Periodic ground state in the nematic phase of DIO due to an intrinsic surface electric field

Abstract

Nematic layers of DIO held in planar aligning cells are found to exist in a periodic ground state, involving polar and azimuthal director deviations, even in the absence of any external perturbing field. The stripe instability appears definitively in thin cells, over a few °C above the antiferroelectric smectic Z_A onset point, weakening progressively with increasing temperature. The pattern wave vector lies practically along the normal to the rubbing direction. In 90°-twist cells, two sets of stripes form with their wave vectors dependent on the substrate rubbing directions, evidencing the instability as a surface phenomenon. We propose for its origin a new mechanism that involves only the usual Frank elasticity and the surface electric field generated by adsorption of ions. The elastic energy of the proposed periodic director deformation field is zero, and it results from the gain in dielectric energy due to the surface electric field overcoming the cost of weakened anchoring energy.

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1. Introduction

Nematic fluids are perhaps best known for their facile structural transformations under external perturbing fields.¹ The prime instance of this is the commercially relevant Freedericksz transition which is laterally homogeneous; it is brought about by the dielectric torque after it overbalances the elastic torque.^2–4 Less universally familiar are the inhomogeneous effects resulting in periodic states that may be static or dissipative in nature. The former arises as an inhomogeneous Freedericksz effect in planar nematics having their twist elastic constant (k₂₂) much smaller than the splay constant (k₁₁).⁵ Another example of the equilibrium periodic instability is the Bobylev-Pikin flexoelectric effect involving repetitive splay-twist director deformations.^6,7 On the other hand, electroconvection in nematics is a nonequilibrium phenomenon originating from dielectric and conductivity anisotropies and manifesting in a rich variety of patterned states.^1,8–11

Electrical responses such as just mentioned are mostly investigated in conventional nematics made of rodlike molecules, like 4,4′-dimethoxyazoxybenzene or 4-cyano-4′-pentylbiphenyl. It is of fundamental research appeal to examine their correspondence to the effects occurring in the nematic states of other molecular systems. The nematic phase formed of rigid bent-core molecules, for example, has been observed to exhibit the so-called nonstandard electroconvection, which is inexplicable by the standard model or its variants.^12,13 In this context, of recent interest are the ferroelectric nematogens such as DIO that form three liquid crystalline phases, namely, the ordinary nematic (N), antiferroelectric smectic Z_A (SmZ_A) and ferroelectric nematic (N_F) phases.^14–17 In all the three phases, electrically excited periodic modulations of the director profile have been reported.^18–20 Some ground state periodic structures have also been discovered in the N_F phase; they are found to involve domains of reverse twist separated by defect walls.²⁰ The present work provides the first example of a ground state periodic instability in the high temperature N phase of DIO that originates from an intrinsic electric field in the absence of any external field.

Observation of ground state periodic structures in nematic samples not subjected to an external stimulus is rather scarce.^21,22 Their analysis is based on elastic constants, which are effective only at or near surfaces. In an experimental study,²¹ the nematic liquid crystal 4-cyano-4′-pentylbiphenyl spread over glycerine surface is found to exhibit a hybrid alignment, with n being orthogonal to the interface with air. In such films of submicron thickness, a periodic structure is seen; it is attributed to the action of the allowed terms k₁₃ and k₂₄ describing curvature elasticity which is linear in the divergence of vectorial deformations of the n-field. The latter naturally integrate out as surface terms, and become effective if the anchoring energy corresponding to the polar angle θ is weak, and that to the azimuthal angle φ is 0. In another study,²² a compound exhibiting nematic to SmA transition held in a 21 µm thick cell treated for homeotropic anchoring has been found to exhibit some small islands with a periodic deformation of n near the surfaces as the temperature is lowered to about 1.5 °C above the transition to SmA. As it is cooled to SmA, the islands also become homeotropically aligned. The authors argue that surface like elasticity corresponding to the polar splay and bend deformations that can have scalar products with the unit vector describing the surface normal produce the structure if the anchoring energy for homeotropic alignment is weak enough in the islands.

In this paper, we report our observation of a periodic instability in the nematic phase, near the two surfaces of planar aligned samples. Unlike in the two earlier reports^21,22 in which the wavelength of the instability is ∼20 µm, in our system it is ∼2 µm. We propose that it arises from a mechanism involving only the Frank elasticity and surface electric field generated by adsorption of ions. We present our findings in Section 3, which is supplemented by four video clips and a related note; the latter also provides some additional optical features; these are available in the SI.

