Variable pitch hydrodynamic electro-optic gratings utilising bent liquid crystal dimers †

Electrohydrodynamic instabilities (EHDI) in liquid crystals form uniform and continuously variable diﬀractive structures when subject to certain material and geometry determined conditions. A one-dimensional grating is one such diﬀractive structure, where the refractive index changes periodically in a direction parallel to the initial liquid crystal director. The period of this structure has been shown previously to vary continuously between the values of the cell gap and half-cell gap approximately, allowing continuous angular modulation of optical beams but with a limited angular range. In this work, the lower pitch limit is shown to also be governed in part by the ratio of the splay and bend elastic constants ( k 11 / k 33 ) of the liquid crystal. A host nematic liquid crystal with standard elastic constant ratios ( k 11 / k 33 o 1) is doped with odd-alkyl-spaced dimeric liquid crystal CB7CB, to create a liquid crystal mixture with a far higher elastic constant ratio ( k 11 / k 33 4 5) than for those previously used in literature EHDI studies. The EHDI gratings formed in this new mixture exhibit pitch lengths significantly below half-cell gap, allowing up to 50% wider angle continuous steering of light. This improves the potential for application in beamsteering and diﬀractive optical devices. the Normal Roll (NR) mode. 3 shows oblique rolls which occur at lower frequencies and have two wave vectors q 1 and q 2 oriented at (cid:2) y Ob to the rubbing direction respectively. 4–6 show prewavy patterns. Here 5 shows a chevron texture, which is formed by the superposition of the NR and prewavy instabilities 51,53 7–9 show the effect of increasing voltage within the conductive regime, where temporal and spatial disorder increases until the Dynamic Scattering Modes (DSM’s 1 and 2) happen. DSMs are highly efficient optical scattering states.


Introduction
Since the 1960s nematic liquid crystals (NLCs) have been used successfully as switchable optical materials in optoelectronic applications, including displays. 1,2 The number of different applications of NLCs has expanded in recent decades to a wider variety of optical devices including spatial light modulators, [3][4][5][6][7][8] lenses, [9][10][11] diffraction gratings [12][13][14][15] and beamsteering devices. 4,[16][17][18] An important goal has been to produce entirely arbitrary phase profiles for incident light when passed through the liquid crystal. Currently, liquid crystal on silicon (LCoS) spatial light modulators are used for such purposes. [3][4][5][6][7][8][19][20][21][22] In LCoS devices, individual electrodes, driven by the silicon back-plane, reorient the NLC to induce the required optical phase profile. However, fringing fields and defects in the nematic director profile limit pixels to a minimum size of B5 mm. 8,16 Given that many pixels are required to create all but the simplest optical features, and the fact that LCoS devices operate in reflection mode only (with much larger minimum pixel sizes for transmissive devices), 5,16 developing alternative methodologies to create high-quality diffractive structures is desirable. In most NLC devices the electro-optic switching is achieved by utilising an external electric field, in order to rotate the average molecular orientation (n, the director), and hence the optic axis (Fréedericksz transition). 23,24 The required electric field is mainly dictated by the dielectric anisotropy (De) and the elastic constants for splay (k 11 ), twist (k 22 ) and bend (k 33 ), [24][25][26] of the NLC. Optimisation of these material parameters is an essential part of ensuring good device performance.
A less conventional method for creating diffractive optical structures in NLCs is by inducing viscous flow. Here, director reorientation is achieved by inducing a periodic material flow, which places a viscous torque on the director, causing it to reorient and form an optical grating. 27,28 An example of such a process is the formation of electrically driven ion flow patterns created by electrohydrodynamic instability (EHDI).
In EHDIs the periodicity, described by pitch length (l C ) of the induced optical grating, is typically dictated by the device spacing, d, the conductivity anisotropy (Ds) and the dielectric anisotropy (De). The material elastic constants play a key role, as for the grating to form the systems elastic inertia must be overcome. [28][29][30] In our previous work, 31 it was argued that, for a given d, k 11 and k 33 influence the aspect ratio and the pitch length of the EHDI rolls, respectively. In this paper, the effect of changing the values of k 11 and k 33 of the NLC on the resulting EHDI is investigated by comparing different materials in several devices of different spacing. The variation in material elasticity is achieved by adding an odd-spaced dimer to the NLC. This increases k 11 /k 33 to an atypical level, which is confirmed through the measurements of elastic constants. This results in shorter pitch EHDI gratings (ohalf-cell gap) than observed previously, thereby allowing wider angle continuous beamsteering steering.

