We report measurements of the bulk flexoelectric coefficient (*e*_{1} − *e*_{3}) of 5CB (4-Cyano-4^{′}-pentylbiphenyl), in the temperature range 20–34 °C, with a relative combined standard uncertainty of 2 %. The chiral flexoelectro-optic method was used with 1 wt % high-twisting-power chiral additive. At 25 °C, (*e*_{1} − *e*_{3}) = 7.10 pC/m with a combined standard uncertainty of 0.14 pC/m.

## I. INTRODUCTION

Flexoelectricity in nematic liquid crystals is a coupling between certain distortions of the average molecular orientation, described by the unit vector **n**, and polarization **P**_{d}.^{1} Two coefficients in the standard Oseen-Frank theory, *e*_{1} and *e*_{3}, characterize the strength of the coupling according to

(using the notation of Ref. 2). Distortion-induced polarization influences the energetics of distorted states, for example near topological defects.^{3,4} The converse effect—polarization-induced distortion—may be exploited in electro-optic devices such as display panels, providing a mechanism whereby the optical properties can be manipulated by an applied voltage.^{5–15} Values of the flexoelectric coefficients are required for inclusion in theoretical models, to test simulations,^{16–19} and to inform the development of devices. Thus, for fundamental and applied reasons, it is important to be able to measure the flexoelectric coefficients of a given material.

There are many reported measurements of *e*_{1} and *e*_{3}, or, more commonly, the combinations (*e*_{1} + *e*_{3}) and (*e*_{1} − *e*_{3}) (e.g., see Ref. 20 for a general review, or Refs. 21–31 for flexoelectric measurements of 5CB in particular). However, comprehensive results with specified uncertainty are still scarce, even for standard materials. There is no consensus regarding standard and accurate flexoelectric measurement techniques. Reported measurements have disagreed as to whether a given coefficient for a given material is positive or negative.^{32} Values have often disagreed considerably, recently by three orders of magnitude.^{33} Clearly, substantial sources of uncertainty exist in some measurement methods.

Rather than attempt to invent a new measurement technique—as has been done many times (e.g., Refs. 21 and 22 and 34–42)—we undertook a detailed study of the uncertainty in an existing one: the standard chiral flexoelectro-optic effect method.^{5,9,43–53} The combination (*e*_{1} − *e*_{3}) is determined, which is known as the bulk flexoelectric coefficient; it governs the flexoelectric response of a system for which the electric field is constant.^{20} The purposes of this paper are: (1) To establish firmly the chiral flexoelectro-optic effect as an accurate method for the measurement of (*e*_{1} − *e*_{3}), where random uncertainties and systematic effects have been analyzed and reduced. (2) To report values of (*e*_{1} − *e*_{3}) for the archetypal nematic material 5CB, at a range of temperatures, with low uncertainty.

## II. METHODS

An electric field applied perpendicular to the helicoidal axis of a short-pitch chiral nematic causes a flexoelectrically-induced deformation that results in a rotation of the optic axis.^{5} For small electric fields and small rotations, the angle ϕ is well-approximated by

where *E* is the applied electric field, *K*_{1} is the splay elastic coefficient, *K*_{3} is the bend elastic coefficient, and *q*_{0} ≡ 2π/*P*_{0} is the wave vector of the undistorted helicoidal pitch *P*_{0}. If ϕ, *E*, *K*_{1}, *K*_{3}, and *q*_{0} are measured, (*e*_{1} − *e*_{3}) may be determined.

The elastic coefficients and the pitch are measured separately, using standard techniques, as detailed below. The method used to measure ϕ for a given *E* has been described elsewhere (e.g., Ref. 50). To summarize: the chiral nematic is confined between two parallel glass sheets such that the optic axis lies in a unique orientation in the plane of the sheets. This is known as the uniform lying helix (ULH) configuration. ϕ is determined by observation of the transmitted intensity between crossed polarizers. The method has a number of favorable features: (1) There are no complications from surface polarization that have affected some previous methods.^{20} (2) The method does not depend crucially on the surface anchoring strength. (3) There are no complications from the effect of ionic screening since an ac signal may be used. (4) The dielectric response does not significantly contribute to the observations. (5) Both the sign and the magnitude of (*e*_{1} − *e*_{3}) can be determined.