2. Experimental section

The compound (2,3′,4′,5′-tetrafluoro[1,1′-biphenyl]-4-yl 2,6-difluoro-4-(5-propyl-1,3-dioxan-2-yl))benzoate (DIO) used in this study had the following phase sequence: isotropic, I (170 °C) N (82 °C) SmZ_A (68 °C) N_F. Most optical observations were made using a Carl-Zeiss Axio Imager.M1m polarizing microscope equipped with an AxioCam MRc5 digital camera and software for recording time-lapse and z-stacked images. For transmitted light intensity measurements, a photodiode (Hamamatsu S2281) with a wideband amplifier (Hamamatsu C9329) and a PC-based digital oscilloscope (PicoScope 4262) were used. For observations in monochromatic light, a He–Ne laser scanning confocal facility (Carl Zeiss LSM5) was used in the transmission mode. The sample cells were of sandwich type, made of ITO coated glass plates, overlaid with polyimide layers; the plates were treated for planar alignment by unidirectional rubbing along R such that the pretilt angle did not exceed 3°; the cells were both parallel and antiparallel rubbed type with the following particulars: parallel rubbed planar cells of 6.8 µm thickness (d) from M/s Instec that were made of soda lime glass with an ITO coating of thickness 0.023 ± 0.005 µm and resistance 100 Ω per square, upon which a layer of polyimide KPI-300B of thickness 0.06 µm is deposited; antiparallel planar cells of thickness 1.6 and 5 µm, and 90°-twisted planar cells of thickness 3, 5, 10 and 20 µm from M/s AWAT that were made of ITO electrodes of resistance 70 Ω per square with a coating of Nissan polyimide SE-130. The sample temperature T, maintained using an Instec HCS402 hot-stage coupled to a STC200 controller, was accurate to ±0.1 °C. Relative temperature T_r mentioned in the text refers to T–T_SmZ with T_SmZ as the onset temperature of the SmZ_A phase. For electric field studies, a Stanford Research Systems function generator (DS345) linked to a FLC Electronics voltage amplifier (model A800) was employed. The rms voltage U and frequency f of the applied field were obtained with a Keithley-2002 multimeter. For convenience, we use the following notations: the orthogonal reference axes x and z define the alignment and observation/electric field directions, respectively; P(α)–A(β) denotes the setting of the polarizer P and analyser A with their axes at angles α and β (degrees) relative to x. Fig. 1 shows the experimental geometry.

Fig. 1 Experimental geometry for quarter turn twist cells. (a) Side-view of the sample cell. The molecular alignment is along y at the bottom substrate and along x at the top substrate; R is the rubbing direction. The twist may be clockwise (as in the right half) or anticlockwise (as in the left half). Field E is taken positive when the bottom electrode is positive. (b) Top view of the sample cell. U is the rms voltage across the cell. P(α)–A(β) denotes Polarizer P and analyzer A with their transmission axes, respectively, at angles α and β relative to x. For untwisted planar cells, R is taken to be along x at both the substrates.

3. Results

3.1. The periodic ground state

In thin sample cells (d < 5 µm) treated for planar alignment with antiparallel rubbing, the director n in nematic DIO is found to be nonuniform. It exhibits a periodic distortion in the absence of any external field. A typical ground state texture observed between slightly uncrossed polarizers is shown in Fig. 2. Notable attributes of the patterned state are the following: (i) it is characterized by nearly periodic stripes with the wave vector directed perpendicular to the rubbing axis R, (ii) the pattern-period is quite small, around 2 µm, which is close to the limit of optical resolution, (iii) the pattern-contrast is low, indicating the corresponding weak distortion as possibly a surface-like effect, not involving the bulk, (iv) the textural visibility and interference color vary over the visual field, and (v) it is almost undiscernible in ordinary light or with a single polarizer in any orientation. Another important feature revealed by z-stacked images is illustrated in Fig. 3 showing select images recorded using laser scanning microscopy. The texture of stripes seen here remains recognizable over a large z-range of 50 µm, starting with the first frame of the stack F₀₁ and lasting till about F₅₂; thereafter, the regularly spaced stripes get progressively blurred out, leading eventually to a patchy texture (see F₈₂). Each of the frames in the series (see Video V1 available in the SI) also shows intensity and visibility variations over the visual field.

Fig. 2 Striped ground state of nematic DIO, as observed in a 1.6 µm thick planar layer at 1 °C above the setting point of the underlying SmZ_A phase. The pattern is revealed readily under polarization contrast, with crossed or slightly uncrossed polarizers [here, P(90)–A(−10)], but is not definitively discerned either with a single polarizer or in ordinary light. Notable are the variations in colour and contrast over the visual field that indicate distortion amplitude as dependent on local conditions. Antiparallel rubbed cell with R as the rubbing or alignment axis.

Fig. 3 Select z-stack images recorded using He–Ne (543 nm) laser scanning microscopy in the transmission mode, showing the appearance of the periodic ground state in monochromatic light; P(0)–A(75); T_r = 0.5 °C; d = 1.6 µm. F_ii refers to the frame number in the z-stack. Successive frames are a distance of 1 µm apart. The colors are false, used to vividly render the intensity map.