Electro-hydro-dynamic instabilities
EHDIs have been the focus of extensive study since Williams first observed them in 1963. Such Williams domains are periodic structures that form when a relatively low-frequency voltage is applied across a layer of NLC. 32,33 These periodic domains are formed by the reorientation of the NLC's optic axis, resulting in a periodic change of the refractive index for light polarised parallel to the axis of the grating vector. This change can be observed using both polarising optical microscopy (POM) 32,34,35 and diffraction spectroscopy. 15,[27][28][29][36][37][38] A basic understanding of the induced periodic domains was provided by Carr 39 and Helfrich 40 using a one-dimensional model based on periodic viscous flow and charge separation, leading to the standard model of EHDIs still used. [27][28][29][30]34,36,[41][42][43][44][45] Fig. 1 (a) Schematic representation of a device that can undergo EHDI. A thin layer of nematic liquid crystal with negative De and positive Ds is placed between two glass plates separated by spacing d. The glass is coated with a transparent conductor, which allows a voltage V to be applied across the material, and an alignment layer, which gives the director (represented by double-headed arrows) an initial orientation. (b) Plot of the textures usually observed in standard EHDI as a mapping of voltage and frequency (see ref. 28). Here the threshold/critical voltage (V C ) for onset of EHDI is shown as the red line below which the texture is homogenous. This behaviour of V C consists of the low frequency 'conduction' and high frequency 'dielectric' regimes. The focus of this work is the narrow region above V C in the conduction regime where well ordered normal rolls exist. (c) POM images of different regions shown in (b) between crossed polarisers where the scale bar in 1 applies to all figures. Here the rubbing direction is shown as either the pink or purple double headed arrow in the bottom left of each Fig. 1 and 2 shows the Normal Roll (NR) mode. 3 shows oblique rolls which occur at lower frequencies and have two wave vectors q 1 and q 2 oriented at AEy Ob to the rubbing direction respectively. 4-6 show prewavy patterns. Here 5 shows a chevron texture, which is formed by the superposition of the NR and prewavy instabilities 51,53 7-9 show the effect of increasing voltage within the conductive regime, where temporal and spatial disorder increases until the Dynamic Scattering Modes (DSM's 1 and 2) happen. DSMs are highly efficient optical scattering states. This standard model is most straightforward for materials with negative dielectric (De o 0) and positive conductivity (Ds 4 0) anisotropies operating in a planar-aligned device, as shown schematically in Fig. 1a. Here, domain formation occurs once the applied voltage, V reaches a critical value, V C . This threshold is expected to vary with frequency in the manner shown in Fig. 1b, where two distinct regimes, 'conduction' and 'dielectric', are observed. 27,28,41,44,46,47 In this work, we focus on the lower frequency conduction regime. In the conduction regime at V C , a one-dimensional phase contrast grating forms, with normal parallel to the initial director (Fig. 1c). Within this normal roll (NR) mode, the period (l C ) decreases with increasing applied frequency. 31,46 When V C is applied, the NR mode is induced by periodic hydrodynamic domains within the material (Fig. 2a). These domains are caused by ions moving to oppositely charge electrodes while the voltage oscillates. This has the dual effect of applying a viscous torque on the director 29,48 and inducing a periodic charge separation in the material creating an in-plane electric field (shown as E ions in Fig. 2b). Both the hydrodynamic motion and the in-plane potential act to deviate the director inside the roll from its initial orientation. The resulting out-ofplane disruption of the director is resisted by the applied voltage across the device (due to negative De) and the material elasticity. 28,29,39,40,45,49 This accounts for the threshold behaviour of V C , where for a given frequency, the combined deforming torques overcome the system inertia allowing the director within the rolls reorient when V 4 V C . At V C , this creates a NR mode phase modulating diffraction grating of the same period as the director modulation. 15,37 The value of V C is seen to increase with applied frequency (Fig. 1a), as ions have shorter time periods to transit the cell.
