Flexoelectricity in Nematic and Smectic-A Liquid Crystals

Flexoelectric effects are observed in both the nematic and smectic-A phases of p-butoxybenzal- p-(J3-methylbutyl) aniline (BBMBA) and p-cyano-benzylidine-p-octyloxyaniline (CBOOA). This is the first reported observation of flexoelectricity in smectic phases. The use of a symmetric interdigital electrode in the homeotropic geometry facilitated the unambiguous separation of linear and quadratic electro-optic effects. Both the interdigital electrodes and those liquid-crystal deformations that are quadratic in the voltage act as optical diffraction gratings with a spacing that corresponds to the repeat distance d for adjacent electrodes. In contrast linear electro-optic effects giv'e rise to diffraction gratings with twice this spacing since adjacent electrodes have opposite voltages. Diffraction maxima due to the linear effects are halfway between the maxima due to the other effects. Using optical heterodyne detection, the intensity of the diffraction maxima believed to arise from the linear effect are indeed observed to be linear in the applied voltage V(w). With homodyne detection the diffracted intensity is proportional to V(w)'. Although previous discussions of flexoelectricity in nematics have been in terms of two flexoelectric coefficients ell and e33, we present theoretical arguments that as long as 'i7 X E = 0 there is only one true volume coefficient and that the other constant can always be included in surface effects. Our measurements of the volume coefficient f = ell + e33 are an order of magnitude larger than previously obtained values for ell and e33' Measured values of f are also nearly independent of temperature, in contrast to previous theoretical models, and of similar magnitude in the smectic and nematic phases. Measurements of flexoelectric signals versus the frequency of the driving voltage obtain relaxation times for splay like nematic fluctuations and undulation-type smectic fluctuations.


I. INTRODUCTION
Following Meyer's suggestion that there can be a linear coupling between nematic curvature and external electric fields, 1 there have been only a few attempts to demonstrate the effect. Experiments by Haas et al. 2 and also by Schmidt et al. 3 were interpreted in terms of the linear coupling mechanism of Meyer, but these interpretations are not universally accepted. Some indication of the current state of affairs is contained in De Gennes's discussion of flexoelectricity4 (the name he assigned to the linear effect). In his book on liquid crystals he reviews the theory and then remarks that the effect has "apparently been observed". The principal purpose of this paper is to describe an experimental technique that we believe unambiguously proves the existence of flexoelectric coupling in both the nematic and smectic phases of the liquid crystals P-butoxybenzal-p-(13methylbutyl) aniline (BBMBA) and pcyanobenzylidene-p-octyloxyaniline (CBOOA). Since the numerical values we obtain for the flexoelectric coupling parameter is only a factor of 10 larger than the numerical values obtained from previous experiments, the interpretation of these previous experiments in terms of flexoelectricity may very well be correct.
Aside from this prinCipal objective, there are a number of other points that we wish to make. First, concerning the phenomenological theory of flexoelectricity, we will argue in Sec. II that, although previous theories introduce two coefficients, there is only one coefficient appropriate to the bulk flexoelectric effect. The difference arises from the difficulty in separating surface from volume effects. Although it is conceivable that a set of experiments on different liquid crystals with different bounding surfaces might be capable of demonstrating experimentally that it is convenient to treat some surface effects in terms of an equivalent-volume free-energy denSity, that has not yet been demonstrated F=~B(VU)2 +~K(VVu) 2, where u represents the usual lattice displacement variables of a crystal. The constant B is the conventional fourth-rank elastic tensor and K is a sixth-order tensor that represents "higher-order" or Frank-Oseen elasticity. In crystalline materials the effects of K are completely obscured by the effects of B. If one neglects relaxation effects, this representation can be applied to nematics by taking B -0, director fluctuations On -Vu, and the K are the Frank constants. 5 Although the application to nematics when relaxation effects are included is somewhat subtler, it has been discussed by Forster et al. 6 and also by Martin,Parodi,and Pershan. 7 For smectics some of the elements of the fourth-rank tensor B vanish, but there are at least three nonvanishing elements for smectic-A phases.
For crystals that lack a center of inversion symmetry, the lowest-order coupling between electric fields and strains are the piezoelectric terms proportional to EVu where E is the electric field. Except for the recently observed chiral smectic phases, 8 all liquid-crystal phases possess an inversion center and the lowest-order linear coupling to the field E has the form of either EVVu or VEVu. These terms are also present in crystals but with nonvanishing first-order elastic constants B the electrically induced strains are proportional to B-1 V E and are small for the usual external fields that vary slowly on an atomic scale. In the nematic phase the only nonvanishing element of B corresponds to. the compressibility and induced strains On -Vu -(KvtlE can become large. Note that the factor V-I is symbolic. It indicates a nonlocal or integral relation between On and E. For nematics this effect was discussed first by Meyer. I Common usage is to call it "flexoelectric" to distinguish it from piezoelectric terms. 4 Reverting now to the conventional notation for nematics the flexoelectric terms are taken as (2.1) where the nematic symmetry axis is taken to be the z axis. In this representation there are two phenomenological flexoelectric coupling parameters ell and e 3 3' Equation (2. 1) is applicable to a linear theory in which one considers only small deviations of the director On from its average direction along z. It is important to appreciate the fact that densities of any kind are never uniquely defined. To put this another way, the energy density is defined in such a way that its integral over the sample volume obtains the correct total energy for the system when it is added to the separately defined surface energies. Thus any two expressions for the free-energy density that differ by a total differential can be considered to be equivalent if their respective surface energies are suitably defined. This argument can be applied to Eqo (2.1). For the case that the electric field E=-V¢, both terms in Eq. (2.1) can be manipulated into the form [On x aE,,/az + On:,> aE/az] plus surface terms. Aside from surface terms, Eq. (2.1) can thus be put in the form F f l, aE" "-~) 1i = + \On"az + Vf£y az ' (2.2) where f = e 11 + e33' The principal conclusion is that there is only one physically independent static flexoelectric coefficient in the nematic phase. As long as V x E = 0 the electric field can be derived from a potential and all forms obtainable from Eq. (2.2) are equivalent.