Observations on 90° twisted-planar samples clearly bring out the periodic distortion as localized in regions close to the substrates. Viewed with parallel polarizers along y, under white light illumination, the field of view presents the texture of two bluish oppositely twisted regions separated by a bright twist disclination; see Fig. 4a. In each of these regions, two sets of stripes are formed; while the wave vector q of one set makes a slight angle with x, that of the other makes about an equal angle with y, such that the stripes tend towards the mid plane director n_M (see later for determining the n_M direction). On a slight rotation of one of the polarizers, expectedly, the reverse twisted regions show a difference in interference color, as in Fig. 4b and c. In Fig. 4b, the central white lines indicate the direction of stripes, and the diagonal arrows, n_M. The significant feature here is that the wave vector of either stripe-set is not aligned with the rubbing direction x or y; this is unlike what is found with a planar untwisted cell (Fig. 3). The inclination of stripes to x or y and the regularity of their spacing are better seen in the laser scanned image (Fig. 4d) displaying an oblique grid texture.

Fig. 4 (a)–(c) Periodic ground state in oppositely twisted regions of a 3 µm thick, 90°-twisted planar layer of DIO at T_r = 0.5 °C, as observed in white light; rubbing axis, determined using a tilt compensator, is along y at the bottom plate and along x at the top plate. While the birefringence color is the same in the two oppositely twisted regions under parallel polarizers along y as in (a), it differs for nearly parallel polarizers, as in (b) [P(100)–A(90)] and (c) [P(80)–A(90)]. Two sets of stripes form with the wave vector closer to x in one and to y, in another. The central white lines in (b) indicate the stripe directions in the two reverse twisted regions; stripes form along lines that tend toward the midplane director. (d) He–Ne-Laser scanned image of a 3 µm thick, 90°-twisted layer in the ground state at T_r = 0.5 °C.

In 90°-twisted samples, the midplane director n_M is readily detected using the normal roll electroconvection (EC) instability exhibited by nematic DIO in the high frequency region. This is exemplified in Fig. 5 showing the diagonal EC rolls in the reverse twisted regions. As is well-known, the wave vector of the roll pattern lies along the midplane director in a quarter turn cell. It is also notable that the spontaneously formed ground, narrow stripes’ state (NSS) continues to exist even in the presence of a high-frequency external field.

Fig. 5 Periodic, wide-spaced, electroconvection rolls in oppositely twisted regions of a 3 µm thick, 90°-twisted planar layer of DIO at T_r = 0.5 °C; P(75)–A(90); 25 kHz, 0.6 V. The wave vector of normal rolls lies along the midplane director.

3.2. Effect of static and low-frequency fields on the periodic ground state

In an increasing dc field of either sign, the ground-state stripes formed in a planar cell along the rubbing axis begin to grow in their visibility above a certain voltage. For example, in a 1.6 µm thick cell at T_r = 0.5 °C, a definitive improvement in contrast is seen at 1.5 V. Thereafter, the distortion continues to increase, so that even in ordinary light or with a single polarizer the patterned state becomes evident. In this process of growth, edge dislocations form increasingly, as in the case of Bobylev–Pikin flexoelectric domains.⁷ Upon reaching a particular higher voltage, the stripes disappear altogether, leaving a uniformly aligned state. This happens at about 2.5 V in our example. Evidently, the uniform higher voltage state is the dielectrically oriented quasihomeotropic state. In Fig. 6 that illustrates all these features, we may note that (i) the period of the stripe pattern is not substantially altered in going from 1.5 V in (b) to 2.5 V in (e), and (ii) the approach of homeotropic Freedericksz state is inferable from the developing extinct regions under P(0)–A(90), as in (f).

Fig. 6 Effect of increasing DC field (b)–(f) on the ground state periodic instability (a) in a 1.6 µm thick layer of DIO at T_r = 0.5 °C; P(75)–A(90). At higher fields, the pattern develops edge dislocations [bright spots in (e)], their number increasing with the voltage U. With the dominance of dielectric realignment, the stripes disappear above 2.5 V.

In low frequency sine-wave fields, pattern modifications that occur in each half cycle bear a correspondence with the voltage varying effects found in a dc field. We describe the results on a twisted cell, which will clearly illustrate the effect of E on the periodic structures. At peak voltages |U_p|, provided the value is sufficiently large, uniform Freedericksz state prevails. Away from |U_p|, while approaching the polarity reversal point, at some instantaneous voltage |U_t| the striped state appears prominently; it decays toward the ground state as |U_t| decreases to 0 V; similarly, while moving away from the polarity reversal point, the ground state stripes continue to grow in contrast until the Freedericksz state takes over completely. Between the dc and low frequency ac field effects, a notable difference is observed in the voltage |U_s| at which the striped state is completely suppressed by the Freedericksz effect. For example, under dc excitation, |U_s| is 2.5 V for a 1.6 µm thick sample at T_r = 0.5 °C; it is below 0.2 V for ac fields of frequency below 1 kHz. In order to exemplify the pattern modifications observed in low frequency fields, we refer to Fig. 7 depicting select frames of a time series (shown as Video V2 in SI) recorded with a 3 µm thick, 90°-twisted layer subjected to a sine-wave field of frequency 20 mHz and rms voltage 0.7 V. The yellowish birefringence color observed at maximum voltages |U_p|, as in Fig. 7a, f and k, corresponds to the uniform Freedericksz state (FS) in which the tilt of nematic director in bulk, θ, toward the E-field, is maximum. As the voltage reduces from its peak value, θ decreases in the FS and the color changes toward violet, as in Fig. 7b, e, g, j and l. With further drop in U_t, close to the 0 V crossing, the striped state develops. In the positive half cycle of U_t, near vertical stripes appear first, as in Fig. 7c; soon after, in about 6 seconds, when U_t is negative, near horizontal stripes replace the vertical stripes, as in Fig. 7d. With rise in negative voltage, FS replaces the patterned state, as in Fig. 7e, f and g; the stripes appearing first in the negative voltage half-cycle are near horizontal (Fig. 7h); after U_t turns positive, near vertical stripes form (Fig. 7i). This sequence of events repeats itself as time advances. The U_t(t) schematic at the bottom shows the instantaneous voltage positions corresponding to the optical textures shown above. At 0 V, with parallel polarizers along x or y, extinction of light is obtained; with patterned states developed around 0 V, a generally bluish texture with alternating hues in successive bands is observed.

