EPR study of the polydomain structure of the twist-bend nematic phase of CB9CB in the bulk

ABSTRACT EPR spin probe spectra of the liquid crystal phases exhibited, in the bulk, by the 1″,9″-bis(4-cyanobiphenyl-4’-yl)nonane (CB9CB) bent-shape dimer showed that, on cooling from the isotropic phase, this material forms a uniaxial nematic phase with a uniform director macroscopically aligned along the spectrometer field. Upon further cooling, a transition into the twist-bend nematic phase is observed after a biphasic region of approximately 4 K. In this lower temperature nematic phase, the director does not appear to be macroscopically aligned. The non-uniform director distribution is modelled as a collection of monodomains tilted with respect to the magnetic field and orientationally distributed around the tilt direction. Graphical Abstract


Introduction
The recent systematic synthesis and phase characterisation of several bent-shape bimesogens [1][2][3][4][5][6][7] has brought the total number of materials known to exhibit the twist-bend nematic phase, N tb , to well over 100 [1]. The N tb phase has been found most often in bentshape bimesogens composed of rigid mesogenic moieties joined by a flexible linker and only in a few rigid bent-core liquid crystals (LCs) [8,9]. This body of work confirms the previous suggestion [8,[10][11][12][13] that the main feature responsible of the stabilisation of the N tb phase is the overall molecular curvature.
The observed strong dependence of the stability of the N tb phase on the bend angle was predicted by generalised Maier−Saupe theories proposed by Greco et al. [14] for rigid bent-core bimesogens and Vanakaras et al. [15] for bimesogens with intramolecular torsion. More recently, Ferrarini extended [16] the previous model [14] to include a distribution of bend angles. The study showed that a broader distribution is expected to decrease the stability of the N tb phase, as supported by recent experimental evidence [7,17,18], and pointed out that the effects of electrostatic nature are not an essential ingredient to explain the existence of the N tb phase.
A generalisation of the work of Greco et al. [14], carried out by Tomczyk et al. [19], showed that the biaxiality of the two mesogenic moieties can affect the N tb phase stability, but the bend angle is still the determinant factor.
If the incidence of the N tb phase formed by bentshape bimesogens needs to be further clarified, the debate over the general structure of the phase has been settled by a recent article from Zhu et al. [20]. This was the first 'direct' measurement of nanometric helical pitch, previously revealed by freeze-fracture transmission electron microscopy measurements [21,22] and electroclinic effect measurements [23], and provided a more direct identification of the heliconical nematic structure, initially proposed by Cestari et al. [24], where the phase is macroscopically uniaxial with a local nematic director that makes a constant tilt angle θ 0 with respect to a macroscopic axis and twists around it in an helical fashion. In their work, Zhu et al. [20] used resonant soft X-ray scattering at the carbon K-edge to study the N tb phase of CB7CB in the bulk and found a spatial periodicity of approximately 8-10 nm of the molecular orientation without electron density modulation. Interestingly, the sample was not macroscopically aligned but appeared to be a collection of well ordered uniaxial domains of variable size and different orientation of the heliconical axis.
Very recently, a work by Stevenson et al. [25] used resonant 'hard' X-ray scattering at the selenium K-edge to investigate the N tb phase of the difluoroterphenyl dimer DTC5C7 in the bulk. The study found a spatial periodicity of approximately 9-12 nm of the molecular orientation without electron density modulation, further confirming the previous evidence in favour of the heliconical nematic structure. Moreover, the heliconical axis of the N tb phase was macroscopically aligned by a magnetic field of 1 T resulting in a hP 2 i of the axis that was estimated to be very close to unity. Owing to this alignment, the authors were able to obtain a molecular-level model of the N tb phase, including a determination of the tilt angle of the mesogens relative to the helix axis.
We recently proposed a model of an orientational distribution capable of describing the non-uniform alignment of the tilted local director with respect to the magnetic field observed in the EPR spectra of the N tb phase of the dimer 1″,9″-bis(4-cyanobiphenyl-4′-yl) undecane (CB11CB) [26]. From the analysis of the spectra of a suitable stable nitroxide free radical dissolved in the LC host, it was possible to recover molecular-level information about the local order in the mesophases, their reorientational dynamics, the tilt angle and the orientational distribution of the heliconical domains in the N tb phase.