The primary drawback of the method is that it requires the material to be intrinsically chiral, or else a chiral additive must be used. In the past, ≈10–30 wt % chiral additive was required and the properties of pure nematic materials could not be measured accurately. However, with the continued development of high-twisting-power chiral additives, it is now possible to create a sufficiently short pitch chiral nematic using only ≈1 wt %. One may intuitively expect that such a small percentage will not prevent an accurate inferred measurement of the pure state. This was shown to be true by Clarke *et al.*^{30} and Salter *et al.*^{51} who reported that chiral flexoelectro-optic measurements, using 1-3 wt % chiral additive, matched those of the pure material measured using a different technique. Clarke *et al.* reported agreement to within ∼10 %, while Salter *et al.* worked to uncertainties in the range 5–60 %. Here we show that the chiral flexoelectro-optic effect is now a viable method to produce measurements with an uncertainty of 2 %.

We used 5CB, whose molecular structure is shown in Fig. 1, supplied by Synthon Chemicals, and the chiral additive R-5011 (Merck KGaA), whose molecular structure is reported in Ref. 54. Where appropriate, we follow the procedures recommended in Ref. 55 regarding uncertainty analysis. Quoted uncertainties correspond to one standard deviation.

### A. Flexoelectro-optic measurement

The aim was to reduce random uncertainties in the flexoelectro-optic measurements to a level commensurate with the uncertainty in the elastic and pitch measurements (≈1 %), and to eliminate systematic effects at this level.

#### 1. Cell construction

Cells for flexoelectric tilt measurements were formed of two, 1 mm-thick, glass sheets with transparent indium tin oxide layers on the inner surfaces. These were stuck together using glue with dispersed spacer beads. The diameter of the spacer beads determined the cell gap. To increase the accuracy of the experiment, the glue pattern shown in Fig. 2 was used. This had two favorable features. First, it was possible to form the ULH without the need for alignment layers, thus removing any potential uncertainty from this source (Eq. (2) is valid in the bulk and ignores surface alignment); the 2 mm wide channel in the glue flow-induced the ULH when capillary filled with the material in its chiral nematic phase under the application of an electric field of 1 V/μm. Second, the glass is unsupported for only a short distance over the channel leading to a stable cell gap.

#### 2. Electric field

A 600 Hz square-wave signal was applied using a signal generator (TG1304, Thurlby Thandar), and measured using an oscilloscope (54600A, Hewlett Packard). This frequency was chosen because it is sufficiently high to make ion flow negligible, yet sufficiently low that a maximum-tilt response was observed for each sample. The random uncertainty on the measurement of the voltage *V* was ∼0.1 %: negligible in relation to other sources of uncertainty. It is believed that there were no significant systematic effects in the measurement of *V*.

To determine *E* from *V*, the cell gap *d* was measured by analyzing the interference in the spectrum of light transmitted through the empty cell before filling. Since there can be some variation in *d* over the area of a typical cell, care was taken to measure *d* at the mid-point of the channel, where the flexoelectric measurements were to be made subsequently. This was achieved using a spectrometer (USB2000, Ocean Optics) mounted on a polarizing microscope (BX60, Olympus). For four repeated measurements of a *d* = 20 μm cell, the experimental standard deviation was 0.04 μm. From this it was estimated that the relative standard uncertainty on a single measurement of *d* is 0.2 %. This is negligible compared to other sources of uncertainty and was thus ignored.

The variation of *d* with temperature (in the range 20–34 °C) was investigated and likewise found to be negligible (<0.4 %). A further systematic effect can arise since *d* will change upon filling the cell with liquid crystal, due to bending of the glass by capillary forces. We found the uncertainty was increased for cells that were made of thinner glass (which is more prone to bending) and/or that had larger regions unsupported by glue or spacer beads. It is believed that the uncertainty was reduced to an insignificant level by using 1 mm thick glass that was unsupported over a distance of 2 mm.

#### 3. Flexoelectric tilt

The temperature of the flexoelectric cell was regulated using a hot-stage (HFS91), and hot-stage controller (TMS93, Linkam). The hot-stage was mounted on a rotatable stage on a polarizing optical microscope (BH-2, Olympus). The transmitted intensity was observed using a photodetector (PDA36A-EC, Thorlabs) and oscilloscope (54600A, Hewlett Packard). The flexoelectric tilt was determined using a vernier scale on the rotating stage, accurate to the nearest 0.1°.

To reduce random uncertainties, ϕ was measured at seven different values of *E*. Typical experimental results are shown in Fig. 3. A linear relationship between tan ϕ and *E* is observed, as predicted by Eq. (2). Fitting a straight line (which was assumed to pass through the origin), the gradient provided a measurement of (*e*_{1} − *e*_{3})/[(*K*_{1} + *K*_{3})*q*_{0}].