Fig. 7 (a–l) Select frames from a time series showing voltage dependent textures in a 3 µm thick, 90°-twisted layer of nematic DIO subjected to a sine-wave field of frequency 20 mHz, and rms voltage 0.7 V, and viewed with parallel polarizers P(90)–A(90). T_r = 0.5 °C. The overlaid sine curve represents the time varying instantaneous voltage, U_t(t). Voltages corresponding to different frames are indicated on the curve.

The foregoing results from the twist cell clearly show that the striped state is associated with periodic distortions confined to thin layers located close to the substrates; with the sample geometry in Fig. 1, the near vertical stripes form in the vicinity of the bottom substrate and the near horizontal stripes, in the vicinity of the top substrate. The reason as to why the distortion amplitude of the two stripe states gets enhanced in a polarity dependent manner will be discussed later.

The voltage varying morphological features are also reflected in the correspondingly varying transmitted light intensity. We present in Fig. 8 the results from light transmission experiments. In Fig. 8a, the minimum of R_o occurs near 0 V crossings at which the low-contrast ground stripe state prevails. At lower voltages at which the Freedericksz state is not realized, as at 0.5 V for example, R_o spikes at voltage peaks due to enhanced amplitude of stripe-state distortion. At higher voltages, as at 1.0 and 2.8 V, during the initial rise in |U_t|, stripes appear with ever increasing contrast leading to the spikes in transmission; further rise in |U_t| causes the transmission to drop due to the onset of dielectric instability; during |U_t| descent from its peak, a second, milder spike in R_o follows, which is well resolved at higher voltages (see R_o3); it is again due to the enhanced stripe state. In Fig. 8b, the minima in R_o are obtained at peak exciting voltages |U_p| at which the sample remains in a uniformly reoriented Freedericksz state. As the nematic director approaches homeotropic alignment with increasing U_t, the time interval of minimum transmission extends around |U_p|.

Fig. 8 Optical response R_o as a function of time (or instantaneous voltage U_t) in a 1.6 µm thick, planar DIO sample at T_r = 0.5 °C at different voltages of the driving 20 mHz sine-wave field. Measurements of transmitted light intensity in (a) and (b) are made with the sample held between crossed polarizers P(0)–A(90) and P(45)–A(135), respectively. Optical outputs R_o1, R_o2 and R_o3 in (a) correspond, respectively, to driving rms voltages 0.5 V (U_t1), 1.0 V (U_t2) and 2.8 V (U_t3); R_o1 and R_o2 in (b) correspond, respectively, to 1.0 V (U_t1) and 2.8 V (U_t2).

3.3. Effect of high-frequency fields on the periodic ground state

As previously mentioned, the NSS state (stripes along the rubbing direction R with q normal to R), that characterizes the ground state in nematic DIO is completely suppressed at higher voltages U_s in dc and low frequency ac fields. We have also studied in detail the stability of the ground state morphology under fields of frequency up to 100 kHz in a 1.6 µm thick planar sample at T_r = 0.5 °C; see Fig. 9. In the low frequency region (<f_L ≈ 9 kHz), above U_s, the texture is that of the pattern-free, homogeneous Freedericksz state (HFS); U_s = U_F. Beyond 9 kHz, U_s corresponds to electroconvective threshold U_EC at which a periodic roll pattern forms with its wave vector q broadly along R. The voltage U_s varies parabolically with f for frequencies below f_L, while the variation is nearly linear above f_L. Below the crossover frequency f_L, U_F < U_EC; above f_L, U_F > U_EC.