At the onset of NR mode, the bend elastic constant, k 33 is the dominant elasticity that must be overcome for the initial deviation of the director to occur. 28,29 Meanwhile, the value of k 11 becomes important at higher deviation. This is analogous to the Fréedericksz transition of a negative De material with the field applied parallel to the initial director. As mentioned previously, this suggests that a low k 33 (and hence high ratio of k 11 /k 33 ) should be advantageous for low pitch high contrast diffraction gratings: larger k 11 implies any deformation will persist further in the z-direction, while a low k 33 implies shorter pitch rolls (l C decreases). 31 While maintaining the uniform NR mode, the range of pitches (l C ) the rolls can exhibit is typically dictated by the device spacing, d. Here, the pitch only occupies values in the approximate range 31,35,37,[50][51][52]  before transitioning into the chevron texture (Fig. 1c). Within the NR mode the magnitude of l C decreases across this range with increasing frequency. In terms of the wave vector, q C , this range becomes or where q 0 is 2p/d. This severely restricts the range of continuously variable optical diffraction angles 15,29,31,38 that is available when using EHDI. Although reducing the cell gap d does result in higher q C as desired, there is a concomitant decrease of diffraction efficiency, due to the reduced optical path length of the rolls. To create wider-angle beam steering devices, therefore, other means for decreasing l C or increasing q C are necessary.
Studies have been carried out in driving bent-core LCs into a state of EHDI 58,61,81-83 and there are a few similar reports on odd-spaced dimeric LCs. [84][85][86] Both bent-core and dimeric LCs tend to exhibit significantly different EHDI behaviour (termed non-standard electroconvection) to those observed in conventional (calamitic) NLCs. [87][88][89] These differences are attributed to the sizeable effect of flexoelectricity in the EHDI of these novel materials, 41,[83][84][85] where the standard Carr-Helfrich model assumes such effects are negligible. 34,39,40 The EHDI of bent-core and dimeric LCs show different frequency dependencies of V C and optical textures, compared to calamitics, making a direct evaluation between standard and non-standard EHDI difficult. In instances where the conductivity is relatively low, low-frequency flexoelectric domains or non-standard rolls appear that are parallel to the director field, contrary to those of standard EHDI. 90,91 Difficulties in comparisons are further compounded as both bent-core and bent dimeric LCs often have nematic phase at elevated temperatures and are usually difficult to align uniformly. Both bent-core and bent dimeric liquid crystals have been shown to exhibit unusual elastic constants when compared to calamitic NLCs. They exhibit values of k 11 /k 33 that are significantly higher than unity, 54,57,60,69,72,76,79,92 where standard calamitics typically exhibit k 11 /k 33 between 0.6 and 0.8. [93][94][95] It has also been shown that doping a standard calamitic liquid crystal with a bent-core or dimeric material can imbue the mixture with some similarly unusual elastic properties 56,92,[96][97][98][99] while still showing a nematic phase. For example, Parthasarathi et al. 97 showed that binary mixtures of the bent dimer CB7CB with calamitic material 7OCB at certain temperatures exhibits k 11 /k 33 4 5. In order to understand the influence of elastic constants on the EHDI formation, our work uses a mixture of a bent dimer (CB7CB) and a standard calamitic NLC (MLC 2081). The permittivities of the resulting mixture were characterised to quantify the changes in dielectric and elastic properties. POM experiments were carried out on the mixture while it underwent EHDI to quantify V C and q C . It was hypothesised that if the outlined arguments of large k 11 /k 33 are correct, this new mixture should exhibit larger values q C than those given in eqn (4). Such large values of q C are beneficial in the goal of achieving wider angle continuous beam steering.
MLC 2081 is a commercial NLC mixture provided by Merck Chemicals UK and was chosen as the host liquid crystal because it exhibits a wide nematic temperature range, a typical k 11 /k 33 (0.7-0.8 at 25 1C), high negative dielectric anisotropy (De = À5.5 at 25 1C) and, importantly, a low inherent conductivity (s > o 10 nS m À1 at 25 1C) to enable conductivity to be controlled through the addition of dopants. The clearing temperature was measured to be 112 AE 2 1C, (using POM in a 10 mm spaced, planar-aligned glass cell on cooling at 0.5 1C min À1 ).
