Temperature dependence of bend elastic constant in oblique helicoidal cholesterics

Elastic moduli of liquid crystals, known as Frank constants, are of quintessential importance for understanding fundamental properties of these materials and for the design of their applications. Although there are many methods to measure the Frank constants in the nematic phase, little is known about the elastic constants of the chiral version of the nematic, the so-called cholesteric liquid crystal, since the helicoidal structure of the cholesteric renders these methods inadequate. Here we present a technique to measure the bend modulus $K_{33}$ of cholesterics that is based on the electrically tunable reflection of light at an oblique helicoidal $Ch_{OH}$ cholesteric structure. $K_{33}$ is typically smaller than 0.6 pN, showing a non-monotonous temperature dependence with a slight increase near the transition to the twist-bend phase. $K_{33}$ depends strongly on the molecular composition. In particular, chiral mixtures that contain the flexible dimer 1'',7''-bis(4-cyanobiphenyl-4'-yl) heptane (CB7CB) and rod-like molecules such as pentylcyanobiphenyl (5CB) show a $K_{33}$ value that is 5 times smaller than $K_{33}$ of pure CB7CB or of mixtures of CB7CB with chiral dopants. Furthermore, $K_{33}$ in CB11CB doped with a chiral agent is noticeably smaller than $K_{33}$ in a similarly doped CB7CB which is explained by the longer flexible link in CB11CB. The proposed technique allows a direct in-situ determination of how the molecular composition, molecular structure and molecular chirality affect the elastic properties of chiral liquid crystals.


I. INTRODUCTION
Elastic constants of splay, twist and bend of liquid crystals define how these materials respond to external forces and boundary conditions [1,2]. There are many well-established methods of measuring elastic constants in the simplest type of liquid crystal, the so-called uniaxial nematic (N), see, for example, Refs. [3,4]. These methods typically use a monocrystalline sample in which the molecular orientation, specified by the director n ( 2ˆ, 1 n n n  − = ), is predesigned to be uniform in space. An external electric or magnetic field is applied to perturb this uniform orientation, and the elastic constants are deduced from the balance of the field strength and the elastic and surface anchoring forces that tend to preserve the initial alignment. These methods are hard to extend to the chiral type of the nematic phase, the cholesteric (Ch) phase, in which the director twists in space, remaining perpendicular to the helicoidal axis and thus forming a rightangle helicoid. The field response of this non-uniform ground state of the Ch phase involves complex structural reorganizations in which the director develops spatially varying twist and also deformations of splay and bend that are hard to separate from each other [1,2]. In absence of the direct measurements, it is usually assumed that the elastic properties of Ch phases are the same as those of their N counterparts. This assumption has never been tested. Thus, there is a clear need for a direct in-situ method to determine the elastic constants of Ch materials. Such a direct method is proposed in this work. It is applicable to Ch materials in which the external field creates a socalled oblique helicoidal structure (ChOH). By measuring the period of the structure as a function of the applied field, one deduces the bend modulus 33 K .
The existence of the ChOH state in a Ch acted upon by an electric or magnetic field has been envisaged theoretically a long time ago [5,6] and confirmed experimentally very recently [7][8][9][10][11].
The ChOH structure occurs in chiral mixtures based on dimeric materials with a small bend constant 33 K as compared to the twist modulus 22 K . Depending on the length of the methylene spacer, the smallest ratio 33 22 / KK for flexible dimers varies between 0.12 [12] and 0. 16 [4]. In the presence of an electric E [7,8] or magnetic [9] field H , the right-angle helicoid transforms into an oblique helicoid, where n is tilted with respect to the helicoidal axis by some cone angle 30    [7], thus forming the ChOH state, Fig.1. The director n in ChOH experiences twist and bend deformations.
The cone angle  and the period P of the structure decrease monotonously with the field increase, while preserving the single-harmonic periodic modulation of the director [7]. The pitch P depends not only on E but also on 33 K which allows us to use the dependence ( ) PE for a direct measurement of 33 K . The approach is illustrated for materials in which the application of an ac electric field E causes the ChOH period P to be in the submicron range; the value of P is then easy to determine by studying selective Bragg reflection of light at the one-dimensional periodic structure of ChOH. The value of 33 K is deduced by measuring the optical response to the varying electric field. The method does not imply any extrapolation of the data from the non-chiral N state and represents a direct in-situ measurement of the bend elastic constant of a chiral liquid crystal.
The proposed method is tested for two types of cholesterics, one formed by a singlecompound 1′′,7′′-bis(4-cyanobiphenyl-4′-yl) heptane (CB7CB) or 1′′,11′′bis(4-cyanobiphenyl-4′yl) undecane (CB11CB) with the flexible dimer molecules doped with a chiral dopant, and another representing a mixture containing a significant amount of rod-like molecules, pentylcyanobiphenyl (5CB), added to the flexible dimers. CB7CB doped with chiral additives yields a temperature behavior of 33 K in the ChOH phase that is very close to the temperature behavior of 33 K in the N phase of pure CB7CB. Namely, 33 K decreases to low values of the order of 0.4 pN near the transition to the twist-bend phase in both N and ChOH. CB11CB doped with a chiral dopant shows an even smaller minimum value, 33 0.2 K = pN. Especially intriguing is the result that 33 K in the chiral mixtures that contain rod-like molecules of 5CB added to the flexible dimers is reduced to 33 0.07 pN K = . 4

