Flexoelectric distortion of developable domains in a tubular discotic liquid crystal

نویسندگان

  • M.B.L. Santos
  • G. Durand
چکیده

2014 We compute the flexoelectric distortion of a developable domain in a tubular discotic liquid crystal phase, submitted to a D.C. electric field. The strong radial polarization, related to the permanent bend of the tubes, does not contribute to the texture distortion, by symmetry. The tubes undergo simply a divergence distortion, the tube circles being shifted from one to another, towards the field. The effect is observed in hexapentoxytriphenylene, close to the transition temperature to the isotropic phase. Interference fringes in polarized light shift away from and towards the core of the developable domain, under the action of a 1 Hz electric field. The ratio of flexo to elastic constants (for the 2D hexagonal tube crystal) is of order 10-6 cgs. The flexo divergence constant is negative, in contrast to classical rod-like liquid crystals. J. Physique LETTRES 44 (1983) L-195 L-200 ler MARS 1983, Classification Physics Abstracts 61.30 77.90 Discotic tubular liquid crystals are often constituted of a 2D hexagonal packing of uncorrelated molecular tubes [ 1 ]. These tubes bend easily into quasi cylindrical developable domains [2]. Such a texture of permanent bend should possess a volume flexoelectric polarization [3] as would any distorted liquid crystalline texture. In principle, this polarization should couple linearly with an externally applied electric field, resulting in a visible mechanical distortion of the developable domains. In this letter, we present the first observation of this new electrooptical effect in a tubular discotic liquid crystal phase. (*) On leave from Universidade Federal de Minas Gerais, Dept de Fisica, 30000 Belo Horizonte MG, Brazil. Partially supported by CNPq Brazil. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:01983004405019500 L-196 JOURNAL DE PHYSIQUE LETTRES Fig. 1. Cross section of a circular developable domain. Full lines represent circular tubes of radius p. The assumed boundary with the isotrope melt is the circle of radius R. The tube lines bend around a rigid core of radius a. Thin lines represent the distorted tube lines in presence of the field E. e : semi-transparent electrodes. Let us first estimate the mechanical distortion of a flexoelectric developable domain in the presence of an electric field E. These domains grow usually from a flat boundary plate (Fig. 1), around a straight cylindrical core of typical radius a 1 Lm, often fixed on the boundary plate [4]. Close to the transition temperature T~ to the isotropic phase, a small vertical temperature gradient allows the developable domain to be in contact with the isotropic melt Here, we make the assumption that the tubes are simple circles of radius p around the core. This approximation is valid as long as p is larger than the small core radius a. We call P the flexoelectric polarization. Under the action of E, the electric torque density P x E locally rotates the tubes, through a small angle ~ (Fig. 1), resulting in a normal tube displacement u(p, 0) ~ p, where 0 is the polar angle. One has obviously : We assume u to be zero on the boundary plate, for 0 = 0 or 9 = 7r and on the assumed rigid core (u(a) 0). The non-uniformity of u creates an elastic reaction of the 2D hexagonal tube packing, described by a term : 2 B(au/ap)Z in the free energy density. B is an elastic constant to be determined. Associated with this lateral tube compression, there should be also a longitudinal tube extension. However, for a small distortion u, the tube extension is quadratic in u, resulting in a fourth-order term in the free energy, which can be neglected. We also neglect the change of the nematic-like curvature energy associated with the distortion u. The nematic curvature I U 2 energy is of the order K + 2 . K is a Frank elastic constant K ~ j8~, where m is a molep p cular dimension. As one has up > m2, the tube compression energy dominates over curvature energy. This last condition is not incompatible with the small distortion condition M/p ~ 1, since m2/a2 is typically of the order of 106. We must now write the expression for P. In analogy with smectic materials [5] P contains two kinds of terms. There should be first a polarization 2 normal to the tubes, proportional to the tube compression Ps = ’2 ~"2’’ There should be also the usual nematic-like terms PN = el n(div n) + e3(curl n) x n, where n is the local unitary vector along the tube line. In the presence of the