Due to its low conductivity and strongly negative De, MLC 2081 did not display a typical EHDI conduction regime at frequencies 41 Hz. To enhance it's conductivity, the ionic dopant TBATPB was added in concentrations of 0.5-1.0% (creating three mixtures M 100 : 0 : 0.5/0.6/1.0 ). TBATPB was chosen as it (and other similar salts) has been shown previously to enhance conduction of LCs. 31,56,100,101 Increase of T NI in mixtures were observed when measured over several months and application of high electric fields, potentially due to the breakdown of one of the lower clearing point components in the mixture caused by the addition of the ions. Hence, comparisons are made at the same reduced temperature (T NI À T) where T NI was checked at the time of each measurement.
The bent dimeric dopant CB7CB (synthon chemicals) is one of a series of dimers that have received much interest in the past few years as it exhibits a novel nematic phase, called the twist-bend nematic, N TB . 63,[66][67][68][69][70][71][72][73][74][75][76][77][78][79][80] In addition to this, it has been shown that when the dimer is mixed with standard calamitics the resultant mixture can also show N TB phases, as well as exhibit distinctive changes in elastic constants. [96][97][98][99] The transitions for the pure CB7CB were measured as Iso 116 1C N 102 1C N TB , in agreement with literature values. 63,67,90 CB7CB has been shown to have a net positive De, due to the dominant effect of the longitudinal dipole moment of the ''hairpin'' conformer over the transverse dipole of the ''banana'' conformer ( Fig. 3b). 70,71 For the EHDI work the mixture composition 60% MLC 2081 and 40% CB7CB (M 60 : 40 : 0 ) was chosen, since this was the composition with the highest concentration of CB7CB with the desired negative De over a wide nematic range (without the N TB phase). Compared to MLC 2081, M 60 : 40 : 0 showed only a slightly reduced T NI of 102 AE 2 1C and remained nematic until 20 1C. The similar nematic temperature range of the mixture components had the additional advantage that both T and T NI À T could be used for material comparisons. The materials were heated above T NI before any measurements were taken and typically all measurements of a particular type (optical or dielectric) were undertaken in a single cooling cycle. It was noted M 60 : 40 : 0 had sufficiently high conductivity to exhibit a EHDI conduction regime without the addition of TBATPB.

Dielectric studies
The dielectric and elastic properties of the materials were characterised as follows. The mixtures were filled into 10 mm spaced devices with bespoke guarded, circular electrodes for accurate permittivity measurements. We assume that there is no change to the elastic constants caused by the addition of TBATPB due to the low concentrations used. The polyimide SE1211 (Nissan Chemicals) was used to produce homeotropic (HT) alignment (pretilt B901), whereas either SE2170 or SE130 (Nissan chemicals) were used for planar homogeneous (PH) alignment, both of which provide low pretilts (B21). 102 Dielectric studies were performed using an Agilent E4980A LCR meter with coupled Linkam hotstage (TMS 95), to measure device capacitance and conductivity as functions of frequency, voltage and sample temperature. The signal was held at 50 kHz for a period of 60 s before each measurement, which was found to be sufficiently long for the capacitance to settle for all temperatures and voltages. This frequency was also found to be far from any relaxation processes and electrode effects. The measured capacitance C m and conductance G m values for the device includes the effect of the alignment layers. These were converted to measured permittivity components e m * using the 10 kHz empty cell capacitance, C 0 : 103 where o is the angular frequency of the applied signal. The empty cell measurement frequency was chosen to be as close to that of the filled cell measurements whilst still avoiding electrode effects. As |De| was found to be very small for some of the samples, it was important to account for systematic errors associated with the dielectric contribution of the alignment layers by including two series capacitances in the analysis: which gives to a first approximation an adjusted LC permittivity (e LC ) from the measured value of the device and reduced drop in potential across the LC layer (V LC ) compared to that applied to the entire device, Here, e LC , d LC and e AL , d AL are the permitivities and thicknesses of the LC material and alignment layers respectively, and A is the electrode area. The high voltage limit of e LC was determined using the technique of Clark et al., 104 and the e LC (V LC ) data was fitted using the method of Welford and Sambles, 105 to give e 8 , e > , k 11 and k 33 . Conductivities were found from frequency scans taken at a fixed voltage of 0.05 V using and a single value s was taken in the frequency-independent range (typically 1 kHz to 100 kHz, depending on material and temperature). It is assumed in eqn (9) that the alignment layers have sufficiently high resistances to not contribute to conductivity in this frequency range.

