2 0, we obtain the following important threshold transcendental equation for the nondimensional parameter Ut/ Uo ( Uo is the threshold for the fixed vanishing boundary condition for the homogeneous layer) : The threshold condition (12) is solved numerically for the following cases : infinite anisotropic interfacial elastic energy of the lower limiting surface (in practice WS1 d 10 4 erg/cm and finite anisotropic interfacial elastic energy WS2 for MBBA at room temperature (T 21 C) and the following values for K13 0 : 0 ; 1.6 ; 3.2 ; 4.8 dyn and the following values of the total flexoelectric coefficient (el + e3x) 104; 3 x 104 ; 6 x 10 4 in cgs units). Case A and B, As > 0 (Fig. 2). A constant electric field applied in the direction of initial orientation (along Z) of a homeotropic nematic layer with positive dielectric anisotropy stabilizes the liquid crystal with respect to As. The same field however destabilizes in a dielectric as well as in a flexoelectric sense the nematic layer homogeneously oriented along the X axis with a positive dielectric anisotropy. The threshold conditions for these two cases are simply obtained from ( 12), (7) and (8) after a substitution of!e, TE, K33, K11 and K13 for ie, J,E, K11, K33 and K13 respectively. The following relative difference in the thresholds is obtained from (7) and (8) (case A, Ae > 0 (for two possible directions of the electric field E) : substitution for the case WS1 oo and WS2 0) after this A numerical tabulation is carried out for formula (13). Here, in the relations between the dielectric and flexoelectric constants given by Helfrich [12] FIG. 2. Polar flexoelectric deformations in nematic layers with a positive dielectric anisotropy : upper cells (case A), lower cells (case B).

9 Frank s elastic constants K 11 and K33 are replaced by the elastic constants of Nehring and Saupe K l 1 and K33 and the sign for inequality is replaced by an equality. Since in nematic liquid crystals the positive dielectric anisotropy changes over a wide range, we have used the following values for s and 8.L covering in our opinion the dielectric and flexoelectric properties of a large number of nematic liquid crystals : 8.L 5, 8 20,15,10. The following data for the elastic coefficients were used for the tabulation : K 10 6 dyne, K ; + 3.2; ± 1.6 x 10 dyne. The numerical values are given on figure 9. In a similar manner for case B, Ae > 0 from (12) after a substitution of Ki 1, K13 and TE for K33, K13 and JE respectively, the threshold condition is obtained showing the electric field for which flexoelectric deformations will arise in this configuration. The calculations analogous to (13) from for WS1 oo, WS2 0 are also shown on figure A discussion of the influence of the physical quantities on the obtained theoretical threshold conditions..1 THE INFLUENCE OF ASYMMETRY IN THE SURFACE ANISOTROPIC ENERGY OF INTERACTION OF LIQUID CRYSTAL MOLECULES WITH THE BOUNDARY WALLS. In the problem under consideration surface energy plays a considerable part. In the consideration of most of the liquid crystal problems it is usually assumed that the surface energy is of considerable value (110 erg/cm ) which in practice assigns a fixed surface deformation angle which remains unchanged under the influence of applied external forces, deforming the liquid crystal even when the cells are very thin (with thickness of the order of 1 um). In the case under consideration considerable surface energy of interaction of the liquid crystal molecules with the walls should be assigned to one of the surfaces only. The sign of the total flexoelectric coefficient e1z + e3x shows which of the surface energies should be larger (e.g. Ws1) under the given direction of the applied electric field (JE or TE). Small surface energy (WS2 WS1) should be assigned to the other surface, which together with the flexoelectric properties of the liquid crystal determines the polarity of the effect. The investigation of the theoretical curves obtained (the most interesting are shown in figures 36 and figure 8) for MBBA leads to the following important conclusions regarding the practical application of this effect (for LC s with a small dielectric anisotropy) : 1. The value of the important parameter FIG. 3. Polar theoretical curves for a homeotropic MBBA layer. may be assumed for the lower limit of a rather weakly fixed nematic liquid crystal. 2. For the upper limit of a rather strongly fixed liquid crystal, one may assume the value FIG. 4. Polar theoretical curves for a homeotropic MBBA layer. 3. If the difference (in the pointed interval) between the two anisotropic surface energies is at least one order, one obtains very clear distinctions in the thresholds for the polar flexoelectric effect.