field, the free energy density can be written as : L-197 FLEXOELECTRICITY IN TUBULAR DISCOTICS We look for a distortion u( p, 0) which minimizes the total free energy of the sample. We must write the electric term with its explicit dependence on u. In the absence of field, the only component of P is PN ~3/p along the circle radius. In the presence of the field, and to lowest order in ulp, it gives to f a contribution : e3 2013( 2013 2013~) cos 0 because of the tube rotation = 2013 2013~. P P 00 P 00 At the same time the local curvature chan es b the uantit a~ _ _ 1 a 2 u resultin in e, changes by quantity 2013~2013 = 2013 "r ~7, resulting Y ~9 C’(7 p2 ( t7 a term . e3 E a2u S in 0 in f These two terms can be integrated exactly over the sample volume rm : 2013 20132013 2013~ s . P OC7 and give no contribution to the equilibrium equation, because of the symmetry of the distortion for 0 = 0 and 0 = 7c. On the other hand, the divergence term PN gives a non vanishing contriel a2u bution 2013 -!a ae E cos 0, linear in u. The smectic-like term Ps gives a contribution to f a2u P P U2lp3 e2 20132013 sin (0 + ~) of order negligible compared to the divergence contribution. Finally, p to lowest order in M/p, the equilibrium equation for u is : with the u = 0 condition on the solid boundaries, and the free force condition ~u/ap 0 on the free surface. For a circular cylindrical developable domain ofradius R, the solution for u is : u is expected to be linear in E, corresponding to a dilation or a compression according to the sign of e 1. The tube lines remain quasi circular with the same radius p but with their centres displaced toward the field E by a quantity u(p). To observe the flexoelectric distortion of developable domains, we use the optical birefringence technique already described [6]. Using suitably polarized light, propagating perpendicularly to the sample, one can observe birefringence fringes parallel to the core, due to the angular change of the extraordinary refractive index related to the circular bend of the tubes. The fringes lateral position xk is defined by a retardation condition : where k is an integer, Å. is the wavelength of light, and Ymax defines the free boundary profile of the developable domain. ne and n. are the extraordinary and ordinary indices of refraction of the discotic texture. In the absence of flexoelectric distortion, ~(9) was a function only of the tube tilt angle 0 = tan-1 (y/x) where x is the lateral distance of the light ray to the central core (see Fig. 1). In the presence of a flexoelectric distortion, the optical retardation L(x, E) depends in addition on the density and thickness changes induced by the tube compression. It is reasonable to assume the linear dependence ~n/n = au/a p for the two indices ne and no. This represents the additivity of optical polarizabilities. The circular profile Y max = p sin 9 is displaced by a quantity Ay = u(R) sin2 0. For a light ray travelling through the core (x = 0), there is no tilt effect on 40, E). As the total optical polarizability is conserved, the compression induced change of refractive index is exactly compensated by the displacement u(R) of the free boundary profile, and 40, E) is unchanged. On the other hand, far from the central core, (x ~ R), the L-198 JOURNAL DE PHYSIQUE LETTRES com ression effect is ne li ible a’~ R 0 . There should be a dis lacement of the intercompression negligible {~2013 (~) = 0 ). displacement ( P ) ference fringes related mostly to the tilt ~ of the tubes. To estimate this effect, we note that the birefringence An = no ne is weak [6, 7]. We can write locally : A~(0) ~ An sin2 0. The variation of the optical retardation is : where the variation of birefringence due to the tilt ~ is : using the expression already calculated for u(p, 0), we obtain : where Omax is the polar angle of the point of abscissa x on the free edge profile. In practice, under the action of the field, the interference fringes should move towards or away from the core, to compensate for aL(x, E). To observe this effect, we prepared developable domains of hexapentoxytriphenylene, which has an isotropic hexagonal discotic phase at Tc = 122 ~C. Our experimental set up has already been described [6]. We open the top of our temperature controlled oven, so that a vertical temperature gradient allows for the domains to grow from the upper plate, and the lower free edge boundary to be in contact with the isotropic melt. We use Nesa-coated semi-transparent electrodes to apply a low frequency electric field across the sample, of thickness