NMR Study of the Twist-Bend Nematic Phase

Jose Pedro Albuquerque de Carvalho1, ∗

1Department of Physics, and CeFEMA, Instituto Superior Tecnico,Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Liquid crystalline dimer systems have been extensively studied in recent years as candidatesfor having the Twist-Bend nematic phase (Ntb), in which the director is distributed in an obliquehelicoid, maintaining a constant oblique angle with the helix axis. The CBC9CB compound exhibitsthe uniaxial Nematic (Nu) phase at temperatures below the isotropic phase range and, at lowertemperatures, a periodically modulated nematic phase which, according to recent studies, seemsconcordant with the Ntb phase. In this work, we will characterize this molecular system in bothnematic phases using proton and deuterium NMR spectroscopy. Simulation of proton spectra fromthe terminal mesogenic groups along with quadrupolar splittings from the carbon-deuterium (CD)bonds in the chain positions adjacent to the mesogenic groups allowed for the determination of theSaupe order tensor for the mesogenic groups. We concluded that, in the Ntb phase, the mesogenicgroup’s nematic order parameter remains approximately constant. The change in the orientationof the most ordered molecular axis, leaving the symmetry plane defined by the aforementioned CDbonds in the Nu-Ntb phase transition, along the verification that the molecular biaxiality assumesa non null value in the Ntb phase are the main factors that account for the change in the molecularorientational order.Keywords: Nematic, Twist-Bend Nematic, Proton and Deuterium NMR spectroscopy, SaupeOrder Tensor

I. INTRODUCTION

So far, in the entire history of liquid Crystals, onlyfour distinct nematic ground states have been found, [1],for achiral molecular systems the uniaxial nematic (Nu)and the biaxial nematic; for chiral molecules the helicalnematic and the blue phases. In 1973, however, Meyersuggested the theoretical possibility that molecular inter-actions that favour bend deformations might lead to atwist-bend nematic phase, in which the director will havean helical distribution with a constant tilt angle between0 and π

2 [2]. This theoretical prediction sparked interestin banana-shaped molecules that may actually exhibitthis new twist-bend phase (Ntb). Molecular dimers atrelatively low temperatures can behave like these bentshaped mesogens and they could be candidates for thisphase. For instance, the liquid crystal dimer CBC7CB asof 1993 had already been reported to have a high tem-perature uniaxial nematic phase followed by a lower tem-perature smectic phase [3], which was odd since similarcompounds only exhibited an uniaxial nematic phase. Inthis way, the authors in [4] made a comprehensive studyof this compound in order to verify if this intermediatephase was actually the twist-bend nematic phase, and ar-rived at results that agree with this interpretation. Morerecently, Robles-Hernandez et al. [5] focused on the studyof the liquid crystal dimer system CBC9CB, which wasalso reported to have a nematic-nematic phase transi-tion [6] and is the compound that is going to be stud-ied in this work, and obtained very similar results thatalso agree with the existence of periodically modulatedstructures. Both papers demonstrate, through deuteriumNMR, that the two carbon-deuterium (CD) bonds at theterminal methylene groups of the linking chain loose their

∗Electronic address: jose.a.carvalho@tecnico.ulisboa.pt

equivalence when entering the Ntb phase. The authorsin [7] went even further and demonstrated that all themethylene groups in the linking chain of the CBC7CBcompound in the Ntb phase have non equivalent bonds,except for the central group because of its two-fold sym-metry.

The existence of the helical distribution in these dimersystem is, however, still under debate. Some authors,[8], argue that even though the helical distribution of thedirector is consistent with the experiments, there is no di-rect evidence of its existence. The authors allege that ifsuch a structure existed, then the spectrum, after a rota-tion of 90 about an axis perpendicular to the magneticfield, should be a superposition of peaks correspondingto the continuous helix. Instead what they obtained wasa typical spectrum of the perpendicular orientation rela-tive to the magnetic field of a uniformly oriented director.If we assume that the diffusion of the molecules is fastenough, then these global heli-conical structures couldstill be possible. However, they propose an alternativeinterpretation of the phase where the molecular pack-ing creates domains that assume predominately twistedand chiral conformations. This interpretation is also ex-panded in [9].

Regarding the potential applications of this phase, theauthors in [10], after studying a possible extension of thetwist-bend nematic model to incorporate electro-optic ef-fects in the CBC7CB compound, found that the directortilt had ultra fast response times (∼ 1µs) to an appliedelectric field. In the same year, there was also a studyin [11] of the optical response of the Ntb phase, where itwas reported that a linear optical response to an appliedelectric field was detected, with a switching time of 4µs,for periodic electric excitation. In this way, it is evidentthat this phase shows promise tit will have technologicalapplications, namely in the display industry.

2

A. Objectives and Outline

The primary objective objective of this thesis is tounderstand the molecular orientational order of theCBC9CB in the Nu and Ntb phase. We will achieve thisby being able to explain, in detail, the 1H NMR spec-tra and the quadrupolar splitting results reported in [8]by A. Hoffmann et al.. In order to achieve this objec-tive, we start by investigating the phase sequence of thiscompound through polarising optical microscopy (POM).Then we will proceed with an preliminary analysis of the1H NMR spectra that expands upon the work presentedand in [12]. In the final section, we will present a nu-merical model that is able to simulate both the 1H NMRspectra in conjunction with the deuterium splittings, pub-lished in [8], along with a discussion of its results and theirimplications on the molecular orientational order of thissystem.