10 monotone as 1021 anisotropy and an applied electric field perpendicular to the layer : again UtTE and UIIE are the threshold voltages for the case WS1 00 and WS INFLUENCE OF THE FLEXOELECTRIC PROPERTIES FIG. 5. FIG. 6. Polar theoretical curves for a homeotropic MBBA layer. Polar theoretical curves for a homeotropic MBBA layer. 4. The threshold conditions obtained for a homeotropic nematic layer with a negative dielectric anisotropy and an applied electric field perpendicular to the layer make it possible to draw the following more important generalizations regarding the values of the anisotropic energies which might ensure a good polar flexoelectric effect (for WS1 > WSZ) : OF THE LIQUID CRYSTALS UNDER CONSIDERATION. The deforming moments due to the flexoelectric properties of a nematic liquid crystal may be in the bulk as well as the surface depending on the formulation of the problem, the type and the direction of the applied electric field interacting with the crystal. In the problem under consideration, the flexoelectric energy is simply added to or subtracted from the anisotropic surface energy. The balance between these two energies determines the type of deformation or with a maximum in the liquid crystal layer well as the polarity of the effect, i.e. the deformation and the threshold value of the applied voltages for a given direction of the electric field. The theoretical curves obtained demonstrate (case A, negative dielectric anisotropy, case B, positive dielectric anisotropy) that the larger value total flexoelectric coefficient assures better polarity of the effect (a larger relative change in the thresholds) for fixed values of the remaining physical quantities, lowering the requirements for the values of the surface energies and increasing the upper voltage threshold. The sign of this coefficient shows, for a given asymmetry in the surface energy (e.g. WS1 > Ws2), for which direction of the electric field the smaller deformation will appear and for which the larger one will (see Fig. 36 and 8). For the case A, As > 0 and B, As 0 flexoelectric deformations are obtained only when the total flexoelectric coefficient has a considerable value. In this case only the surface flexoelectric moments are deforming and a relation exists between the dielectric, flexoelectric and elastic coefficients of the liquid crystal which forbids the appearance of the flexoelectric deformation (see Fan [13]). For case A For case B U UE and UltE are the threshold voltages for WS1 oo and WS2 0. In a similar way, the following inequalities (with WS1 > Ws2) are given for the respective limitations on the values of the surface energies in the case of a homogeneous nematic layer with positive dielectric These are the only known relations between the elastic constants of Frank, of Nehring and Saupe and the flexoelectric coefficients of R. Meyer at present. The larger value total flexoelectric coefficient e1z + e3x determines a flexoelectric deformation even for a larger anisotropic surface energy and at the same

11 The Relative 1022 case considered ( WS1 oo, WS2 0), shown on figure 9 (the inequalities (14) show that the theoretical curves on figure 9 for real values of e lz + e3x must be translated in the direction of smaller As) that for nematic liquid crystals with As > 0, the relative change in polarity is reduced on a small scale with the increase in anisotropy (case B, As > 0). In case A, As > 0, the ratio UI,IUO is also reduced on a considerable larger scale, but with decrease in anisotropy. FIG. 7. angle 0 at the interface for K13 0, Ag 0 (case A) and K13 > 0, As > 0 (case B) and various values of WS2 d U*(elz + e3x) (WS1 (0). FIG. 9. change in polarity (case B, As > 0) and U, I E / UO (case A, As > 0). For case A, AE 0 and case B, As 0 the mirror images around the Z axis of the curves are valid E 11 + e,, 8, (The theoretical curves for real values of el, + e3x must be translated in the direction of smaller As.) FIG. 8. Polar threshold theoeretical curves for a homogeneous MBBA layer. time it gives the larger threshold above which these deformations would take place. 