De Gennes discussed flexoelectricity in the smectic-A phase in terms of three coefficients el = e11> e 2 = e22 and e3 = e33' 9 Taking u as the layer displacement variable, if -V"u and -Vyu are substituted into Eq. (2.1) for On" and On Y ' respectively, one obtains two of De Gennes's three terms. The additional one has the form (a 2 u/az 2 )E z ' We will argue below that negligible errors result from neglect of the flexoelectric contribution to the displacement vector and taking V • eE = O.
With this approximation, it is straightforward to convert this term into the same form as Eq. (2.2). Thus, even in the smectic phase there is only one bulk flexoelectric coefficient f'" el1 + e 33 -e22 (EVE~) where the superscript E implies the low-frequency dielectric constant at zero strain.
Although it is possible that for either short wavelengths or very near the nematic-to-smectic phase transition the substitution might be inappropriate, we neglect complications of this type in the following analysis.
The magnitude of f might be crudely estimated from its dimensions of charge/unit length as one electronic charge (4.8 x 10-10 esu) divided by a molecular length of 20 A, that is, f=2.4xI0-3 esucm-I • This is considerably larger than an estimate obtained by dividing the expected dipole moment of typical nematic molecules by the square of the molecular length. This obtains numbers of the order off-lO-5 esucm-t that agree with predictions of both Helfrich l and Meyer. I The experimentally determined number that we obtain here is midway between these two values, of the order of 3 xl 0-4 esucm-I •

III. EXPERIMENTAL CONSIDERATIONS
The measurements to be described here consist of applying a voltage </>(w,qz) that is periodic in both time (angular frequency w) and one spatial direction (spatial frequency q,,) to an aligned liquid-crystal sample and measuring the light-scattering effects due to this voltage. The combination of Eq. (2. 2) and the conventional linear elastic theory for either nematic or smectic liquid crystals leads to the prediction that the optical frequency dielectric tensor will be modulated at the same spatial and temporal frequencies as the voltage. In contrast a second mechanism for coupling the electric potential to optical effects is through dielectric alignment. 4 This phenomena is quadratic in the applied voltage and leads to modulations in the optical dielectric tensor at (0, 2w, 0 and 2q,,). The two effects are easily separable in the geometry to be described below.
A periodic electrostatic potential is realized by means of the interdigital electrodes illustrated in on a glass slide. An Si0 2 layer was then put down on top of the electrodes. Application of a voltage V(w) to the oPPOsite electrodes produces a voltage ¢(w, q:xJ with 27Tq;1", 5 Jlm in the center of the pattern. Although the voltage is not really sinusoidal in space, the effects of this can be separated experimentally and will be discussed later. The scattering effects due to ¢(w,q:xJ are illustrated in Fig. 2 I I I I I I I I I I I I I I   I (a) unperturbed sample homeotropic geometry with V= 0; (b) dis tortion expected with a linear coupling (V '" 0); (c) distortion expected with a quadratic coupling (V '" 0); and (d) direction: O-incident beam, ± 1-light scattered from linear flexoelectric effect, ± 2-light scattered from the quadratic dielectric alignment effect. 2(b). Due to the optical anisotropy of the l~quid crystal, this results in an optical perturbation with wave vector qx that diffracts the incident wave into directions 6 =sin-1 (±q"A/27T). In contrast the quadratic, or dielectric alignment effects, illustrated in Fig. 2(c) produce an optical inhomogeneity that varies as 2qx and which diffracts light at 6=sin-1(±qxA/7T).
The metallic electrodes themselves diffract light, but if one masks off the "ends" of the electrodes, this light is concentrated at the diffraction maxima for the quadratic effect. Under these circumstances, the only light at 6=sin-1 (±q"A/27T) is due to the linear effect. On the other hand by allowing some light to diffract from the ends of the electrodes, it is possible to obtain a controlled amount of background light at the diffraction maxima of the linear effect and this can be used as a reference wave to obtain heterodyne detection. Higherorder diffraction from the metallic electrodes represent a slight complication to the quantitative interpretation of the results but, as we will discuss below, it is not serious.