In this article, the EPR spin probe technique [27][28][29] is employed to study the mesophases exhibited by the odd symmetric LC dimer 1″,9″-bis(4-cyanobiphenyl-4′-yl)nonane (CB9CB) to show that the above molecular-level information can also be consistently recovered for this homologue and to help establish the EPR technique as a tool that can provide very useful structural and dynamic information in a relatively straightforward and efficient manner that appears to be well-suited for a systematic study of the increasing number of liquid crystal dimers, oligomers, and possibly, polymers exhibiting the N tb phase.
The structure of the article is as follows. The next section reports the experimental details. The Theory section briefly describes the models used to analyse the EPR spectra. In the Results and Discussion section, the information recovered from the analyses are exposed, and finally, in the Conclusions section, a summary of the main findings is presented.

Experimental
The LC dimer CB9CB is formed by two cyanobiphenyl groups linked by an alkyl chain of nine carbon atoms, as first reported in [30]. It was synthesised in the Department of Chemistry at the University of Hull (UK) and was used without further purification. Its molecular structure is shown in Figure 1.
Upon cooling from the isotropic (I) phase, the CB9CB LC exhibits a transition to the ordinary, uniaxial nematic phase (N u ) and, on further cooling, to the N tb phase.
The phase sequence of this material was studied by Tripathi et al. [31] via high-resolution adiabatic scanning calorimetry at the low scanning rate of 0.15 K/h and it was N tb -377.22 K -N u -392.92 K -I. More recently, Robles-Hernández et al. [32], using modulated DSC at 1 K/min, found the following transition temperatures: Crystal -357.6 K -N tb -379.09 K -N u -394.92 K -I, where the N tb -N u and the N u -I transitions were approximately 2 K higher.
The nitroxide spin probe used for doping the CB9CB LC was the 3β-doxyl-5α-cholestane free radical (Aldrich, hereafter, referred to as CSL). This was employed in a number of previous studies [26,28,[33][34][35] where it proved to be a reliable probe to monitor the order and the dynamics of the LC system due to its size, morphology and rigidity, which results in a strong orientation by the LC host. The CSL structure is shown in Figure 2 together with the chosen ordering (x,y,z, solid line) and magnetic (x′,y′,z′, dashed line) molecular frames and the indication of its two main reorientational motions, tumbling and spinning, with the corresponding components of the rotational diffusion tensor: D ? (reorientation of the molecular long axis) and D k (rotation around the long axis), respectively.
CSL was added to CB9CB in the I phase at a typical concentration of approximately 1 Â 10 À4 g CSL =g CB9CB , which is below the limiting concentration typically suggested [27] to avoid Heisenberg spin exchange distortion effects.
Samples of CSL-doped CB9CB were inserted into glass capillaries of 1.8 mm internal diameter for the EPR measurements. EPR spectra were acquired with a Bruker EMX spectrometer equipped with an ER 041XG microwave X-band (9.5 GHz) Gunn Diode bridge and a rectangular ER 4102ST cavity. The samples were thermostated with a nitrogen flux through a variable temperature unit Bruker B-VT 2000. The temperature was monitored with a calibrated type T thermocouple (Comark Ltd.) kept in close proximity with the sample and showed a stability better than AE 0.05 K.
The molecular magnetic frame (x 0 ; y 0 ; z 0 ) was chosen according to the standard system of coordinates for the N-O paramagnetic moiety with the x 0 axis along the N-O bond [35,36] and the z 0 axis perpendicular to the five-membered ring, i.e. parallel to the p z orbital containing the unpaired electron. According to a standard approach, the z axis of the molecular ordering frame is considered parallel to the principal axis of inertia of the probe (its 'long axis'), and, to simplify the rotation which takes the ordering into the magnetic frame [35,36], the y axis is considered parallel to the z 0 axis. To reduce the correlation among variable parameters, the Euler angles, β and γ, between the molecular frames, were fixed in the fittings to 15 and 90 , respectively, in agreement with previous results obtained in related systems [28,[33][34][35][36].