4.3 INFLUENCE OF THE DIELECTRIC ANISOTROPY. In all cases investigated the dielectric anisotropy worsens the polarity of the effect (the dielectric deforming or stabilizing moments are quadratic with respect to the applied electric field) and changes the voltage thresholds by lowering them (see W. Helfrich [2]). The dielectric anisotropy however plays a positive role as well since the lowering of the upper threshold lowers the operating voltage of this effect. Variation of this voltage in a wide range may therefore be accomplished with the change of Ag. The influence of the dielectric anisotropy on the participation of the flexoelectric properties in the polar effects is considerably more complicated. It is difficult to evaluate the influence of the various parameters in case the larger dielectric anisotropy leads to a larger difference in the flexoelectric coefficients as well. The conclusion can be drawn from the 8jj and 81 remain unchanged in a wide range for the widely investigated nematic liquid crystals with negative dielectric anisotropy and therefore the comparison between Helfrich s threshold (3 V) and our thresholds (calculated for MBBA) demonstrates to a large extent the influence of the dielectric anisotropy in these crystals on the polar effects under consideration. For liquid crystals of nematic type with large negative dielectric anisotropy the mirror images around the Z axis of the curves given on figure 9 are valid (81 > E E II El). 4.4 INFLUENCE OF SECONDORDER ELASTICITY. The presence of elastic energy of the second order brings about a change in the bulk deformation energy and a deformation surface energy strongly dependent on the type of deformation of the bulk liquid crystal layer. The theoretical results demonstrated that for anisotropic surface energies Ws1 oo and WS2 0. (Fig. 9) : a) For positive and negative values of K13 in case B, Ag > 0, the relative change in polarity is increased or

12 1023 lowered respectively. (For case A, A8 0 the reverse is true.) b) The negative values of K13 for case A, A8 > 0 (dielectrically stable liquid crystal) considerably change the ratio U, IEI Uo. For case B, A8 0 this is valid for the positive values of K13. The theoretical results also unexpectedly demonstrated (case A, A8 0, case B, A8 > 0) that for some Critical Values of the Anisotropic Elastic Surface Energy of interaction of the liquid crystal molecules with the walls (strongly dependent on the values of the total flexoelectric coefficient e1z + e3x) the deformations start with a jump. The case WS d (e 1 _, + e3x) U* is considered for a simple example (infinite anisotropic interfacial energy at the other wall). For these values of WS a monotone solution with a jump is obtained (K13 > 0, case B, A8 > 0) or a solution with a maximum in the liquid crystal layer also with a jump (K13 0, case A, A8 0). These results are shown on figure 7. They are tabulated for isotropic elasticity K 7.5 x 10 dyne, K x 10 dyn. It is evident from the figure that for some values of WS considerable deformations may be established in the cell for one direction of the applied electric field. For the reverse direction of the field the nematic layer is stable thus assuring a much better polar flexoelectric effect. Similar deformations may be obtained for values of over the threshold for > 0 in case A the electric field considerably As 0, Kl 3 or when K13 0,As > 0 for case B as well. These results are mathematically explained by the constraint on the derivative d8/dz at the walls (z d, 0) coming from the boundary conditions and the presence of surface flexoelectric energy. Physically this constraint means that for a smooth change in the electric field for Critical values of the surface energies, a smooth transition from a monotone deformation to a deformation with a maximum in the liquid crystal layer cannot take place. References [1] MEYER, R. B., Phys. Rev. Lett. 22 (1969) 918. [2] HELFRICH, W., Appl. Phys. Lett. 24 (1974) 451. [3] NEHRING, J., SAUPE, A., J. Chem. Phys. 54 (1971) 337. [4] NEHRING, J., SAUPE, A., J. Chem. Phys. 56 (1972) [5] DERZHANSKI, A. I., PETROV, A. G., VI Internat. LC s conf., Kent, U.S.A. (1976) Abstracts A2. [6] PETROV, A. G., VI Internat. Conf. Spectroscopy, Sunny Beach, Bulgaria (1974) Abstracts 248. [7] DERZHANSKI, A. I., PETROV, A. G., I Internat. LC s conf. of soc. countries, Halle, DDR (1976) Abstracts 78. [8] DERZHANSKI, A. I., HINOV, H. P., Phys. Lett. 56A (1976) 465. [9] BARRATT, P. J., JENKINS, J., J. Phys. A : Math. Nucl. Gen. 6 (1973) 756. [10] DAFERMOS, C. M., SIAM J. appl. Math. 16 (1968) [11] LESLIE, F. M., J. Phys. D : Appl. Phys. 3 (1970) 889. [12] HELFRICH, W., Mol. Cryst. Liq. Cryst. 26 (1974) 1. [13] FAN, C., Mol. Cryst. Liq. Cryst. 13 (1971) 9.