In practice the incident light is not usually normal to the glass slides. Figure 3 illustrates the general geometric conditions. Maintaining the axis defined in Fig.  2 relative to the electrodes, the wave vector (k j ) of the incident laser beam is at an angle {3i with respect to the z axis. If k i is projected onto the plane of the electrodes, its projection J4 makes an angle O'i with the x axis. By virtue of the periodic electrodes there will be an optical inhomogeneity Oe(q) whose spatial Fourier components will only be nonzero for q in the xz plane. Thus there will be scattered light with wave vector k' that makes an angle rss with the z axis and whose prOjection on the xy plane ~ makes an angle as with the x axis. Since the scattering condition fixes ~=k!±q;, and since the scattering is elastic for all practical purposes, the magnitude of k S is fixed by the polarization of the scattered radiation and the laser frequency. Fixing both the magnitude of k' and its projection ~ allows just one value for qz=k~k!. Thus by varying either {3i or a i it is possible to probe the dependence of Oe on q" for fixed qx' Note that the only component of Oe to vary linearly with ¢ (w, q) is Oexz.
The actual experimental details are illustrated in Fig. 4. The incident wave is obtained from a Spectra Physics 133 He-Ne laser. The beam is focused onto the electrodes with a small solid angle such that the spread in wave vectors k i at the electrodes does not exceed the uncertainty in k i due to the finite beam size. The scattered light is detected by an RCA 1P28 photomultiplier that is masked by a pinhole in order to define the scattered wave vector kS. For homodyne detection of the flexoelectric signal, the ends of the electrodes are masked off and the only light detected at the appropriate angle is the light scattered by the flexoelectric effect. If the voltage is modulated at a frequency W/21T, the homodyne signal It (2w) is at twice this frequency. For heterodyne detection a small part of the edge of the electrode pattern is exposed to the incident light. This region scatters light into exactly the same direction as the flexoelectric effect and a signal It (w) proportional to ErefE.cat(w) cos¢ is observed. The angle ¢ is the optical phase difference between the reference light (E ref ) and the flexoelectric signal E.cat(w). Although the phase angle is not adjustable, it can be measured by comparing I 1 (w), I 1 (2w), and the intensity of the ref erence beam.
The photomultiplier output is fed into a PAR phase sensitive detector in which the reference signal is locked to either the voltage applied to the electrodes [to measure It(w)) or its second harmonic [to measure I 1 (2w)). The intensity of the direct or undeflected beam It is monitored by UDT 500 photodiode and the ratio 1/1 0 is obtained from a Data Precision 2540 A1 digital voltmeter in the ratio mode. The sample itself is in the hom eo tropic geometry and it is sandwiched between the top surface of the electrodes (Fig. 1) and a second glass FIG. 4. Block diagram of experimental apparatus. The laser beam is slightly focussed on the cell, through the lenses LI and L,., and polarized by P. The intensity in the zero order of diffraction is monitored with the photodiode PD. The pinhole PH allows the selection of kS, and the photocurrent given by the phototube PM is analyzed in the lock-in amplifier. The ratio of the two signals is obtained using a digital voltmeter as described in the text.
slide. The sample thickness is 500 J.Lm. The entire simple assembly is contained in a two-stage oven in which the temperature is controlled to a few m °C.
The first and most basic qualitative observation is the appearance of the expected intensity at the angles corresponding to diffraction by 21Tq;1 = 5 J.Lm. This signal was observed in both the nematic and smectic phases of BBMBA and CBOOA. Figures 5 and 6 show typical observations of the expected linear and quadratic voltage dependence of the heterodyne Signal at w and the homodyne signal at 2w. Detectable effects were observable for voltages as low as 10-3 V and linearity was confirmed over three decades. The qualitative observation that these signals vanish. for incident light with a i = 0 and polarization in the y direction confirm the origin of the scattering as due to  For q z = 0 the scattering condition demands k~ = k~ and thus k! = -k~. These equations can be combined to show that k S x ps", 0 or that for this geometry ps is parallel to k' and cannot radiate.
The small discrepancy between the condition qz=O and the vanishing of the signal is due to the fact that since ¢ is not strictly sinusoidal in x, the flexoelectric modulation of incident light that is first diffracted into higher orders by the metallic grating also contribute a small intensity at the observation point. We will show theoretically, however, that near q z == 0 the only signal linear in q z = k~ -k~ is the one due to flexoelectric modulation of the undiffracted beam. The linear dependence on qz thus provides an experimental method for eliminating effects due to the fact that the voltage is not strictly sinusoidal in x. A further confirmation of this would be the results for the smectic phase. Because of the simultaneous presence of both first-and second-order elasticity, the response function for OE is sharply peaked near q z = O. In the smectic phase these higherorder diffraction effects will thus be unimportant and in the smectic the intensities should vanish at q z == 0 to within experimental uncertainty, that is, to within a few mdeg of arc. Although this measurement could not be achieved with BBMBA, Fig. 8 illustrates this effect for CBOOA.
Because of this effect, the geometry with a i == 0 is not really optimum for studying the flexoelectric effect in the smectic phase. For nonvanishing values of a i incident light polarized as an ordinary wave has a component of E in the x direction proportional to sinai. The inhomogeneity DE"z produces a polarization in the z direction that radiates as an extraordinary wave in the direction of k S • Thus, in this geometry the flexoelectric effect produces depolarized scattering at a q z = k; -k~ that depends on {3i, ai, and the average dielectric tensor for the uniaxial material. By rotating the samples around the z axis (i. e., varying a i ), one sweeps out a range of q z. The peak intensity, corresponding to q z = 0, corresponds to scattered light being on the crescent described in Refs. 11 and 12. Figure Fig. 9 the halfwidth of the curve is determined by the sample thickness. For softer materials this technique would be a sensitive probe of K/B. 11,12 Note also that the striking features of the angular dependences for the observed signals could be understood in terms of general symmetry or geometric ideas and without any discussion as to whether the driving force was truly a bulk flexoelectric effect or a surface effect in which the bulk distortion was produced by a linear electric effect at the surface. We will elaborate on this point in Sec. IV, but the principal point is that the q z dependence does not really prove the existence of a volume flexoelectric effect.