Equation (2) assumes a bulk response and ignores surface/confinement effects which, if significant, would cause the comparison of data with Eq. (2) and the subsequent measured values of (*e*_{1} − *e*_{3}) to be invalid. Such effects should be revealed as a systematic variation in flexoelectric tilt with a variation in *d*. We repeated the tilt measurements with four different cell gaps (*d* = 5, 9, 20, and 43 μm). No systematic variation in the extracted value of (*e*_{1} − *e*_{3})/[(*K*_{1} + *K*_{3})*q*_{0}] was observed. These four measurements were used to provide our estimate on the uncertainty of the flexoelectro-optic measurements by statistical calculation of the mean and experimental standard deviation; the relative standard uncertainty on a single measurement of (*e*_{1} − *e*_{3})/[(*K*_{1} + *K*_{3})*q*_{0}] was 1.4 %. We used *d* = 20 μm for all subsequent flexoelectric measurements.

The measurement method should produce results that are, in theory, independent of the alignment quality of the ULH (though some degree of alignment is required for a useful signal to be observed, and a well-aligned ULH will result in a cleaner signal). To confirm this experimentally, the tilt of a given cell was measured after forming the ULH four times with varying uniformity. No systematic variation in the extracted value of (*e*_{1} − *e*_{3})/[(*K*_{1} + *K*_{3})*q*_{0}] was observed above the 1.4 % random noise. It is concluded that there is no significant uncertainty in our measurements from this source.

To investigate the possible uncertainty due to the addition of a small amount of high-twisting-power chiral additive, we carried out tilt and pitch measurements on five different mixtures with varying concentrations of chiral additive R-5011 (1, 1.25, 1.5, 1.75, and 2 wt %). The pitch was measured using the method discussed below. No systematic variation in the extracted values of (*e*_{1} − *e*_{3})/(*K*_{1} + *K*_{3}) was observed above the random noise. It is concluded that there is no significant systematic uncertainty in our measurements due to the chiral additive.

An uncertainty could potentially arise due to Eq. (2) itself. Its derivation relies on the assumption of spatially uniform ϕ, which is known to be unjustified if *K*_{1} ≠ *K*_{3}.^{56,57} We have carried out numerical calculations which can account for non-uniform ϕ, and for *K*_{1} ≠ *K*_{3} (in addition to nonzero dielectric anisotropy and non-uniform helicoidal twist).^{58} For the typical material parameters of 5CB, and relevant values of *E* (up to 0.4 V/μm), it was found that this, more refined, model predicts tilts that vary from Eq. (2) by <0.5 % (and the average ϕ for a given *E* varies much less). Thus, it is conlcluded that Eq. (2) is fit for purpose, and there is no significant uncertainty from this source.

### B. Elastic coefficients

Measurements of the elastic coefficients for 5CB have been widely reported (e.g., Refs. 59–63). We carried out our own measurements, however, to ensure the most direct comparison possible, and to produce an uncertainty estimate appropriate to our needs. *K*_{1} and *K*_{3} were measured using the standard capacitance technique on a planar aligned nematic cell undergoing a Fréedericksz transition.^{63–67} The capacitance was measured using a Precision Component Analyzer (6440A, Wayne Kerr Electronics) at a frequency of 600 Hz—the same frequency at which the flexoelectro-optic measurements were taken. The cells used here were 20 μm thick with planar alignment, supplied by Instec (LC2-20.0). The temperature was again regulated using a hot-stage (HFS91, Linkam), and hot-stage controller (TMS93, Linkam). The capacitance measurements were divided by that of the empty cell to give the effective relative dielectric permittivity. *K*_{1} and *K*_{3} were extracted by comparing this data with theoretically calculated permittivity curves (see Appendix for details). We employed an automated curve fitting program written in MATLAB which automatically computed—and provided the standard uncertainty on—*K*_{1} and *K*_{3}.

The standard uncertainties, as computed from the program, were ≈0.4 % on *K*_{1} and ≈1.0 % on *K*_{3}, producing a typical uncertainty on the combination (*K*_{1} + *K*_{3}) of ≈0.6 %. To check for systematic effects, we compared our values for (*K*_{1} + *K*_{3}) with those reported by Bradshaw *et al.* who measured *K*_{1} and *K*_{3} using two different methods.^{63} We interpolated and averaged their two data sets (Table I in Ref. 63) and found that the resultant values of (*K*_{1} + *K*_{3}) differ from ours by, on average, 0.8 %. This suggests that our measurements of (*K*_{1} + *K*_{3}) are relatively free from significant systematic effects, and our value of 0.6 % is indeed a reasonable estimate of our combined standard uncertainty.