Theory
The explicit form of a simple probability density distribution, Pðα; β d Þ, capable of describing the non-uniform alignment of the tilted local domain director with respect to the magnetic field observed in the EPR spectra of a bent-shape bimesogen in the N tb phase has been obtained in a previous work [26], where λ d is a positive constant and f ðα; β d Þ ¼ sec 2 θ 0 cos β d À cos α tan θ 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi cos 2 α sin 2 θ 0 À cos 2 β d þ cos 2 θ 0 q 2 1 þ cos 2 α tan 2 θ 0 ð Þ 2 : Here, β d is the angle between the local domain director and the magnetic field, θ 0 is the tilt angle and α is the azimuthal angle around the tilt direction whose distribution, for simplicity, has been assumed to be uniform. The EPR spectrum of the CSL spin probe in the N tb phase, I N tb ðω À ω 0 Þ, is modelled as a superposition of uniaxial nematic domain spectra, Iðω À ω 0 ; β d Þ where Iðω À ω 0 ; β d Þ is the unsaturated, high-field EPR spectrum at frequency ω, which is calculated using the classic Stochastic Liouville Equation approach of Freed and collaborators [27,37,38] which predicts that Figure 2. Molecular structure of the CSL spin probe together with the chosen ordering (x; y; z, solid line) and magnetic (x 0 ; y 0 ; z 0 , dashed line) molecular frames, the Euler angles, β and γ, between the molecular frames and the principal components, D ? (reorientation of the molecular long axis) and D k (rotation around the molecular long axis), of the rotational diffusion tensor.
in the assumption that the CSL spin probe reorients in a locally uniaxial nematic domain whose director, n, forms with the spectrometer uniform magnetic field an angle β d . The central frequency, ω 0 , at the spectrometer field, B 0 (0.339 T), is obtained from the g factor, g 0 , and the Bohr magneton, β e : ω 0 ¼ g 0 β e B 0 = h. L is the Liouville superoperator obtained from the orientation dependent spin Hamiltonian,Γ is the diffusion superoperator describing the reorientational motion of the probe, jvi is a vector containing spin transition moments averaged over the equilibrium ensemble and I is the identity. A brief account of the general approach of Freed and collaborators [27,37,38] has been presented in a previous work [28]. To reduce the total number of model parameters, the reorientational motion of the probe is assumed to take place in a uniaxial, mean field ordering potential where λ 20 is the strength of the potential, k the Boltzmann constant, T the temperature in Kelvin and P 2 ðcos βÞ is a second rank Legendre polynomial. The local order is described by the orientational order parameter, hP 2 i, defined as where β is the probe orientation with respect to the local domain director. The order of the local director with respect to the tilt direction is described by the orientational order parameter, hP 2 i d , which has been defined in the previous work [26]. A value of hP 2 i d ¼ 0 represents the limiting case of an isotropic distribution of the domain directors, whereas in the other limit of hP 2 i d ¼ 1, the local director is uniformly oriented along the tilt direction, i.e. at a fixed angle θ 0 .
The integration of equation 3 along both α and β d was carried out numerically using the adaptive multidimensional integration Fortran routine DCUHRE [39]. The EPR spectra simulation program employed was based on a set of Fortran routines implementing the slow tumbling theory for a spin probe reorienting in a LC, developed by Freed and collaborators [27,37,38], combined with a software package [40] that optimises the fit parameters using the Gauss −Newton−Marquardt non-linear least squares method [41].

Results and discussion
In previous studies, we have shown that CSL is a reliable probe that can accurately monitor the local molecular order, the reorientational dynamics and the local director distribution of various LCs and, in particular, of the 5CB LC [42,43]. A good agreement between the order parameter of the probe and the LC host was also reported by Oganesyan et al. via molecular dynamics simulations of the CSL probe in 5CB [44]. Unfortunately, despite the chemical similarity of the CB9CB dimer to 5CB it is not possible to assume that CSL will be also able to correctly report the molecular organisation of CB9CB. This ability depends essentially on the local chemical environment experienced by the probe in the LC host. Information about the chemical environment in which an EPR spin probe is located can be obtained from a rigidlimit spectrum. Such a spectrum is typically recorded at a temperature at which the sample is solid and molecular reorientations are frozen on the experimental time scale. In these conditions, the spectral lineshapes and the positions of the peaks are determined only by the hyperfine tensor which, in turn, depends on the polarity of the environment. A rigid-limit spectrum of CB9CB was recorded in the bulk at 173.2 K and is reported in Figure 3   chemical environment experienced by CSL in the CB9CB dimer is very similar to that in 5CB, therefore, it is reasonable to expect that CSL will be also a reliable probe of the molecular organisation of the dimer.