Fig. 9 Frequency variation of the suppression voltage U_s at which the narrowly-spaced stripe state disappears completely. Inset: Enlarged view of the U_s(f) plot in the lower frequency region. Below the red line, the ground state morphology is preserved; above this line, the uniform Freedericksz state is found up to frequency f_L; beyond f_L, in its stead, the normal roll state of electroconvection is observed. d = 1.6 µm. T_r = 0.5 °C. NSS, HFS, ECS and WSS stand for Narrow Stripes’ State, Homogeneous Freedericksz State, Electroconvection State and Wide Stripes’ state, respectively.

In samples of thickness 5 µm or more, the ground state is characterized only by thermal fluctuations with no regular stripes revealed. However, there is reason to believe that the weak NSS distortions may be present, possibly masked by bulk thermal effects, since a static or low-frequency external electric field is found to render the NSS visible in thick DIO layers. By way of demonstration, the NSS developed in a 10 µm thick TN layer subjected to a dc field is shown in Fig. S1 of SI; the same instability driven by a square wave (f = 40 mHz) field is depicted in Fig. 10 (insets); the transmitted intensity profiles show that at polarity reversals, strong R_o variations occur briefly due to the reconfiguration of the director field. At the peak in R_o following a polarity reversal, narrow stripes, oriented near vertically in +U and near horizontally in −U, occur with maximum contrast. The visibility of the NSS continually drops thereafter, but importantly the pattern remains discernible until the next polarity switch. This prolonged pattern persistence indicates a correspondingly protracted process of ion-reorganization.

Fig. 10 Optical response R_o of a 10 µm thick twisted nematic (TN) layer as a function of voltage U_t of the applied square wave field of frequency 40 mHz. Insets show the narrow stripes patterns observed in the two half-cycles following polarity switch. T_r = 0.5 °C.

In high frequency fields, in thin cells, as we have seen in Fig. 5, the ground NSS remains unaffected even after formation of the WSS that involves periodic distortions originating in the bulk. Likewise, in thick cells, the uniform ground state (UGS) texture continues till the WSS is induced. An interesting feature of the WSS is that, beyond a certain thickness-dependent frequency f_q, it undergoes a wave vector switching transformation; more specifically, q switches from the direction of midplane director-normal n_M⊥ at a lower voltage to that of midplane director n_M at a higher voltage. This phenomenon is exemplified in Fig. 11 (as also in Video V3 available in SI). In the 5 µm thick layer, f_q is determined as 280 kHz. Below f_q, the bifurcation from the ground state is to the WSS1 in which the texture is similar to that of normal rolls in standard electroconvection (Fig. 11a). Above f_q, the primary bifurcation is to the WSS2 with the wave vector turned by 90° relatively (Fig. 11b). A secondary bifurcation into the WSS1 occurs at an elevated voltage (Fig. 11c and d). In Fig. 12, we present the phase diagram in the U–f plane for sample thicknesses of (a) 5 and (b) 20 µm. Notably, the frequency f_q at which the wave vector switching takes place is thickness dependent. The instability WSS1, going by the direction of its q relative to the midplane director, is possibly ascribable to electroconvection. The origin of WSS2 is not that obvious.

Fig. 11 Wide stripes’ states (WSSs) formed in a 5 µm thick TN layer under the action of high frequency electric fields. The texture in (a) is typical of the instability found up to about 280 kHz (f_q) in the 5 µm thick layer. (b) The WSS2 found for frequencies above f_q at the first bifurcation from the uniform ground state. (c) and (d) The WSS1 into which the WSS2 transforms at a higher voltage; an early stage of this transformation is seen in (c), while the eventual steady state is represented in (d). T_r = 1.0 °C.

Fig. 12 (a) Phase diagram in the U–f plane for nematic DIO at T_r = 1 °C; n_M-stripes and n_M⊥-stripes belonging to the wide stripes state (WSS) appear along the midplane director and director-normal, respectively. Squares are for a 5 µm thick TN layer and circles, for a parallel rubbed 6.8 µm planar cell. UGS denotes uniform ground state. Inset shows the frequency dependence of the threshold voltage difference of the two WSSs in the TN cell. Continuous red line is a linear fit; black and blue lines are parabolic fits. (b) Phase diagram similar to (a), but for the sample thickness of 20 µm; FS denotes Freedericksz state with the threshold U_F.

The narrow stripe state may also be induced in a thick sample by doping it with a surfactant. For example, a sample of DIO admixed with 3 wt% of 2-octadecoxypropanol (mixture referred as DOP3), held in a 9 µm thick Instec cell shows the stripes in its ground state. On the other hand, pure DIO without doping does not exhibit the stripes. This is demonstrated in the Video V4, available in SI. As we shall see later, this result is significant in terms of the anchoring strength as a factor determining the stability of the patterned ground state.

The striped ground state is also affected by temperature, appearing with progressively decreasing visibility while heating the sample above the SmZ_A-N transition point. The pattern continues to be visible for T_r up to about 10 °C. While cooling, the stripes instability disappears once the SmZ_A phase sets in. In Fig. S2 of SI, the temperature variation of pattern contrast is illustrated.