### C. Pitch

The pitch was measured using a variation of the standard wedge cell technique.^{68–70} Repeated measurements using four different cells produced a random uncertainty estimate on a single measurement of 1.2 %. It is believed that systematic effects were reduced to insignificant levels in our experimental setup. The temperature was regulated using a hot-stage (LTS350, Linkam), and hot-stage controller (TMS93, Linkam).

### D. Sign of (*e*_{1} − *e*_{3})

The convention implicit in Eq. (2) is that positive ϕ corresponds to a rotation in the positive sense around an axis in the direction of *E*. Experimental observation of the sign of ϕ, for a given sign of *E*, showed that, in all cases, (*e*_{1} − *e*_{3})/[(*K*_{1} + *K*_{3})*q*_{0}] > 0. The chiral additive R-5011 is known to produce a right-handed helix,^{52} such that *P*_{0} > 0 and *q*_{0} > 0. Since *K*_{1}, *K*_{3} > 0, this implies that (*e*_{1} − *e*_{3}) > 0.

### E. Temperature

For each measurement (tilt, elastic coefficients, pitch), the clearing temperature *T*_{c} was determined for the given cell and measurements were taken at consistent reduced temperatures *t* ≡ *T*(K)/*T*_{c}(K). This reduces uncertainties due to absolute temperature variation across different cells, different hot-stages, and due to the introduction of chiral additive. The uncertainty on the reduced temperature, calculated using the hot-stage display readout, is negligible. To deduce temperatures in °C, the reduced temperature is combined with the the value *T*_{c} = 35.5 °C, reported by the supplier of the 5CB, which is assumed to be accurate to the nearest 0.1 °C.^{71}

## III. RESULTS

The measurement results are listed in Table I and plotted in Fig. 4. To infer the values of (*e*_{1} − *e*_{3}) at an arbitrary set of given temperatures in °C, a fourth-order polynomial fit was made to the data, as plotted in Fig. 4. For reference, the interpolated values at 20, 21, 22, …, 34 °C are provided in Table II. Given the probable existence of correlated systematic effects (which are generally known to be smaller than the random uncertainty on a single measurement, but otherwise unknown), as a first approximation, we retain the same relative uncertainty on the interpolated values as was found for the individual measurements.

. | $\frac{\left(e_1-e_3\right)}{q_0\left(K_1+K_3\right)}$ $e1\u2212e3q0K1+K3$
. | (K_{1} + K_{3})/2
. | q_{0}
. | $\frac{\left(e_1-e_3\right)}{\left(K_1+K_3\right)}$ $e1\u2212e3K1+K3$
. | (e_{1} − e_{3})
. |
---|---|---|---|---|---|

t
. | (× 10^{−2}μm/V)
. | (pN) . | (μm^{−1})
. | (× 10^{−1} V^{−1})
. | (pC/m) . |

0.950 | 6.54(9) | 8.66(6) | 7.55(8) | 4.94(9) | 8.55(17) |

0.955 | 6.49(9) | 8.23(5) | 7.56(8) | 4.90(9) | 8.08(16) |

0.960 | 6.50(9) | 7.80(5) | 7.57(8) | 4.92(9) | 7.68(15) |

0.965 | 6.48(9) | 7.35(5) | 7.57(8) | 4.91(9) | 7.22(14) |

0.970 | 6.44(9) | 6.86(4) | 7.58(8) | 4.88(9) | 6.70(13) |

0.975 | 6.47(9) | 6.35(4) | 7.59(8) | 4.91(9) | 6.24(12) |

0.980 | 6.51(9) | 5.81(4) | 7.60(8) | 4.94(9) | 5.75(11) |

0.985 | 6.43(9) | 5.23(3) | 7.61(8) | 4.89(9) | 5.12(10) |

0.990 | 6.44(9) | 4.57(3) | 7.61(8) | 4.90(9) | 4.47(9) |

0.995 | 6.36(9) | 3.76(2) | 7.62(8) | 4.85(9) | 3.64(7) |

. | $\frac{\left(e_1-e_3\right)}{q_0\left(K_1+K_3\right)}$ $e1\u2212e3q0K1+K3$
. | (K_{1} + K_{3})/2
. | q_{0}
. | $\frac{\left(e_1-e_3\right)}{\left(K_1+K_3\right)}$ $e1\u2212e3K1+K3$
. | (e_{1} − e_{3})
. |
---|---|---|---|---|---|

t
. | (× 10^{−2}μm/V)
. | (pN) . | (μm^{−1})
. | (× 10^{−1} V^{−1})
. | (pC/m) . |