The phase behaviour exhibited by the CB9CB LC was studied by recording a series of EPR spectra, on cooling, starting at a temperature of 413.2 K and cooling to 343.2 K. In this temperature interval, the observed phase behaviour, discussed in detail below, was as follows: N tb -373.2 K -N u -389.2 K -I, with a downshift of the transition temperatures of approximately 4 K with respect to the values determined by Tripathi et al. [31]. Previous studies of the related LC CB7CB in presence of a small amount of a dopant [45,46] also found downshifts of the transition temperatures of a few K. The N u -I transition temperature was determined as the mean between the lowest temperature in the I phase (390.2 K) and the highest temperature in the N u phase (388.2 K). The N tb phase was found to coexist with the N u phase in the range 375.2-371.2 K. The centre of this interval was taken as the N tb -N u transition temperature, T N tb ÀN u .
The isotropic model was employed to fit in a consistent manner the EPR spectra recorded in the temperature interval 413.2-390.2 K (see Figure 4, black line: experimental, red line: fit). These are clearly fast motional spectra exhibiting three well-resolved hyperfine lines typical of an isotropic phase. The distance between the central and the outer peaks is quite constant across the phase and corresponds to 14.7 G (1 T = 10 4 G) at 413.2 K, which is very similar to the value of 14.6 G that we have found in the fast motional regime of the isotropic phase of 5CB in the bulk.
The distance between the central and the outer peaks showed a marked decrease at the temperature of 388.2 K, suggesting the occurrence of a phase transition. In the temperature interval 388.2-377.2 K, the lineshapes exhibited small and consistent changes expected for a uniaxial nematic phase forming a monodomain uniformly aligned along the magnetic field. As expected, spectra in this temperature interval could be consistently fitted to the uniaxial nematic monodomain model. EPR spectra of this uniaxial nematic phase are shown in Figure 5 (black line: experimental, red line: fit).
At temperatures lower than 377.2 K, it was no longer possible to obtain a consistent fit to the uniaxial nematic monodomain model. By a careful inspection of the spectra, the uniaxial nematic spectral contribution appeared to decrease while a new contribution was appearing, suggesting the onset of a new phase coexisting with the N u phase. A small biphasic region, across the N tb -N u phase transition, has been previously observed by Tripathi et al. [31] for pure CB9CB and its mixtures with 5CB and is consistent with the observed weakly first order nature of the N tb -N u transition. Previous works on the related LC CB7CB [45,46] also showed the presence of a biphasic region, induced by a small amount of a dopant.
Following these findings, spectra recorded at 377.2 K or below were fitted to a model formed by two spectral contributions. One contribution, due to the N u phase, was calculated according to the same uniaxial nematic monodomain model employed in the interval 388.2-377.2 K; the other contribution, due to the N tb phase, was calculated according to the distributed tilt (DT) model presented earlier [26].
The extent of the temperature interval of the biphasic region was initially roughly estimated by observing that in  spectra recorded below 371.2 K the signature lineshapes of the uniaxial nematic spectral contribution were no longer present and that spectra recorded below 371.2 down to 366.2 K appeared to be fairly similar, suggesting that the system was not experiencing further phase changes.