The frequency dependence of 11 (w) at fixed temperatures and qz is shown in Fig. 10 for both the nematic and smectic phases of BBMBA. The most striking feature is the order of magnitude of the frequency of which the signal falls off. There is a change by a factor of 10 3 in going from the nematic to smectic phases. Deferring a detailed discussion to Sec. IV, we point out that in the nematic phase the relaxation frequency for a pure splay mode would be K lI q; (yf}-l,   the anisotropic elastic effects lead to a value given by 2, where Kl1 is the usual Frank constant and yt is an effective splay viscosity parameter. It is defined explicitly following Eq. (4.11). The quantity B is the smectic elastic constant for layer compression, II is a shear viscosity, and D is the sample thickness. Taking K I1 -10-6 dyn at 45 DC obtains a reasonable value of YI -0.7 P. 13 On the other hand, if II is taken to be 1 P, the deduced value of B -8 X 10 9 erg cm-3 is also reasonable. Equations enabling a quantitative study of the response function versus w will be obtained in Sec. IV. These will enable us to draw specific conclusions regarding the boundary conditions in the sample and to obtain a numerical value for f = ell + e 33 • Figure 11 illustrates the lOW-frequency dependence of the heterodyne Signal observed in the nematic phase of CBOOA. Below about 60 Hz the amplitude falls off in just the manner one would expect from a volume effect if the electric fields within the bulk were being screened by ionic impurities. The relaxation time for this effect can be estimated from the known magnitude of diffuSion constants [) for small dye molecules in MBBA. 14 The resultant [)qU2rr is just the correct magnitude to explain the effect. Similar results were also obtained for BBMBA except that since lower temperatures could be used, the relaxation time was approximately a factor of 10 larger, i. e., the fall off began about 3 or 4 Hz. If we assume that the bulk effect is being screened and the surface effects are not, we can conclude from the shape of the frequency dependence thatf?-10p, where p is the coefficient for surface polarization to be defined below [see Eq. (4.8)]. AU o sentially proportional to the elastic constants, 3 this result must be examined quite carefully. Both K11 and K33 have striking temperature dependences but according to this result f = e 11 + e33 varies only slightly.
The temperature dependence of 11 (w) is shown in Fig.  14 for the smectic phase of BBMBA. One possible uncertainty involves the correct assignment of the nematicsmectic phase transition temperature T NA' The T NA shown here was identified as that temperature at which one first observed the crescent pattern that is characteristic of static light scattering from smectic materials. Since this pattern is only expected in well-aligned samples for which deviations from perfect alignment are not too large, it is not obvious that this is the correct choice. Nevertheless, it does coincide with the temperature at which the characteristic nematic fluctuations disappear. The problem with this choice of TNA is that the temperature dependence of the flexoelectric signal does not seem to change at T NA • 'Rather, the significant decrease in the flexoelectric signal that is predicted to occur when Bqlq;2 »K11q~ (see Sec. IV) does not happen until T -T NA '" -O. 15°C. Since the nematicsmectic phase transition for BBMBA is first order, we do not believe this can be attributed to the temperature dependence of B. Although a systematic error in temperature could reconcile the two observations, we do not believe our temperature measurements are this bad. A more likely explanation would involve the temperature dependence of both boundary conditions and the ease of nucleating defects in the smectic structure. This difficulty aside, the principal observation is that for T -T NA ' :5 -O. 15°C the flexoelectric signal decreases very rapidly with decrease in temperature. AntiCipating the theoretical results in Sec. IV and further experimental results to be presented in Sec. V, this decrease is interpreted as an increase in the smectic elastic constant B with decreasing temperature.
Finally, we would like to describe an interesting effect that should be investigated in more detail. Figure  15   constant and increase the light scattering intensity. 17 For this interpretation the relaxation time for the "elastic dipole" density would have to be of the order of 10-4 sec, so that for frequencies above 10 4 Hz the elastic dipoles could not move and both aged and new samples would behave identically. This seems somewhat fast if the relaxation mechanism is the diffusion of impurities of the same size as the broken Schiff's base.
On the other hand, elastic dipole relaxation could also arise from either reorientation or other chemical effects. Alternative interpretations involving "elastic monopole" effects could also contribute to the increased low-frequency signal. 17 There is no point in further speculation since additional experimentation is clearly required.

IV. THEORETICAL DISCUSSION
There are three separate catculations that must be carried out in order to make quantitative use of the type of data that was illustrated in Sec. III. First, it is necessary to obtain a reasonable description of the electrostatic potential ¢(x, z) that results from the electrodes shown in Fig. 1. Second, we need to calculate the optical inhomogeneity OE that results from this potential if we assume the flexoelectric coupling described by Eq. (2.2). Finally, we must calculate the scattered light intensity due to the predicted OE.