0.950 | 6.54(9) | 8.66(6) | 7.55(8) | 4.94(9) | 8.55(17) |

0.955 | 6.49(9) | 8.23(5) | 7.56(8) | 4.90(9) | 8.08(16) |

0.960 | 6.50(9) | 7.80(5) | 7.57(8) | 4.92(9) | 7.68(15) |

0.965 | 6.48(9) | 7.35(5) | 7.57(8) | 4.91(9) | 7.22(14) |

0.970 | 6.44(9) | 6.86(4) | 7.58(8) | 4.88(9) | 6.70(13) |

0.975 | 6.47(9) | 6.35(4) | 7.59(8) | 4.91(9) | 6.24(12) |

0.980 | 6.51(9) | 5.81(4) | 7.60(8) | 4.94(9) | 5.75(11) |

0.985 | 6.43(9) | 5.23(3) | 7.61(8) | 4.89(9) | 5.12(10) |

0.990 | 6.44(9) | 4.57(3) | 7.61(8) | 4.90(9) | 4.47(9) |

0.995 | 6.36(9) | 3.76(2) | 7.62(8) | 4.85(9) | 3.64(7) |

T
. | (e_{1} − e_{3})
. | T
. | (e_{1} − e_{3})
. |
---|---|---|---|

(°C) . | (pC/m) . | (°C) . | (pC/m) . |

20 | 8.56(17) | 28 | 6.17(12) |

21 | 8.28(17) | 29 | 5.84(12) |

22 | 8.00(16) | 30 | 5.48(11) |

23 | 7.70(15) | 31 | 5.09(10) |

24 | 7.40(15) | 32 | 4.66(9) |

25 | 7.10(14) | 33 | 4.17(8) |

26 | 6.80(14) | 34 | 3.62(7) |

27 | 6.49(13) |

T
. | (e_{1} − e_{3})
. | T
. | (e_{1} − e_{3})
. |
---|---|---|---|

(°C) . | (pC/m) . | (°C) . | (pC/m) . |

20 | 8.56(17) | 28 | 6.17(12) |

21 | 8.28(17) | 29 | 5.84(12) |

22 | 8.00(16) | 30 | 5.48(11) |

23 | 7.70(15) | 31 | 5.09(10) |

24 | 7.40(15) | 32 | 4.66(9) |

25 | 7.10(14) | 33 | 4.17(8) |

26 | 6.80(14) | 34 | 3.62(7) |

27 | 6.49(13) |

## IV. CONCLUSIONS

We now attempt to compare our measurements with those previously reported for 5CB in the literature. A meaningful comparison would require previous measurements of (*e*_{3} − *e*_{3}) for 5CB, at a specified temperature, with a specified uncertainty. Unfortunately little relevant data exists. While there have been many reported measurements of the flexoelectric coefficients of 5CB,^{21–31} most report the combination (*e*_{1} + *e*_{3}). Few report the combination (*e*_{1} − *e*_{3}).

Link

*et al.*^{27}appear to report a measurement of (*e*_{1}−*e*_{3}) = 11 ± 1 pC/m, however no temperature is specified. Moreover, the assumption in Ref. 27 that Δε = 2 for 5CB—and dielectric effects may therefore be ignored in their experiment—is questionable.At a shifted temperature of

*T*−*T*_{c}= −5°C, Clarke*et al.*^{30}report (*e*_{1}−*e*_{3}) = 4.8 pC/m measured using the hybrid aligned nematic cell technique, and (*e*_{1}−*e*_{3}) = 4.2 pC/m measured using the flexoelectro-optic effect method. No further indication of uncertainty is provided.The measurement (

*e*_{1}−*e*_{3}) = 10.3 pC/m is reported in Ref. 72 at*t*= 0.9815. No uncertainty estimate is provided.The ratio (

*e*_{1}−*e*_{3})/*K*_{2}= 2.4 V^{−1}at*t*= 0.9844 is reported in Ref. 24 with an estimated uncertainty of 10 %. Here*K*_{2}is the twist elastic coefficient. To infer (*e*_{1}−*e*_{3}), a separately reported value of*K*_{2}is required. Using*K*_{2}= 3.1 pN at the same reduced temperature, extracted from Fig. (3) of Ref. 61 (which has a reported uncertainty estimate of 4 %), the inferred value is (*e*_{1}−*e*_{3}) = 7.4(8) pC/m. This may be compared against our interpolated value of (*e*_{1}−*e*_{3}) = 5.21(10) pC/m at the same reduced temperature. The values are discrepant.