Spectra recorded in the temperature interval 377.2-366.2 K were then included in a global analysis to reduce the correlation among the increased number of fit parameters. This was performed by adopting a globalisation scheme based on the assumption that the order parameters are constant with temperature when the N u and the N tb phases are in equilibrium. Accordingly, the local order parameter, hP 2 i, of the N u and the N tb phases, the order parameter of the domain directors, hP 2 i d , and the tilt angle, θ 0 , of the N tb phase were considered as global parameters. Moreover, in a preliminary analysis, the local order of the N u and the N tb phases in equilibrium was assumed to be the same, therefore, the local hP 2 i was fitted to a single variable parameter. Given the increased complexity of a model formed by two spectral contributions, the choice of the initial values of the fit parameters (guess) was determinant to ensure a reliable and meaningful analysis. Guess value for the local hP 2 i of the N u and the N tb phases was 0.44 which corresponded to the best-fit value of hP 2 i of the spectrum recorded at 377.2 K analysed using the uniaxial nematic monodomain model. Guess value for the order parameter of the domain directors, hP 2 i d , was 0.5 which corresponded to the typical value of hP 2 i d found in the analyses of spectra of the N tb phase of the related LC CB11CB presented earlier [26]. Guess value for the tilt angle, θ 0 , was 1:0 according to the result of Robles-Hernández et al. [32], where θ 0 was zero at the onset of the N tb phase. Guess value for the molecular-level reorientational dynamics, represented by the perpendicular component of the rotational diffusion tensor, D ? , was 0.027 ns À1 which corresponded to the best-fit value of D ? of the spectrum recorded at 377.2 K analysed using the uniaxial nematic monodomain model. This value was used as guess for both the N u and the N tb phases across the temperature interval 377.2-366.2 K. Guess values for the fractional contribution of the spectral component due to the N tb phase, f tb , were obtained from a series of preliminary global analyses, carried on without optimisation, of the spectra recorded between 377.2 and 366.2 K. In these analyses, all other parameters were kept fixed while the f tb values, at the various temperatures, were systematically varied manually until a satisfactory visual fit was obtained. A further preliminary global analysis of the spectra recorded in the temperature interval 377.2-366.2 K, this time with optimisation, was then performed. Best-fit parameters showed a consistent temperature behaviour with an overall good quality of the fits (not shown). This analysis narrowed the range of the biphasic region by showing that f tb was essentially zero at 377.2 K, as expected, and was very close to unity at 368.2 K and below. Best-fit parameters of this global analysis were then used as guesses for a further global analysis, with optimisation, of spectra between 377.2 and 368.2 K. In this analysis, the local order parameters hP 2 i of the N u and the N tb phases were optimised independently. Best-fit parameters showed only small differences from those recovered from the previous analysis. In particular, f tb was very close to zero at 377.2 K and very close to unity at 368.2 K, further narrowing the biphasic region in the interval between 375.2 and 371.2 K. The hP 2 i of the N u phase was 0.43 and was similar, as expected, to the value of 0.39 obtained for the N tb phase. The hP 2 i d was 0.55 and the tilt angle, θ 0 , was 0:1 i.e. close to zero, as expected. The molecular-level reorientational dynamics, represented by the perpendicular component of the rotational diffusion tensor, D ? , of the N tb phase (shown below) was slower than that of the N u phase. This result is in agreement with our previous observation on the molecularlevel reorientational dynamics of the related LC CB11CB [26], however, arguably, is not sufficient, by itself, to explain the very large decrease of the fieldinduced director dynamics observed in the N tb phase of the CB11CB LC [47] and in the homologues CB9CB [48] and CB7CB [49]. This interesting point is further discussed below when the temperature dependence of D ? across the N u and N tb phases is presented.
The previous global analysis was then repeated using as guesses for the tilt angle, θ 0 , values ranging from 0:5 to 10 . All analyses gave very similar best-fit parameters, within 1% of the values presented above, assessing the robustness of the fit. EPR spectra in the biphasic region are shown in Figure 6 (black line: experimental, red line: fit). The temperature dependence of the fractional contribution of the spectral component due to the N tb phase, f tb , across the biphasic region of CB9CB, obtained from the above global fit, is presented in Figure 7.