A. Electrostatic potential due to interdigital
Since the electrodes sketched in Fig. 1 are of much greater extent in the y direction than their periodicity in the x direction, we approximate this problem by one of infinite extent along y. Also both the thickness of the metallic deposit and the covering Si0 2 layer are small compared to the periodicity and we take both of these to be zero. The over-all sample thickness is also larger than the periodicity and we assume the liquid crystal is of infinite extent along the + z direction. For the moment we will also neglect the dielectric anisotropy of the liquid crystal.
We now wish to solve for the potential ¢ satisfying \l2¢ = 0 and the following boundary conditions on the regions illustrated in Fig. 16 since the regions above and below the electrodes are not equivalent dielectrics. The solution to this problem is a straightforward exercise in conformal mapping. 18 Using the transformations ~=x+iz and ~'=f2sin(7T~/d) followed by e' =F(l/f2, ~), the region in Fig. 16(a) is mapped into the interior of the rectangular illustrated in Fig. 16 (4.2b) o The electric fields are easily obtained in terms of dQ/ d~ using the Cauchy relations The electric fields is expected to be periodic in x = Re~ with a period of 2d. This follows from Eq. (4.4) Since Ex -0 as z -00, the potential ¢ is directly obtained from ¢ = -f: dz E z • Another interesting quantity in relation to the quadratic dielectric coupling is "=0 x sin[27T(2n + 1) xd-1 ]. (4.5) Although the above was derived after explicitly neglecting the dielectric anisotropy of the liquid crystal, that is easily corrected. In place of V2 ¢ = 0 one wants the solution E~ a 2 ¢/ax 2 +E~ a 2 ¢/az 2 ==:0 where the superscript distinguishes the low-frequency dielectric tensor EO from the optical-frequency tensor E. The correct solution is then obtained by substituting z (EVE~)1I2 for z in the above equations. To be completely rigorous, the effect of including the linear flexoelectric effect on the displacement vector D must also be included. That is, V oD is not simply equal to -V • (E • V ¢) with a constant E. In principle such effects could be included in the definition of E, however the dielectric tensor is often tacitly taken to represent the relation between D and E when on ==: O.  assumes a boundary in which the minimum energy is with the director normal to the surfaces. The term F~ reflects the fact that there could be a surface polarization that would couple directly to the external field. Since there is no physical way to separate that terms that arise from surface interactions from terms that are left over after Eq. (2.1) is converted into Eq. (2.2) by partial integration, we define a single phenomenological parameter p that includes both. The term F; thus has the form (4.8) The tacit assumption that the upper and lower surfaces are identical has been made. Minimizing the total free energy with respect to On at constant E obtains the volume equation  In practice the experiments have been done with voltages that were Sinusoidal functions of time. The same equations hold except that q~ is replaced by q~ =q~(1 + jww(jl), where Wo = K l1 q;(yf)-l and yj is an effective viscosity parameter that describes the relaxation behavior of the nematic splay mode. Using the notation of Forster et al. 6 (yj)-l = yjl + t(1 _ A)2V;1.

Smectic phase
The situation in the smectic phase is not significantly different from the nematic. In place of Eq. (4.6) the bulk contribution to the free energy is (4.12) Equation (4.12) is only applicable in the hydrodynamic approximation in which the director variation I'mx is uniquely related to the layer displacement -"xu. Elsewhere in this paper we show data near the nematicsmectic phase transition where this may not be strictly true. However, we make this approximation here. In a similar vein we assume the boundary conditions can be given in terms of the layer displacement and take the surface energies as The solution to the time-dependent problem is trickier in the smectic case because at a finite ratio of q / qx the overdamped undulation or splay like mode couples to the shear velocity so as to form an underdamped 2308 J. Appl. Phys., Vol. 47, No.6, June 1976 propagating shear mode. The conditions under which this actually happens depend on the relative magnitude of a number of different variables and we will not go into that here. Rather, we confine ourselves to the case that q z is small and assume a single overdamped nematiclike mode. In this case 19 the time-dependent problem is approximately solved by the substitution q~ =q~(1 +jw/wo) where wo"'K l1 qYV 3 , where V3 is a shear viscosity in the notation of Martin et al. 7
For each value of qx the amplitude of the scattered E field will be shown below to depend on q z = k~ -k~ according to I'm(q z) = J f I'm(z) exp(iq.z) dz. Since the general expressions for on (q z) are lengthy, we will not write them out, but rather concentrate on specifiC examples.
First, observe that KD -5 X 10 2 and since hf -exp(-KD), there is negligible error in taking h~ -O. Second, in the nematic phase qe -q", -10 4 cm-1 and since A can be estimated 20 to be of the order of 10-4 erg cm-2 or less, we obtain q s -10 2 cm -1. The result is that Iq", ID» 1 and also Iq", I »qs. Furthermore, if we restrict experimental observations to small qz=k~-k~ such that K l1 q; » K 33 q;, (4. 17a) where [E",(q,,)]z=o is obtained from Eqs.  The prediction is that I'm(q z) is relatively insensitive to q z in the nematic phase and its frequency dependence is essentially Lorentzian. It is also worth mentioning that, although the above numerical estimates result in a "free boundary condition" in which A is neglected, these same values predict a "rigid boundary" in typical Frederickz geometries. Alternatively, a larger value for A in which qs» q '" obtains (4.18) As in the other example, Eq. (4.18) predicts that I'm(q,) is insensitive to q z == k! -k~ so that the angular dependence of the scattering cannot be used to distinguish between the two cases. The predicted frequency dependence, however, is sufficiently different to distinguish between the two. Figure 10(a) described typical Lorentzian behavior in the nematic phase and we take this to confirm q s« qx in the present geometry. Anticipating the results for the predicted relation between observed light scattering and on(qz)' the measurements described above should obtain a measure of {f +p)Kil in the nematic phase.