Devices
Several devices were used to assess the effects of elastic constants on NR diffraction gratings. These are listed in Table 1. Testing a range of devices allowed the effects of spacing, temperature, conductivity and permittivity on the V C and q C values of the NR mode be characterised. In all the devices where an NR mode was observed, the pitch q C was recorded using POM with a stage micrometre. Each frequency was set, and the applied voltage slowly increased from zero until the instability pattern was visible and the onset voltage V C

Soft Matter Paper
Open Access Article. Published on 07 October 2020. Downloaded on 11/12/2020 10:21:37 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.

View Article Online
was found. Within the NR regime, V C and q C were described by empirical fits: 31 and where f crit,V and f crit,q are critical frequencies at which the values of V C and q C are twice their low-frequency values.
The changes in grating pitch with frequency was confirmed through measurement of the diffraction angles as a function of frequency of the applied field. Results were taken using a HeNe 633 nm laser (Thorlabs) with the polarisation aligned parallel to the x-direction as shown in Fig. 2b, so that the incoming wavefronts experienced the periodic phase modulation. As in previous works, 31,37,38 m = 2 gave the highest intensity of the non-zero order diffraction peaks.

Material characterisation
MLC 2081 was found to have a negative De from T NI (measured 112 AE 2 1C) to room temperature. The temperature dependence of the measured e > , e 8 and e is shown in Fig. 4a, where, On filling, all samples of MLC 2081 with TBATPB (M 100 : 0 : 0.5/0.6/1.0 ) had a slightly higher values of T NI in the range 115 AE 3 1C and did not have significantly different measured permittivity values. 31 The mixture of MLC 2081 and bent dimer CB7CB (M 60 : 40 : 0 ) had a T NI = 102 AE 2 1C and De which changed sign from weakly positive between 102 1C and 63 1C, to weakly negative below 63 1C. This behaviour is due to the high gradient of permittivities versus temperature for the odd dimer mixture, due to the change in distribution of the conformational state from the positive De hairpin conformer to the negative De banana conformer. 66,69,79,80,106 The values of e > , e 8 and e iso of M 60 : 40 : 0 are shown in Fig. 4b, where statistical errors from repeated measurements are used (found to be AE0.5% for e > , e 8 , 1% for e iso ). To confirm the changing sign of De, the homeotropic (HT) M 60 : 40 : 0 sample was cooled from T NI + 10 1C with a 30 V-(10 kHz) voltage continuously applied across the device. Above 63 1C, the sample remained dark since the electric field supported homeotropic alignment where the material has a positive De, while below this temperature a Schlieren texture was observed, due to the material undergoing a Fréedericksz transition.