II. THEORETICAL MODEL
The peak wavelength Bragg  of Bragg reflection of light at a uniform ChOH structure is determined by the pitch P , The pitch P and the cone angle  are both tunable by the applied electric field E [7], is a critical field at which ChOH transforms into an unwound state, 0  = , 0 P is the equilibrium pitch of Ch in absence of the external field. Equation (4) suggests that 33 K can be determined from the dependence ( ) PE; the latter can be measured, for example, by using selective reflection of light.
Equations (4) and (5) have been derived assuming an ideal uniform ChOH. Such an ideal structure can exist in an infinitely thick sample, in which the surface anchoring of the director at the boundaries can be neglected. In cells of a finite thickness d , surface anchoring distorts the twist-bend director configuration and causes spatially-varying dielectric properties at the boundaries and redistribution of the electric field within the cell [10]. Selective reflection of light 5 in these cells is determined by the central bulk region, in which P ,  , and the acting electric field bulk E are coordinate-independent [10]. Thus, in Eqs. (4) and (5) Using Eqs. (1) and (2) and accounting for the finite cell thickness (6), the wavelength of the reflection peak Bragg  for the known applied electric field av E can be written as ( ) where ( )

III. EXPERIMENTAL METHODS AND MATERIALS
We studied four chiral mixtures. All contain flexible dimeric molecules that are known to form bend conformations and as a result, induce the so-called twist-bend nematic phase with nanoscale director modulation [13][14][15], the ChOH state [7,8] and a small 33 K in the nematic phase [4, 12,16]. All these mixtures feature a Ch phase and a chiral analog of the twist-bend nematic phase ( * TB N ) [13,14,17]. The sinusoidal ac field of frequency 3 kHz was applied using a DS345 waveform generator (Stanford Research) and 7602-M wideband amplifier (KROHN-HITE Co.). The electric field E induces the oblique helicoidal ChOH structure with its axis t along the field, ˆ| | tE . At a fixed temperature, the ChOH pitch P is tunable by the field in a wide spectral range including the visible part, Fig.7(a) [7,8].
The Bragg reflection was recorded using a tungsten halogen light source LS-1 and

B. Bragg reflection spectra of ChOH
The ChOH state forms under an applied electric field at temperatures above the * TB N -Ch phase transition. The Bragg reflection spectra are measured as a function of the electric field The reflection peaks were collected from well equilibrated (over 10 min) ChOH states.
An example of the field-controlled spectral properties of M4.2 is shown in Fig.7(a). These spectra are used to determine the wavelength