In the smectic phase the magnitudes are somewhat different. First, qe == (B-IK)1I2q~ -100 or 10 1 cm-I • Second, the surface energy [Eq. (4.13)] involves constants A that are either as large as the elastic constants of the cell walls or otherwise involve severe distortions of the smectic ordering near the walls. According to the definition, values of qs= B-IA ?;qx"" 10 4 _10 5 cm-I are not unreasonable. Thus, the smectic phase satisfies "rigid boundary conditions" in which qs» qe. Assuming that q s is also large compared with qx In the limit that w -0 and q z -0 this simplifies to

OO(qz)"" (2BtY(EVEY)" 2 Dqx[E;,:(q x J]z=o.
(4.20) The most striking difference between the results for the smectic and nematic phases is that OO(q z) is very sensitive to q. in the smectic phase, while it is relatively insensitive to the nematic. This arises from the extreme anisotropy of the smectic elastiCity. In practice the q z dependence of Eq. (4.19) is sensitive to sample thickness and typical half-widths Ilq z are of the order of rr/D.

c. Light scattering amplitudes
Although many authors have considered light scattering phenomena from both anisotropic media and dielectric slabs, we have not found explicit formulas for the examples discussed in this paper and we have had to derive the results ourselves. The methods are straightforward and in lieu of a step-by-step derivation we will just outline the procedure.
Using the results of Secs. I1-IVB we assume an optical inhomogeneity OE", linearly related to 00. For a given incident optical electric field, there is an inhomogeneous optical polarization oP = (471)-1&. consistent with the appropriate boundary conditions on the anisotropic dielectric slab. The solutions are approximate only in the sense that they do not include the effects of either reflected or multiple scattered beams in the fields that give rise to the OP term. Multiple scattering effects would not be linear in OE and their neglect is not serious. Adequate formulas to include effects due to multiple reflections of the incident laser beam would result in small corrections. Similarly multiple reflection corrections for the scattered radiation are also neglected. The present problem is simplified by the conservation of the wave vector in the xy plane. The general solution for the E and M wave outside the dielectric slab has both a fixed k; = kf, + ~ and I k' I .
These in turn fix k~. Thus we assume that for each polarization there is one plane wave for arbitrary amplitude and phase propagating away from the dielectric slab above the slab and a second one below. Inside the slab we assume the E and M wave is the superposition of one plane wave with fixed kt and k~ > 0 and a second with k~ < 0 that both satisfy the homogeneous Maxwell's equations plus an "inhomogeneous" solution.
The inhomogeneous solution is facilitated by expressing oP(z) as a Fourier series in exp[±i(n71z/D)]. The only subtle point is to ensure that the requirement V· E + 4710P) = 0 is satisfied for the inhomogeneous solution.
This, however, is a standard problem in nonlinear optics. 21 Together with application of the usual E and M boundary conditions, this procedure reduces the problem to a linear 4 x 4 matrix problem in which the complex amplitudes of the four optical electric fields are determined. Recalling the tacit assumption that OO(x) is both periodic and of infinite extent along x, the quanti ty 00 (q z) appearing in the following results for the amplitude of the scattered E fields is defined such that for specific k~ -kf, = qxx, OO(q z) is the coefficient of exp(+iqxx) in the expression for ffn(x,z) exp(-iqzZ)dz. This is consistent with a sign convention that takes the E fields to be the real part of waves exp[i (wt -k· r)].

D. Incident beam-ordinary polarization
The scattered wave is entirely extraordinary with amplitude where k ext is the wave vector for the scattered light outside the dielectric, k O == 271Aol, and Ei is the incident optical electric field. Geometric considerations obtain /2 , (4.23) where k~=EJ.k~ and k~=E"k~. The results previously obtained indicate that large-amplitude will be observed in the smectic phase only near qz=O. This obtains (4.24) The most convenient experimental geometry was a i "'h or qx"'k!(EIIE~I_1)1I2. This maximized the scattered intensity and kept the algebra as simple as possible. Variation of 0' i at constant (3i also varies q z and for small angles; " ( I ) [k2(ki)-2 1]-112' I" i vqz= E,l/EIl qx; ,l p - SInO' va . (4.25) Typical numerical values obtain Oqjqz'" 10-2 for 00' '" O. 05 rad and demonstrate that the values of qz can be set with relatively high precision.

E. Incident beam with extraordinary polarization
Scattered light is observed in both ordinary and extraordinary polarizations. The result for the ordinary polarization (i. e., the depolarized light scattering) is (4.27) In our samples Ell> El and the condition for q z '" 0 is only satisfied if cosa l < 0 or 0'1 > ~7T. The maximum intensity obtained from Eq. (4.26) is smaller than obtainable from Eq. (4.22) because in the present case 0' =~7T cannot be obtained. Although we did not make extensive measurements to verify the equations, typical intensities were found smaller for this geometry as compared to the intensity in the geometry appropriate to Eq. (4.22). Note also that both Eqs. (4.22) and (4.26) are proportional to sinal. The qualitative effects of this factor are easily observed in either geometry.