As MLC 2081 was negative throughout its nematic range, the elastic constants was obtained by fitting e(V) in the HT device at a variety of temperatures. Due to the change in sign De for dimer mixture M 60 : 40 : 0 at a crossover temperature of 63 1C, a HT device was used to determine elastic constants below and a PH (planar homogeneous) device was used above this temperature, respectively (S1 includes examples of the fitted data). The values of the elastic constants obtained are shown in Fig. 4c, where errors are taken from the combined error in values of |De| and V C . Here, as De became very small closer to the crossover temperature of 63 1C, only a limited region of the entire Fréedericksz transition was studied. This resulted in a considerably higher experimental uncertainties around this region (shown as hollow symbols). In Fig. 4c, both k 11 and k 33 of MLC 2081 and k 11 of M 60 : 40 : 0 are fitted to the square of the nematic order parameter estimated from the Maier-Saupe theory, calculated using: 107 where k 0 ii are the values of the elastic constant as temperature T tends to zero kelvin and T* is the temperature from which second order phase transition behaviour is calculated. The values U = 0.225 and T* = 1.023. T NI were found by fitting to numerical predictions of the Maier-Saupe theory. 108,109 (Table S1, ESI † gives fitting values). It is evident in Fig. 4c that the addition of CB7CB has decreased the magnitude of both k 11 and k 33 , and that k 11 /k 33 is greatly increased. The k 33 for M 60 : 40 : 0 does not follow the Maier-Saupe relation (eqn (13)) and is found to have the opposite temperature dependence of decreasing magnitude with increasing order parameter. Similar behaviour has been observed in other bent dimeric compounds and their mixtures with calamitic materials previously. 60,70,90 The origin of this is sometimes considered to be caused by stiffening of the alkyl spacer unit, which leads to the banana shaped conformer of the dimer molecules becoming favoured and hence dielectrically negative. Correspondingly, the banana-shaped conformer favours bend deformations thereby reducing k 33 . 66 Table 1 Devices used for measurements of V C and q C . The table lists the material, spacing, alignment layer used, whether the Normal roll mode EHDI is observed (Y = Yes, N = No) and the T NI at the point of testing. Devices 1 and 2 are pure MLC 2081 with different cell gaps; devices 3-5 have with varying amounts of TBATPB, and devices 6 and 7 have mixtures with 40% bent dimer CB7CB. The polyimides used are all low pretilt   a Device 4 has a higher T NI due to the increase in T NI over time and application of field. All other devices were tested immediately on filling.
As discussed in earlier sections, for an NLC to adopt the NR mode of EHDI, sufficient ion flow is required to produce a torque that destabilises the uniformly aligned director. 34,36,39,40 Fig. 4d shows plots of s as a function of temperature for pure MLC 2081 TBATPB and CB7CB doped samples (M 100 : 0 : X materials and M 60 : 40 : 0 ). It is clear from the data that the addition of the salt TBATPB dramatically increases the s values compared to pure MLC 2081 as intended, thereby enabling mixtures with similar conducitivies to be compared. As reported previously, the TBATPB appears to degrade HT alignment layer meaning values s 8 were not possible to find accurately. 31 It can be seen that in M 60 : 40 : 0 the addition of CB7CB increased the conductivity of the mixture compared to pure MLC 2081, this conductivity was sufficient in M 60 : 40 : 0 to undergo EHDI without the addition of TBATPB. It was also noted that above around 75 1C, the mixture M 60 : 40 : 0 changed sign Ds from positive to negative. This behaviour is interesting and may be due to the same change in ratio of hairpin and banana conformer that leads to the change in sign De at 63 1C and the anomalous temperature dependence for k 33 of the mixture.

EHDI characterisation
The NR mode EHDI behaviour of devices 3-7 was investigated in detail. The critical voltage and wave-vectors V C and q C were measured as functions of frequency for temperatures across the nematic range where De is negative. Although EHDI was observable in the dimer mixture M 60 : 40 : 0 while De was positive, the patterns were highly transient over time as found previously, 106 due to the competition between Fréedericksz reorientation and the formation of EHDI rolls. In this case, the situation is further complicated by the crossover in Ds. In the NR mode, the values of V C and q C in all samples behaved in the manner described by eqn (10) and (11). Here at a given frequency, f diel , the onset pattern (at V C ) would change from the NR mode into the chevron pattern (see Fig. 1c). As has been previously observed this transition appears to consistently occur in standard calamitics at a frequency where l = d/2, 31,50-53 leading to the limits of continuous variation given in eqn (4). A possible reason for such behaviour could be a similar effect to that described by Penz and Ford, 45 where due to a reduction in roll pitch a favourable hydrodynamic solution can be attained by partially overlaying two rolls above one another. Further work to confirm such a possibility within the structure of the standard model, is required. Fig. 2d shows the hypothesis leading to this paper, where by creating a bent dimeric mixture with unusually low elastic constant k 33 the rolls will become more