4. Discussion

The alignment of the apolar director (n) field of a typical nematogen in cells treated for specific anchoring conditions is quite simple. If the walls are treated for planar anchoring, usually obtained by unidirectional rubbing of the plates coated with an appropriate polymer, the n-field is uniform if the rubbing directions of the two walls are antiparallel, and twisted by 90° in the two degenerate senses if the rubbing directions are orthogonal. The high temperature nematic phase of DIO has this expected response in commercial cells obtained from Instec, USA. On the other hand, in cells procured from the AWAT, Poland, the medium exhibits a new type of relatively weak periodic structure with its wave vector normal to the rubbing direction if the plates have antiparallel rubbing, and a somewhat oblique two-dimensional lattice structure in the optical field of view in cells with walls rubbed in orthogonal directions. The latter observation indicates that the periodic structure occurs near the two surfaces of the cell. The fact that the structure occurs only for cells from a particular source also shows that the surface attributes are of vital importance for its formation.

We may note that weak surface anchoring energies for tilt and/or azimuthal orientations of the director by themselves cannot lead to a spontaneous periodic distortion of the director near the surface. The enhanced visibility of the periodic structure near a given surface by a low DC electric field of a specific sign (Fig. 6 and 7) shows that a surface electric field is the origin of the structure. Such a field is generated by selective adsorption of ions of one sign by the polymer used to favour the planar alignment of the director. The charged surface in turn results in a counterion cloud which screens the field over the Debye length. In a nematic medium with positive dielectric anisotropy (Δε) the surface field weakens the anchoring energy,²³ and can also distort the director field along the field direction,²⁴ costing an elastic energy. None of the earlier investigations on the effect of surface electric field point to a periodic distortion that we have described earlier. As for the director field corresponding to the ground state, the optical characteristics of the patterned state clearly indicate it to be similar to that of the Bobylev–Pikin flexoelectric instability.^6–8

The screened electric field has an exponential decay from the surface over the Debye length L_D, and assuming that the anchoring energy is weakened sufficiently, as discussed in ref. 24 the dielectric coupling with a nematic having positive dielectric anisotropy can be expected to produce a small distortion of the polar angle θ ≪ 1 rad given by

in which Z is measured from the surface. In our case, as there is a periodic structure, we modify the distortion field to involve the azimuthal angle φ as well, and propose the following:
andwhere l_y is the pattern wavelength.

The free energy density is given by

Using eqn (2), the free energy density takes the form

in which E(z) = E₀ exp(−z/L_D), where E₀ is the electric field at the surface given by σ/(2ε₀ε_⊥), σ being the areal charge density arising from adsorbed ions and θ₀ is assumed to be ≪1 rad. Remarkably, with the specific director structure given by eqn (2), ifthe net splay is 0, thus the relevant elastic energy also goes to 0. As already noted in ref. 25, the splay cancellation for this distortion is total. Even more remarkably, if θ₀ = φ₀, the net twist also goes to 0, and the result is that for this specific combination of director structure given by eqn (2), which results in a periodic director distortion, and the spatial periodicity given by eqn (5), with the amplitudes of the tilt and azimuthal distortions being equal, the structure does not cost any elastic energy!

The competition is hence only between (i) the negative dielectric energy gained by the medium with positive dielectric anisotropy under the action of the electric field near the surface as described above, which causes the nonzero θ(y, z) profile described by eqn (2), and (ii) the positive anchoring energy cost of generating nonzero values of θ₀ and φ₀ at the surface. For a cell with antiparallel rubbing directions of the walls, the dielectric energy integrated over d/2, i.e., half the thickness of the cell and averaged over the wavelength l_y of the periodic structure has to be compared with the anchoring energy averaged over l_y. The dielectric energy is given by

Assuming for simplicity that the anchoring energies corresponding to both the tilt θ₀ and azimuthal φ₀ deviations are equal to W, the anchoring energy is given bywhich takes a very simple form for θ₀ = φ₀ required for the elastic energy to vanish. Then the energy of the cell per unit area is given byThe structure with a periodic splay-twist deformation of the director field near the surface becomes energetically favourable when F_area becomes negative. A large positive value of the dielectric anisotropy is a basic requirement, and indeed DIO is reported to have Δε ∼ 100 as the nematic to SmZ_A transition point is approached.¹⁴ A strong adsorption of ions with a specific sign is another requirement, as it will give rise to a large surface electric field E₀, which also helps to reduce the effective anchoring energy W.²³ This, of course, depends on the particular polymer used for the surface treatment and only the commercial cells from Poland fulfil this requirement while those from Instec do not, as our experimental results show. Mixing DIO with a surfactant can be expected to alter the surface conditions, in particular reducing the anchoring energy. The periodic structure is observed in such mixtures even in cells made by Instec.