With both incident and scattered light of extraordinary polarization, the theoretical result for the scattered amplitude is E S = ikoEl(k; + Elk~xttIAEOn(qz)Ei x (k;sin{31 cosy + k; cos{31 cosa l ) x (El sin 2 {31 + ETI cos 2 (31 t 1l2 , With a small amount of algebra, Eq. (4.28) can be shown to have the form E S -(qz/qx)1in. The intensity ratio of polarized to depolarized scattering when the incident light is extraordinarily polarized is roughly (q j q x)2. In the smectic phase on(q z) is sharply peaked at q z -0 and the two effects combine such that polarized scattering is considerably weaker than depolarized. Typically, scattering can be observed at small nonzero values of qz -7T ID, and for D -500 J.lm and 27Tq;l", 5 J.lm the ratio of polarized to depolarized intensities is of the order of 10-4 • Polarized scattering is clearly not important in the smectic phase.
In the nematic phase this geometry does have practi-cal application. For one thing qz and qx; can be of comparable magnitude so the intensity ratio is not as unfavorable for polarized scattering as in the smectic. Second, as we mentioned earlier, the experimental electrodes do not provide voltages ¢(x,z) Fig. 7 for the nematic phase and in Fig. 8 for the smectic. Note that in the smectic 11/'''' 0 as expected. In order to evaluate the experimental ratio flKl1 in the nematic phase, we used this effect.

A. Nematic phase
One result of the theoretical analysis in Sec. IV was the recognition that the q z dependence of the flexoelectric signal that was displayed in Fig. 7 can be used to separate higher-order diffraction effects from the principle first-order effect. The relevant theoretical formula is Eq. (4.28). In the special case that at =7T and as = 0, with the angles {3i and {3s both small [of magnitude qx(2k,lt 1 or smaller], and both waves polarized as extraordinary waves, several approximations are possible. The most subtle involves k; sin{31 cosy + k~ cos{3i cosa l which is rigorously equal to I k" I sin({31 -(3S) when cosy = 1 and cosa i =-1. For small angles and near to the condition qz=O, qz=k;-k~'" 2-1(EJEII)lkll[({31)2-({3S)2] and since qx; '" I k" I ({3s + (3i), these can be combined to ob- For experimental purposes it is more convenient to express this result in terms of the value for {3i outside the material ({3i)ext", (E,l)1/2{3t and since o{3i must equal -o{3s if qx; is fixed, we can write (o{3i)ext '" +EIIE~1/2(qjqx;) and In particular with K11 = 1. 9 X 10-6 dyn, the de-ducedvalueofj+p=2.5X10-4 esucm-1 . Further, if we assume j» P as previously argued on the basis of the screening effects observed below a few Hz, this is the experimental value of j. Both previous experimental results for MBBA3 and theoretical estimates of e11 and e33 1,10 obtained similar magnitudes. The uncertainty in the present numerical result could be as large as 40-50% if one includes all of the independent possibilities for error. Note also that the maxima tilt angle I:in(x,z) in these experiments was only of the order of 2° and this should be well within the range in which the linear theory is applicable. For CBOOA the magnitude of j was also of the order of 10-4 esu cm-I .
We can also give a tentative assignment to the sign of j + P for BBMBA. For the geometry appropriate to Fig.   7, the electromagnetic field scattered by the flexoelectric effect can be assigned a phase from Eq. (5.3). Note that the phase appearing in Eq. (5.3) is dependent on (1) the convention in which E is given in terms of the real part of complex waves that propagate as exp [i (wt -k 'r) and (2) the origin of the x axis relative to the electrode position. For example, Fig. 16 specifies the origin of the x axis as coinciding with the center of the space between the two electrodes. For a positive voltage V, the amplitude and phase of the fundamental components in the Fourier series given by Eqs. (4.4) are fixed. A shift of origin from between the electrodes to the center of an electrode would convert the cosine series for Ex into a sine series and the coefficient of  in that series would differ from the coefficient in the cosine series by a factor of i. The phase factors in Eq. (5.1) result from these conventions as well as the physical effect of flexoelectric phase modulation of the incident light.
As discussed above, heterodyne detection of the lowest-order flexoelectric signal is possible because near the ends the metallic electrode pattern has the same periodicity as the flexoelectric deformation. Although the flexoelectric deformation produces a phase modulation, the scattering due to the opaque electrooes is the result of spatially amplitude modulating the incident light. If the symmetries of the spatial amplitude and phase modulation relative to origin of the x axis were identical, the two signals would be i1T out of phase and heterodyning would not occur. In fact the symmetries are not the same. The phase modulation is symmetric about the space between electrodes, while the amplitude modulation is symmetric about the electrode center. Flexoelectric phase modulation is described by a cosine series in x with imaginary amplitudes, while the spatial amplitude modulation from the ends of the electrodes is described by a sine series in x with real amplitudes.
For a particular scattering direction the coefficients of exp(iqxX) are both purely imaginary.