Therefore, while our results generally agree to within a factor of two or so with previously reported values, it cannot be said that there is quantitative agreement. (Neither is the previous literature quantitatively consistent with itself.) We suggest this further motivates the report of our measurements herein.

The data presented in Table I indicates that the ratio (*e*_{1} − *e*_{3})/(*K*_{1} + *K*_{3}) is independent of temperature to within the experimental uncertainty of ≈2 %. This implies that the temperature dependence, and hence the order-parameter dependence, of (*e*_{1} − *e*_{3}) is the same as that of (*K*_{1} + *K*_{3}) to this level of uncertainty.

The combination (*e*_{1} − *e*_{3}) is appropriate for incorporation in theoretical calculations using the Oseen-Frank theory for situations in which the electric field is constant, or approximately so. In addition, it can provide the relevant coefficient in Landau-de Gennes theory in the standard one-flexoelectric-coefficient approximation (for which *e*_{1} = −*e*_{3}).^{4,73}

We believe our results are the most accurate reported for any flexoelectric coefficient (or combination thereof), for any material, using any technique. We conclude that the chiral flexoelectro-optic effect may be used to measure the bulk flexoelectric coefficient of nematic liquid crystals with low uncertainty. The results presented herein for the archetypal material 5CB may be useful for testing theoretical models and other measurement techniques.

## ACKNOWLEDGMENTS

This work was carried out under the COSMOS project which is funded by the Engineering and Physical Sciences Research Council UK (Grants No. EP/D04894X/1 and No. EP/H046658/1). S.M.M. acknowledges The Royal Society for financial support. F.C. thanks R. James for useful discussions.

### APPENDIX

The equations governing the director in the Fréedericksz transition experiment, from minimization of the Oseen-Frank free energy and a Maxwell equation (assuming a linear dielectric), were

where θ is the angle of the director with respect to the surface of the cell, ε_{0} is the permittivity of free space, ε_{∥} is the parallel component of the relative permittivity, ε_{⊥} is the perpendicular component, Δε ≡ ε_{∥} − ε_{⊥}, *u* is the electric field potential, and overdots denote differentiation with respect to the normalized position through the cell 0 ⩽ *z* ⩽ 1. The governing equations were solved subject to the weak-anchoring boundary conditions

where θ_{0} is pretilt and *w*_{s} is the surface anchoring strength. For given parameter values, the effective relative dielectric permittivity was determined from the director profile.

The effects of flexoelectricity in our Fréedericksz transition experiment^{35,74} were investigated and found to be negligible at the level of uncertainty to which we worked. Flexoelectricity was thus ignored in the final fitting method reported here.

The fitting (least squares non-linear) was found to be insensitive to plausible variations of *w*_{s} to within the required accuracy. *w*_{s}*d* was subsequently set as a fixed parameter with the reasonable value of 2.5 × 10^{−7} J/m. For a given temperature, ε_{⊥} was determined from the low-voltage capacitance data, and set as a fixed parameter. To determine the pretilt, {θ_{0}, ε_{∥}, *K*_{3}, *K*_{3}} were initially set as free parameters and fitting was carried out for the full set of reduced temperatures. From these ten fittings, it was determined that θ_{0} = 1.84(5)°. Subsequently, the data at all temperatures were refitted with θ_{0} set as a fixed parameter of value 1.84°, and with {ε_{∥}, *K*_{1}, *K*_{3}} free. Typical experimental data and a fitted curve are shown in Fig. 5, which illustrates the quality of the agreement between experiment and theory.

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*The flexoelectro-optic effect in cholesteric liquid crystals*, Ph.D. thesis,

*Evaluation of measurement data – Guide to the expression of uncertainty in measurement*, 1st ed. (

*Highly flexoelectric liquid crystals*, Ph.D. thesis,

*K*

_{2}= 3.3 pN was used (at a reduced temperature of ≈0.98 to be consistent with our other input values) from Ref. 61: although the essential result is not particularly sensitive to the exact value of

*K*

_{2}used.

*T*

_{c}as determined using a hot-stage and hot-stage controller with a Fréedericksz cell was 35.51 °C, which supports the supplier's value.