Spectra in the N tb phase, below the biphasic region, in the temperature interval 368.2-343.2 K, were analysed with the distributed tilt (DT) model presented earlier [26]. A series of preliminary analyses showed a relatively large correlation between the local order parameter, hP 2 i, and the tilt angle, θ 0 . To limit this correlation, all the spectra in this temperature interval were included in a global analysis by assuming hP 2 i independent of the temperature and treated as a global parameter. This assumption appears to be reasonable according to previous studies [32,48,50] where the CB9CB hP 2 i exhibited a relatively small variation across the N tb phase. Guess values for the hP 2 i and the hP 2 i d were 0.39 and 0.55, respectively, obtained from the global fit in the biphasic region for the N tb phase. Guess values for the tilt angle, θ 0 , were taken from the Haller-type θ 0 temperature dependence obtained by Robles-Hernández et al. [32], with the zero of the Haller function shifted at 368.2 K, which corresponds to the first temperature recorded in the N tb phase below the biphasic region. Guess value for the perpendicular component of the rotational diffusion tensor, D ? , was taken as the best-fit value at the lowest temperature of the biphasic region.
Using this globalisation scheme, the overall quality of the fits was good and the recovered best-fit parameters showed a small correlation and a consistent and meaningful temperature dependence, with a best-fit hP 2 i of 0.61, in keeping with previous results [32,48,50], and a hP 2 i d of 0.50. Six EPR spectra (black line), between 363.2 and 348.2 K, and the corresponding fits to the DT model (red line) are shown in Figure 8.
The temperature dependence of the local hP 2 i across the full temperature range studied is presented in Figure 9, open squares.
The dependence of the molecular-level reorientational dynamics, represented by the perpendicular component of the rotational diffusion tensor, D ? , on the reduced temperature, T Ã ;T=T N tb ÀN u , across the N tb and N u phases, is shown in Figure 10 (squares), compared to the values obtained for the related LC CB11CB [26] (circles). In the N u phase, values for the CB9CB LC (open squares) were essentially identical to those for the CB11CB LC (open circles). Across the biphasic region, the reorientational dynamics of the N tb phase (solid symbols) was slower than that of the N u phase (open symbols). Finally, in the N tb phase, the reorientational dynamics of both LCs had values of the same order of magnitude and showed a further, almost linear decrease with the temperature. This significant reduction of the dynamics in the N tb phase at the molecular-level, which is related to an increase of the microscopic rotational viscosity of the phase, is nonetheless quite small when compared to the decrease of several orders of magnitude of the field-induced director dynamics, which corresponds to a very large increase of   [31] for pure CB9CB. In the biphasic region (series of five points centred on the N tb -N u transition), hP 2 i best-fit value was 0.43 for the N u phase (data points shown) and 0.39 for the N tb phase (not shown).
CB11CB [26], N tb CB11CB [26], N u   Figure 11. Dependence of the tilt angle, θ 0 , on the reduced temperature, T Ã ;T=T N tb ÀN u . The values were recovered from fits of the spectra recorded in the N tb phase, below the biphasic region, by adopting two different globalisation schemes (open and solid squares) and are compared to the values obtained for the related LC CB11CB [26] (open circles) and to the Hallertype θ 0 temperature dependence obtained for the CB9CB LC by Robles-Hernández et al. [32] (solid line) with the zero of the Haller function shifted at 363.2 K, which corresponds to the first temperature considered in the N tb phase, below the biphasic region. In the first globalisation scheme (open squares), the local hP 2 i was assumed to be constant across the N tb phase, below the biphasic region. In the second (solid squares), the local hP 2 i had the empirical Haller-type temperature dependence of equation 7 (T NI ¼ 389:2 K, hP 2 i 0 ¼ 0:97, β ¼ 0:21), shown as a solid line in Figure 9 (see text for details). the macroscopic rotational viscosity, observed in the N tb phase of the CB11CB LC [47] and in the homologues CB9CB [48] and CB7CB [49]. A very large increase of the macroscopic viscosity accompanied by only a relatively small increase of the microscopic viscosity suggests that, when the system goes from the N u to the N tb phase, the molecular scale environment experiences only a small change whereas, on a larger scale, the system undergoes a much larger transformation. The difference between microscopic and macroscopic viscosity in the N tb phase has been recently discussed by Cifelli et al. [49]. In this work, the authors measured the molecular translational self-diffusion coefficient in the N u and N tb phases of CB7CB via NMR diffusometry and the macroscopic rotational viscosity of the phases via field-induced director dynamics. The self-diffusion coefficient showed a decrease, with no discontinuity, on going from the N u to the N tb phase, whereas the macroscopic viscosity increased by several orders of magnitude.