Assuming that the ions are monovalent, the Debye length

where k is the Boltzmann constant, ρ is the number density of the ions, and e is the charge of an electron. For T = 360 K, ε_⊥ ∼ 30, and ρ ∼ 10²¹, i.e., about one ion per million molecules, L_D ∼ 0.3 µm. If the adsorption of ions is strong, we can assume σ ∼ 10⁻⁴ C m⁻², and E₀ ∼ 2 × 10⁵ V m⁻¹. For Δε ∼ 100, the second term in the square brackets of eqn (8) is ∼10⁻⁶ J m⁻². The wavelength of the periodic structure l_y = 2πL_D ∼2 µm, is similar to the experimental value. The anchoring energy should be weaker than this for the periodic structure at the surface to become energetically favoured, as has been found in DIO in the AWAT cells. When the two plates of the cell are oriented with their rubbing directions mutually orthogonal, the weak anchoring results in the director actually rotating by less than 90° in either clockwise or anticlockwise sense, leading to the oblique two dimensionally periodic structures seen optically.

In conclusion, the mesogen DIO has been found to exhibit weak periodic distortion of the director field in the apolar (paraelectric) nematic phase. Our experiments in both planar and twisted nematic cells clearly show that surface electric field arising from adsorbed ions gives rise to the structure. Such a field weakens the anchoring energy and destabilises the planar alignment for the sample with a relatively large positive dielectric anisotropy. We have developed a simple model to show that a periodic structure with small deviations from the planar structure does not cost any curvature elastic energy, and the gain in dielectric energy can overcome the anchoring energy cost to stabilise the structure. It would be interesting to look for this novel mechanism in other nematic LCs with large dielectric anisotropy. In closing, we note that, in some recent experiments, the nematic phase of DIO has been observed to involve helical conformers that separate into reverse twisted domains.^26,27 That, instead of chiral domains, we detect a periodic distortion in the ground state is seemingly indicative of the crucial role played by surface anchoring conditions in determining the orientational states of nematic DIO. It would be interesting to see if the modulated ground state observed in the N phase of DIO is a general feature of the paranematic phase of other antiferroelectric mesogens such as reported in ref. 28.

Conflicts of interest

There are no conflicts of interest to declare

Data availability

Data will be provided upon request.

The data supporting this article have been included as part of the supplementary information (SI). Supplementary information is available. See DOI: https://doi.org/10.1039/d5sm00920k.

Acknowledgements

The authors are thankful to Prof. B. L. V. Prasad, Director, Centre for Nano and Soft Matter Sciences, Bangalore for the experimental facilities, and to Dr D. S. Shankar Rao for his help in preparing the samples.