ovular, allowing greater values q C than those given in eqn (4). To examine this, the devices were tested at several temperatures across their nematic range to observe the NR mode at a wide range of k 11 , k 33 and their ratio. Fig. 5 shows an example comparison of V C and q C values for the NR mode in devices 4 and 7, which are filled with the ion-doped M 100 : 0 : 0.5 and bent dimer doped M 60 : 40 : 0 respectively. V C and q C are shown at several temperatures as functions of frequency while maintaining the NR state. These two devices were chosen for comparison because of the similar spacing (B20 mm) and conductivity (B100 nS m À1 at 30 1C) yet they display significantly different values of q C and V C . The lines in the figure represent best fits of the data points to eqn (10) and (11), respectively. As the temperature is increased, the divergences shift to higher f crit values. This shift was observed previously and attributed to a higher number and mobility of the available charge carriers in the material at higher temperatures. 31,56 Comparing Fig. 5a, b and c, d, it is suggested that the manipulation of elastic constants caused by the addition of the CB7CB has led to significantly different values of V C and q C . From the examples given in Fig. 5 it can be seen that the dimer doped system has significantly lower V C values in the range 4 AE 2 V while the ion doped system is around 3 times this at 12 AE 2 V. Similarly the dimer doped mixture displays significantly larger maximum q C values as was predicted. This is considered in more detail in the Discussion section. Fig. 6 shows the plots of the largest (q max ) and smallest (q min ) values observed for an NR grating in each of the devices in Table 1. The data demonstrate the causal link between the changes in elastic constants and the observed q C values. Here each quantity is normalised to q 0 (= 2p/d) and the ratios plotted against the elastic constants and k 11 /k 33 . From these plots, it is clear that only the mixture including the bent dimer (M 60 : 40 : 0 ) is outside the limits given in eqn (4), with new limits of approximately,

Effect of elastic constants on EHDI pitch
The new limits indicate the possibility of wider angle steering in the mixtures containing the dimer material. To confirm this diffraction analysis of the devices 5 (no dimer sample) and 7 (with dimer) was carried out. Fig. 7 is the plot of the steering angle for the dominant n = 2 diffraction order. It is clear from the plot that in the dimer doped device both the angles (y min , y max ) and their range (Dy) have increased due to larger values of q C . Further, Fig. 6a and b show that lower k 33 values tend to exhibit the most considerable disagreement with eqn (4), while higher values are always inside the limits. Fig. 6c and d, show how the q C ranges change with, k 11 , where the effect is less clear, possibly due to the changes in q C being dominated by k 33 . Fig. 6e and f, show this data as functions of k 11 /k 33 , which seems to indicate a trend of higher q C values and range as k 11 /k 33 is increased as previously predicted. 31 From each of these graphs, it is apparent that the concentration of TBATPB does not change values of q C displayed, provided there are sufficient ions for the system to undergo NR mode EHDI. Similarly, the device spacing does not change the limits of q C in either the CB7CB or the TBATPB doped MLC 2081, once the results are normalised by q 0 .

Discussion
So far, we focussed on the effect of varying material elastic constants to increase the values of q C and reduce V C . However, the results also show that the addition of CB7CB also change the material's dielectric and resistive properties (e and s in Fig. 4), in addition to probable changes in unmeasured viscosity. To establish that the effects mentioned are due to change in elastic constants, let us consider the likely effect of these other factors on the behaviour of V C and q C , separately.
Behaviour of V C Firstly to assess the observed changes in V C consider the model of Kramer and Pesch, 44,47 where a 2 is the Leslie-Ericksson flow viscosity coefficient that is approximately the negative of the rotational viscosity g 1 , 107 and De eff and Ds eff , K eff , are the effective dielectric and conductivity anisotropies and elastic constant respectively. Although conductivity is included in eqn (15) it is unlikely that it is playing a dominant role in behaviour of V C for the dimer doped mixture. In the ion doping experiments (e.g. . In contrast to conductivity, dielectric anisotropy is likely to play an important role in the reduction of V C . This is due to the lower value of |De| in M 60 : 40 : 0 reducing the stabilising torque imposed by the applied electric field on the director compared to that of MLC 2081 (reducing the negativity of the first term in the denominator of eqn (15)). Finally, although difficult to conclude about the effect of viscosity without a direct measurement, the addition of the dimer should increase the mixture viscosity and hence increase V C . This, in addition to the fact that samples were examined at a large range of temperatures with fairly small changes in V C indicates the role of the strongly temperature dependent viscosity, is small. These observations lead us to conclude that the reduction in V C is likely dominated by the dual effects of lowering the elastic term k eff (mainly due to the reduction of k 33 ) and reduction of the negative |De|.