d = 50 ~.m. The lower electrode is grounded. As previously explained, we observe the birefringence fringes under a polarizing microscope, with white light. In the absence of a field, we measure the positions Xk of the fringes. From these static data, we can deduce [6] the mean birefringence [ An [ 0.13 and the local thickness [8] of the discotic phase 7! ~ 30 ~.m. We now apply across the sample a square wave electric signal at 1 Hz, with alternate amplitudes ± 100 V. We do observe a lateral shift of the fringes synchronous with the applied field. The shift is outward, for positive (downward) field, and inward for negative Fig. 2. Side-by-side microphotographies of a developable domain; interference polarized white light fringe pattern under the action of an electric field E = V/d. 2a : V 0. 2b : V > 0. The core of the developable domain is the central straight white line. Scale : 5 J.1m per division. The fringe shift allows the estimate of the flexoelectric distortion of the developable domain. L-199 FLEXOELECTRICITY IN TUBULAR DISCOTICS one. The picture shows two side by side photographs of the fringes, corresponding to the two situations. This synchronous distortion is proof of a linear flexoelectric coupling with E. All other effects, for instance dielectric coupling, hydrodynamic flow from space charges, a.s.o., would be quadratic in E, i.e., would appear at D.C. or at twice the applied frequency. The effect is readily observable when the mean sample temperature is adjusted to a few degrees below T~. On cooling down, the developable domains grow over the whole thickness, the discotic phase becomes more rigid and the effect disappears. Close to Tc, we observed that the fringe shift is an increasing function of the field. For a ± 100 V voltage we can estimate from the pictures the amplitude of the field induced distortion, writing A~/~ ~ AL/L 0.13 for the third fringe (k = 3, L = 3 ~ 1.5 ~m for x = 27 ~m). This results in BIJ el 1 2.2 x 106 cgs. With the estimate I el I 2.2 x 10-4 cgs, this gives a very small B 102 cgs. This extremely small value of B, compared to a typical value of 10’ cgs for standard smectics, is obviously related to the temperature being very close to 7~. On decreasing the frequency, the fringe shift is always visible as a transient effect followed by a quasi total relaxation. We understand this effect as a plastic relaxation, due to the motion of defects. In fact, the domain shape changes irreversibly under the action of a D.C. field. It is probably because of the existence of a large density of defects (tube dislocations for instance) that the discotic sample appears so soft. Increasing the frequency, the effect remains visible up to 10 Hz, above which the eye cannot follow the fringe shift. We are now studying in more details the high frequency behaviour. More interesting is the sign of the flexo coefficient el, which is negative, i.e. opposite to the one found in usual liquid crystals made of rod-like instead of disc-like molecules. This result must be compared with the corresponding change of sign of the optical birefringence, ne no > 0 for rod-like molecules liquid crystals and ne no 0 for discotic phases [9]. The sign of el , mostly related to the electric quadrupolar moment of the molecules [10], seems then connected to the molecular symmetry of the mesophase. To conclude, we have observed the flexoelectric coupling of developable domains of a discotic material with a low frequency electric field. The field induced distortion is large enough, close to T~, to result in a visible birefringence shift. The ratio of the flexo to the tube transverse elastic compression constant is found to be anomalously large (e, /B ~ 106 cgs, close to Tc)’ probably because of the existence of a large density of defects. The sign of e is negative, i.e. opposite to that of rod-like molecule liquid crystals. The bend flexoelectric term e3 does not contribute, because of the symmetry of the problem, exactly as demonstrated before [11] in chiral smectic C*. As with nematics, this flexoelectric distortion is only visible because of the free boundary condition for the tube (the director) lines. As it is well known, strong anchoring conditions inhibit most of the volume flexoelectric effect [ 12], except for the case of large initial distortions [ 13, 14]. Acknowledgments. We thank the Bordeaux Group (Drs. Destrade, Tinh and Gasparoux) for the supply of discotic material, and Dr. Martinot-Lagarde for useful discussions.

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تاریخ انتشار 2016