References

1. S. A. Pikin, Structural Transformations in Liquid Crystals, Gordon and Breach Science Publishers, New York, 1991.
2. V. Freedericksz and A. Repiewa, Theoretisches und Experimentelles zur Frage nach der Natur der anisotropen Flüssigkeiten, Z. Phys., 1927, 42, 532–546,  DOI:10.1007/BF01397711.
3. V. Freedericksz and V. Zolina, Forces causing the orientation of an anisotropic liquid, Trans. Faraday Soc., 1933, 29, 919–930,  10.1039/TF9332900919.
4. H. J. Deuling, Deformation of Nematic Liquid Crystals in an Electric Field, Mol. Cryst. Liq. Cryst., 1972, 19, 123–131,  DOI:10.1080/15421407208083858.
5. F. Lonberg and R. B. Meyer, New Ground State for the Splay Freedericksz Transition in a Polymer Nematic Liquid Crystal, Phys. Rev. Lett., 1985, 55, 718–721,  DOI:10.1103/PhysRevLett.55.718.
6. Y. P. Bobylev, V. G. Chigrinov and S. A. Pikin, Threshold flexoelectric effect in a nematic liquid crystal, J. Phys., 1979, 40(Suppl. 4), C3331,  DOI:10.1051/jphyscol:1979364.
7. Y. P. Bobylev and S. A. Pikin, Threshold piezoelectric instability in a liquid crystal, Sov. Phys. JETP, 1977, 45, 195–198.
8. L. M. Blinov and V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials, Springer, Berlin, 1994.
9. W. Helfrich, Conduction-Induced Alignment of Nematic Liquid Crystals: Basic Model and Stability Considerations, J. Chem. Phys., 1969, 51, 4092–4105,  DOI:10.1063/1.1672632.
10. E. Dubois-Violette, P. G. de Gennes and J. Parodi, Hydrodynamic instabilities of nematic liquid crystals under A. C. electric fields, J. Phys., 1971, 32, 305–317,  DOI:10.1051/jphys:01971003204030500.
11. E. Bodenschatz, W. Zimmermann and L. Kramer, On electrically driven pattern-forming instabilities in planar nematics, J. Phys., 1988, 49, 1875–1899,  DOI:10.1051/jphys:0198800490110187500.
12. D. Wiant, J. T. Gleeson, N. Éber, K. Fodor-Csorba, A. Jákli and T. Tóth-Katona, Nonstandard electroconvection in a bent-core nematic liquid crystal, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2005, 72, 041712,  DOI:10.1103/PhysRevE.72.041712.
13. N. Éber, Á. Buka and K. S. Krishnamurthy, Electrically driven structures in bent-core nematics, Liq. Cryst., 2022, 49, 1194–1222,  DOI:10.1080/02678292.2021.1973129.
14. H. Nishikawa, K. Shiroshita, H. Higuchi, Y. Okumura, Y. Haseba, S.-I. Yamamoto, K. Sago and H. Kikuchi, A Fluid Liquid-Crystal Material with high Polar Order, Adv. Mater., 2017, 29, 1702354,  DOI:10.1002/adma.201702354.
15. R. J. Mandle, S. J. Cowling and J. W. Goodby, A nematic to nematic transformation exhibited by a rod-like liquid crystal, Phys. Chem. Chem. Phys., 2017, 19, 11429–11435,  10.1039/c7cp00456g.
16. X. Chen, V. Martinez, E. Korblova, G. Freychet, M. A. Glaser, M. Zhernenkov, C. Wang, C. Zhu, L. Radzihovsky, J. E. Maclennan, D. M. Walba and N. A. Clark, The smectic Z_A phase: Antiferroelectric smectic order as a prelude to the ferroelectric nematic, Proc. Natl. Acad. Sci. U. S. A., 2023, 120, e2217150120,  DOI:10.1073/pnas.2217150120.
17. S. Brown, E. Cruickshank, J. M. D. Storey, C. T. Imrie, D. Pociecha, M. Majewska, A. Makal and E. Gorecka, Multiple Polar and Non-polar Nematic Phases, ChemPhysChem, 2021, 22, 1–6,  DOI:10.1002/cphc.202100644.
18. K. S. Krishnamurthy, S. K. Prasad, D. S. S. Rao, R. J. Mandle, C. J. Gibb, J. Hobbs and N. V. Madhusudana, Static and hydrodynamic periodic structures induced by ac electric fields in the antiferroelectric SmZ_A phase, Phys. Rev. E, 2025, 111, 045420,  DOI:10.1103/PhysRevE.111.045420.
19. B. Basnet, S. Paladugu, O. Kurochkin, O. Buluy, N. Aryasova, V. G. Nazarenko, S. V. Shiyanovskii and O. D. Lavrentovich, Periodic splay Fréedericksz transitions in a ferroelectric nematic, Nat. Commun., 2025, 16, 1444,  DOI:10.1038/s41467-025-55827-9.
20. P. Kumari, B. Basnet, M. O. Lavrentovich and O. D. Lavrentovich, Chiral ground states of ferroelectric liquid crystals, Science, 2024, 383, 1364–1368,  DOI:10.1126/science.adl0834.
21. O. D. Lavrentovich and V. M. Pergamenshchik, Stripe Domain Phase of a Thin Nematic Film and the K₁₃ Divergence Term, Phys. Rev. Lett., 1994, 73, 879–982,  DOI:10.1103/PhysRevLett.73.979.
22. I. Lelidis and G. Barbero, Novel surface-induced modulated texture in thick nematic samples, Europhys. Lett., 2003, 61, 646–652,  DOI:10.1209/epl/i2003-00120-y.
23. B. Valenti, M. Grillo, G. Barbero and P. T. Valabrega, Surface anchoring energy and Ions Adsorption: Experimental analysis, Europhys. Letters, 1990, 12, 407–412,  DOI:10.1209/0295-5075/12/5/005.
24. G. Barbero, L. R. Evangelista and N. V. Madhusudana, Effect of surface electric field on the anchoring of nematic liquid crystals, Eur. Phys. J. B, 1998, 1, 327–331,  DOI:10.1007/s100510050190.
25. K. S. Krishnamurthy, D. S. Shankar Rao, M. B. Kanakala, C. V. Yelamaggad and N. V. Madhusudana, Saddle-splay-induced periodic edge undulations in smectic-A disks immersed in a nematic medium, Phys. Rev. E, 2020, 101, 032704,  DOI:10.1103/PhysRevE.101.032704.
26. N. Yadav, Y. P. Panarin, W. Jiang, G. H. Mehl and J. K. Vij, Spontaneous mirror symmetry breaking and chiral segregation in the achiral ferronematic compound DIO, Phys. Chem. Chem. Phys., 2023, 25, 9083–9091,  10.1039/d3cp00357d.
27. N. Yadav, Y. P. Panarin, J. K. Vij, W. Jiang and G. H. Mehl, Optical activity and linear electrooptic response in the paraelectric subphase of an achiral ferroelectric nematic compound, Liq. Cryst., 2023, 50, 1375–1382,  DOI:10.1080/02678292.2023.2219238.
28. P. Nacke, A. Manabe, M. Klasen-Memmer, C. Xi Chen, V. Martinez, G. Freychet, M. Zhernenkov, J. E. Maclennan, N. A. Clark, M. Bremmer and F. Giesselmann, New examples of ferroelectric nematic materials showing evidence for the antiferroelectric smectic-Z phase, Sci. Rep., 2024, 14, 4473,  DOI:10.1038/s41598-024-54832-0.

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