The dependence of the tilt angle, θ 0 , on the reduced temperature in the N tb phase, obtained with the globalisation scheme described above, is presented in Figure 11 (open squares). Values obtained at the first two temperatures close to the biphasic region showed a relatively larger correlation between the local hP 2 i parameter and θ 0 and have been removed. To study the effect of this correlation, the previous global fit was repeated with a second globalisation scheme for hP 2 i whose temperature dependence was described by the following empirical Haller-type [51] function where T NI , the temperature of the observed N u -I transition, was fixed at 389.2 K whereas hP 2 i 0 and the exponent, β, were obtained by fitting the Haller equation 7 to the hP 2 i temperature dependence, in the range 389.2-343.2 K, presented in Figure 9 (open squares). Best-fit parameters were hP 2 i 0 ¼ 0:97 and β ¼ 0:21. The corresponding Haller function is shown as a solid line in Figure 9. The overall quality of the fits obtained from this second globalisation scheme was similar to the previous one. The recovered temperature dependence of θ 0 was only slightly different from the previous one and is also shown in Figure 11 (solid squares), compared to the values obtained for the related LC CB11CB [26] (open circles) and the Haller-type θ 0 temperature dependence obtained for the CB9CB LC by Robles-Hernández et al. [32] (solid line) with the zero of the Haller function shifted at 363.2 K, which corresponds to the first temperature considered in the N tb phase, below the biphasic region.

Conclusions
Upon cooling from the isotropic phase, EPR spectra of the CSL spin probe dissolved in the CB9CB liquid crystal confirmed the presence of a uniaxial nematic phase. Upon further cooling, the CB9CB liquid crystal exhibited a new phase that was no longer a monodomain. The EPR spectra clearly showed a non-uniform local director organisation that could be modelled as a collection of monodomains, tilted with respect to the magnetic field and orientationally distributed around the tilt direction. This result is compatible with the commonly accepted description of the twist-bend phase [20,24,25] in a bulk sample in which the heliconical axis is not uniform due e.g. to a relatively weak magnetic field, not able to fully align the local domains, resulting in the observed non uniform organisation of the local tilted directors. Between the two nematic phases, a biphasic region of approximately 4 K was found. EPR spectra in this region could be consistently analysed assuming a temperature independent value of the molecular order parameter, hP 2 i, of the order parameter of the domain directors, hP 2 i d , and of the tilt angle, θ 0 . At the lower end of the biphasic region, a discontinuity in the value of hP 2 i was observed. This result is in keeping with the first-order nature of the N tb -N u transition. The value of the hP 2 i recovered in the N tb phase is in line with earlier research for this system [32,48,50] whereas in the N u phase slightly lower values were recovered. The dependence of the reorientational dynamics on the reduced temperature in the N u phase is essentially identical to that exhibited by the related dimer CB11CB obtained in a previous work [26]. This result appears to be quite reasonable given the similarity of the two molecules and the fact that the temperature interval of the N u phase of the CB11CB LC has a width very similar to that of the CB9CB LC and is shifted by only approximately 5 K to higher temperatures. In the N tb phase, the dynamics slows down significantly, indicating a larger microscopic rotational viscosity compared to the N u phase, in agreement with our previous result [26]. Interestingly, this increase of the microscopic viscosity is quite small when compared to the increase of the macroscopic viscosity of several orders of magnitude observed in the N tb phase of the homologues CB11CB [47], CB9CB [48] and CB7CB [49]. This observation suggests that, when the system goes from the N u to the N tb phase, the macroscopic molecular organisation changes substantially, however, the molecular scale environment experiences only a small change.
Finally, the tilt angle is close to zero at the N tb -N u transition, in agreement with a previous result obtained for the CB9CB LC [32] and for the related dimer CB7CB [52,53]. Across the N tb phase, the tilt exhibits an almost linear temperature dependence which is clearly different from the Haller-type one determined previously [32]. In particular, the tilt reaches values in the range 25-30ᵒ which is in line with results obtained for the dimer CB7CB [52,53] and for the dimer DTC5C7 [25]. Values of the tilt angle in the range around 30ᵒ were also obtained in a previous work on the dimer CB11CB [26], but with a different dependence on the reduced temperature.