Behaviour of q C Similarly to the arguments for V C given above, it is unlikely that observed changes to q C range were caused by changing conductivity as this was not observed when examining different ion concentrations, with orders of magnitude variation in conductivity. 31 Unlike the effect on V C , the reduction of minimum feature size defined by q C is not caused by changes to De.  Table 1 respectively). Despite similar spacings, the device with CB7CB demonstrates considerably wider angle steering.
This has been shown in previous works, 29,35 where it had been noted that a smaller |De| is likely to decrease q C values. This may allow q C for M 60 : 40 : 0 shown in Fig. 6 to go below the lower limits of eqn (4), but can not lead to rolls that surpass the upper limit. This supports the argument that the observed large q C values seen in the dimer doped mixtures are dominated by the changes to the elastic properties. Samples without dimer were tested at many temperatures and ionic dopant levels, covering a wide range of values of s, e and viscosity but none of them showed q C 4 2q 0 .

Role of flexoelectricity
Another potential material parameter that can influence the formation of EHDIs is the flexoelectric coefficients. CB7CB has been reported to have forty times higher flexoelastic coefficient values compared to standard calamitics 108 so the mixture M 60 : 40 : 0 is likely also to have significantly increased flexoelectric coefficients (particularly e 3 ). The EHDI analysis by Tavener et al. 29 using the standard model does not include flexoelectric terms, and without such modelling methods, prediction of their effects on observed V C and q C values is difficult. However, it is unlikely that they are responsible for the observed change in the conduction regime, where such effects should be magnitudes smaller than the ionic effects. [83][84][85]109,110 Flexoelectric rolls are observed in both bent-core and odddimer liquid crystals. [111][112][113] However, it is unlikely the rolls investigated here are flexoelectric in origin since they depend strongly on sample conductivity, 81 have the normal parallel to the director, 82 and have a different frequency and voltage dependence of the onset behaviour. [114][115][116][117] All mixtures studied here follow the predictions for the NR mode, indicating these flexoelectric terms are unlikely playing a significant role in the observed behaviour. 58,61,[81][82][83] Conclusion This paper demonstrates the influence of LC elastic constants on the pitch of the grating structures in nematic EHDIs. New nematic mixtures are designed by adding an odd-spaced bent dimeric nematic liquid crystal to a typical host calamitic NLC to deliberately change the elastic properties while maintaining negative De. EHDI are induced in a variety of mixtures and a systematic study of the influence of material properties on the grating formation is provided. In our previous work, 31 it was predicted that a possible methodology of reducing l C values would be to create materials with significantly lower k 33 . The current work supports this hypothesis and demonstrates the ability of such low k 33 values to increase q C values to higher than 2q 0 , which a standard calamitic NLC cannot usually achieve. This work acknowledges that, although the change in k 33 is the principal candidate for the observed effects, there is also the possibility that changes in the material's viscosities, permittivities, conductivities or flexoelectric coefficients could also contribute to these changes. However, arguments are presented that suggest the change is dominated by the large reduction of k 33 associated with the odd-spaced dimer.
Although LC systems undergoing EHDI have been considered unsuited in their ability to be applied to real-world devices compared to systems such as LCoS, this work demonstrates their ability to create well-ordered grating structures with sub 4 mm features, which are comparable to the limits of LCoS reflective devices, with the additional advantages of operating in transmissive mode and being continually variable. We demonstrate that as the frequency is increased, in a 10 mm spaced device, l varies from 7 to 3.5 mm. This created a continuous angular variation from B9 to 211, although reduction in efficiency at higher steering angles was observed. 31 It should also be noted that the demonstrated technique of lowering k 33 could also have application in other types of grating formation based on a hydrodynamic phenomenon, or in increasing the stability of structures that have a similar desired director conformation.

Conflicts of interest
There are no conflicts to declare.