Of particular interest is the different symmetries from the ends of the electrode at y > 0 and y < O. The transmission functions from the two ends are identical except that one is shifted by d relative to the other. For qx = 1T/ d this means that the phase of the heterodyne signal relative to the voltage V will differ by 1T depending on whether the reference signal is taken from the end of the electrodes with y > 0 or y < O. This effect has been observed. The optical phase angle rp predicted by this calculation is either 0 or 1T. In practice, the observed angle in the nematic phase typically satisfied I cosrp I '" O. 7. Further, by comparing the relative phase of the heterodyne signal and applied voltage with the phase predicted by the above analysis, we conclude thatj+p > O. Since we previously argued j» P on the basis of the low-frequency effects, this impliesj> O.

B. Smectic phase
For T« TNA the smectic flexoelectric signal was too small to be studied quantitatively by homodyne detection and we had to rely exclusively on the heterodyne method. The experimental geometry used 011 = 90° and OI s = 0° and we take (5.4) The optical phase angle rp was chosen to satisfy I cosrp I = 0.7. This value was conSistently found over a range of temperatures and angles {31, 011 whenever both heterodyne and homodyne signals could be measured simultaneously. At room temperature, with {3i =0. 24, 011 = t1T rad, and V = 2 V, we obtained a value of I E S I I EI I-I -3.5x10-6 • UsingEq. (4.22), the following numerical result is obtained: I:in (q z) '" 2. 6 x 10-11 cm for q z '" O. This number can be inserted into Eq. (4.20) and with [Ex (qx) ]z=o = -2-1 1TK-I Vd-1 [note that EAqJ is defined to be the coefficient of  in the series defined in Eqs. (4.4)], one obtainsj/B"'3xl0~15 esucm 2 erg-l. If we take B-l0 10 ergcm-3 , we obtainj-3. 5X 10-4 esu cm-1 in agreement with the value obtained for the nematic phase. Although this number is less precise than the nematic value because of the extra uncertainty associated with the optical heterodyne effiCiency (i. e., the value of cosrp) and the fact that the range of possible values for B is greater than for K, it is quite clear that its magnitude is not significantly different from the nematic value. Assuming the director tilt is simply related to the layer displacement, the above value of I:in(qz) corresponds to a layer displacement of the order of l:in(qz)(DqJ-I -2X10-6 A.

VI. SUMMARY
The most important result of this paper is the unambiguous proof of the existence of flexoelectricity as a bulk phenomena in both nematic and smectic liquid crystals. Although earlier experiments were consistent with bulk flexoelectricity, the nature of these experiments was such that alternative interpretations could not be discounted. In particular, Helfrich's experiment 3 could have been interpreted as a pure surface effect.
A second result is the demonstration that the flexoelectric coefficient j = el1 + e33 does not depend on temperature in the manner that has previously been suggested on theoretical grounds. In fact, f is essentially the same in the nematic and smectic phases of BBMBA. This is particularly striking in that it implies significant flexoelectric effects in regions of large smectic curvature like focal conic lines and small smectic vesicles. De Gennes makes the point in his work on liquid crystals that, to the extent flexoelectricity is a strong effect, the conventional local elastic theories are not correct. Of course for most macroscopic elastic experiments, flexoelectric effects are probably weak enough that they can be forgotten and both the Frank theory of nematics and the conventional elastic theories of smectics are valid. There are, however, regions of large curvature near the core of disclinations and smectic focal conic lines where flexoelectric effects may not be ignored. For examp Ie, consider a disclination where the director n = (xx +yy)(x 2 +y2r1l2. Assuming there is no free charge, one can demonstrate the existence of an electric field E -f(x 2 + y2)-1 /2. Taking f-1O-4 esu/cm and a radius (x 2 +y2)+112_10-6 cm obtains an electric field of 10 esu/ cm 2 = 3 x 10 4 V / cm. It is not clear that no free charge and thus V 0 D = 0 is the most realistic condition and if one allows charges to build up, the material near the disclination obviously will have a different impurity content than elsewhere in the sample. Since compOSitional inhomogeneity can have Significant elastic effects, the effect of flexoelectricity could significantly modify calculations that attempt an estimation of the "core" energy of defects.
Helfrich has also pointed out the role that curvature elasticity can have on the shape of red blood cell erthrocytes. 22,23 In view of the fact that the flexoelectric coefficient in smectic BBMBA is of the same magnitude as in nematic BBMBA, we are led to ask if the magnitude of the flexoelectric coefficient in lyotropic smectics (i. e., lamina phases of lipids) might not lead us to a flexoelectric origin for red blood cell curvature. Experiments exist that demonstrate that the outer and inner surfaces of erthrocytes have different biochemical properties. 24 Other experiments have also demonstrated that charged ions like Ca++ have specific effects on the shape of erthrocytes and thus the curvature of red blood cells membranes. [25][26][27][28] Although there is no clear evidence that flexoelectric phenomena play a Significant role in these effects, the symmetry in these problems is such that it could. We believe experiments on flexoelectricity in lamina lipid phases should be undertaken in order to evaluate the flexoelectric magnitudes.

ACKNOWLEDGMENTS
One of the authors (Jp) acknowledges the friendly atmosphere he has found during his stay at Harvard University. It is also a pleasure to mention the gracious-