C. Bend elastic constant of ChOH
According to Eq. (10), 33 K is determined by the fitting parameter 1 a of the electric field dependence of the Bragg wavelength Bragg  and by the dielectric anisotropy   . Using the data collected for all four chiral mixtures, we determined the temperature dependencies of 33 K , Fig. 8 KT is slightly shifted down along the temperature axis in the pretransitional region, as compared to a nematic CB7CB, Fig. 8. The pretransitional increase in 33 () KT in materials such as CB7CB is usually associated with the appearance of pretransitional molecular clusters of the twist-bend phase; in these clusters, equidistance of one-dimensional director modulations hinders deformations of twist and bend of the twist-bend axis [13,16,23]. The chiral dopant might suppress the development of these clusters, thus reducing the pretransitional increase of 33 K in CB7CB:S811 as compared to that of pure CB7CB in Fig. 8 KT shows a similar non-monotonous behavior. A rather unexpected feature is that in these mixtures that contain 5CB rod-like molecules, the minimum value of 33 K is extremely low, about 0.07 pN, i.e., 5.4 times smaller than in CB7CB:S811 and 2.3 times smaller than in CB11CB:S811. The exact mechanism accounting for this difference is not clear. Tentatively, it may be associated with the geometry of the 5CB molecules that represent approximately one half of the CB7CB and CB11CB molecules. In the mixtures, 5CB molecules might serve as broken CB7CB/CB11CB dimers that facilitate the bend of the latter. Obviously, this argument cannot be extended to the pure nematic phase of 5CB, in which 33~1 0 pN K [24]. Finally, it is of interest to note that at 3C TB TT  +  , 33 K in a strongly chiral mixture M4.2 is slightly smaller than in the weakly chiral M1.8.

V. CONCLUSION
We propose a direct method of measuring the bend elastic modulus 33 K in chiral nematics. K shows a nonmonotonous dependence on the temperature near the transition to the chiral analog of the twist-bend nematic phase, with a pronounced minimum and an increase as the temperature is lowered. This feature is universal for both nematic and chiral nematics that show a transition into the twist-bend state and can be associated with the appearance of pretransitional molecular clusters of the twist-bend phase; in these clusters, equidistance of onedimensional director modulations hinders deformations of twist and bend of the twist-bend axis [13,16,23]. The dimer CB7CB doped with the chiral additive S811 exhibits a temperature behavior of 33 K that is very close to that in the nematic phase of pure CB7CB. The longer methylene-linked CB11CB dimer doped with S811 shows 33 K with the minimum value 2.4 times smaller than its counterpart CB7CB:S811, which might be related to a lower resistance to bending of the long methylene bridge in the CB11CB molecule. Our data on the bend constant 33 K in the mixture 19 CB11CB:S811 are significantly lower than 33 K measured previously by Balachandran et al [18] for pure CB11CB. In our case, 33 K is in the range (0.16 ÷ 0.55) pN, depending on temperature, Fig. 8. In contrast, Ref. [18] reports 33 K in the range (5.8 ÷ 10.5) pN, at least one order of magnitude higher. Note here that CB11CB shows the NTB phase, the existence of which requires a very small value of 33 K . In fact, the first measurements of the elastic constants in NTB-forming flexible dimeric materials [3,4,12,13,16] show that 33 K is below 1 pN within a range of about 5 -10 C  above the N-NTB phase transition. Our data for the CB11CB:S811 mixtures are in line with these previous measurements of 33 K in flexible dimeric materials. Rather surprisingly, we also observe a dramatic decrease of 33 K in the chiral mixtures that contain rod-like 5CB molecules added to flexible dimers; in these mixtures, the minimum value is 33 0.07 pN K = , which is 5.4 times smaller than the minimum value in CB7CB:S811. A tentative explanation is that the relatively short 5CB molecules serve as structural "fillers" that facilitate the bend of CB7CB/CB11CB dimers. The detailed mechanisms of how molecular composition, molecular structure and chirality influence the observed features of macroscopic elastic properties of the chiral nematic phases are not known and deserve further studies.