II. THEORETICAL BACKGROUND

Matter presents itself in different states of aggregation,the more common ones being the crystalline solid (Cr),the isotropic liquid (Iso) and the gas state. The liquidcrystal (LC) state shares mechanical properties with theliquid state since it is a fluid phase, however this phasepossesses optical, electric and magnetic properties char-acteristic of crystalline solids. The molecules in all liq-uid crystal phases diffuse much like molecules in liquids,however, they maintain some degree of orientational or-der (nematic LC phases) and sometimes even positionalorder (smectic and columnar phases) [13].

The amount of orientational order in a LC is smallerthan that of a crystal and it stems from the fact thatthe molecules have a slight tendency to orient themselvesin a direction, defined by a pseudo vector n. This ori-entational order is only possible because molecules thatpossess these phases are anisometric. This orientationalorder causes some physical properties of a LC systemalong the direction of the director to be different fromdirections perpendicular to it.

A. Order in Nematic LCs

As previously mentioned the orientational order in liq-uid crystal is not nearly as perfect as in a solid thereforewe can define a nematic order parameter S that quantifiesthis degree of molecular alignment:

Isotropic Liquid (no alignment): S = 0Nematic: − 1

2 ≤ S ≤ 1Crystalline Solid (perfect alignment) S = 1.

Each molecule in a mesogenic compound can have theorientation of its most ordered axis defined by a set ofangular variables (θ, φ, ψ) as represented in figure 7c. Toobtain the nematic order parameter S we have to com-pute the ensemble average (represented by the 〈, 〉 sym-

FIG. 1: Representation of the angles that describe the orien-tation of a molecule in a nematic liquid crystal.

bols) of:

S ≡⟨

3

2cos2 θ − 1

2

⟩. (1)

The temperature dependence of this order parameter inuniaxial LCs not close to the critical temperature, TC ,where the transition from liquid crystal to isotropic liquidoccurs, can be approximated by the Haller Law: [14]

S = S0

(1− T

T†

)α, (2)

where the parameter S0 is a scaling factor and α is amaterial constant. The parameter T† is temperature atwhich the order parameter would reach zero if its contin-uous decrease were not aborted by the discontinuity dueto the first-order transition to the isotropic phase [14].It is noteworthy to point out that the temperature T† isvery close to the temperature TC .

The distribution of orientations in the system can bequantified by a distribution function f(θ, φ, ψ), knowingthis function it is possible to calculate S directly:

S =

∫Ω

(3

2cos2(θ)− 1

2

)f(θ, φ, ψ)dΩ. (3)

Orientational order plays a fundamental role in LCs inrelation between the tensorial molecular (mol) propertiesand the corresponding macroscopic (mac) tensorial prop-erties. Generalizing the concept of order parameter, wearrive at the Saupe order matrix,

Sijαβ =

⟨3

2cos θiα cosβjβ −

1

2δijδαβ

⟩, (4)

which allows that relation to be expressed as:

Amacij =

(2

3Sijαβ +

1

3δαβδij

)Amolαβ , (5)

where Amacij and Amolαβ are the tensorial components ofsome physical properties in the laboratory and molecularframes, respectively and i, j represent the axis in the lab-oratory frame and α, β the axis of the molecular frame.In the Super order matrix cos θiα are the direction cosinesrelating the molecular and laboratory frames. In the par-ticular case of NMR in a high magnetic field the quanti-ties that will determine the spectra only depend on the zz

3

components in the laboratory frame (where the magneticfield lies) of specific tensorial variables. In this contextthe Saupe order matrix, Szzαβ , becomes the relevant or-der variable. This tensor describes, in a molecule fixedframe, the order of the laboratory fixed frame’s frame zaxis to which the director is parallel in oriented samples.Note that Szzzz describes how the axis of greatest order isordered relative to the director, which coincides with ourdefinition of nematic order parameter. In its principalframe this matrix is diagonal and only dependent on Sand an additional parameter D = Szzxx − Szzyy that definesthe molecular biaxiality:

Szzαβ =

D−S2 0 00 −D+S

2 00 0 S

(6)

Since the orientation of this frame is usually unknown wemust consider a rotation from a molecular frame we de-fine in the molecule to this principal frame. This changeof frame is carried out through a congruent transforma-tion using the Euler angles (α, β, δ) defined according to[15]1. In this way the Euler angles also represent molec-ular parameters of the system.

B. Nuclear Magnetic Resonance

NMR is an experimental technique that allows us tomeasure the interaction of a system of nuclear spins, withintrinsic magnetic moment and angular momenta, withan external magnetic field, mediated by the absorptionan re-emission of electromagnetic radiation. This processdepends on the magnetic environment of the spins andsince this environment is dependent on the characteris-tics of the electronic cloud, measuring these interactionsand analysing the energy spectrum of the NMR Hamil-tonian allows us to probe the molecular structure of agiven sample which is the focus of this work. Addition-ally, though it will not be the focus of this work, theenergy transfer between spins and between spins and thelattice is associated with relaxation times, that can alsoprovide valuable insight regarding molecular dynamics.

In high field NMR, the Zeeman term, Hz, of theHamiltonian ends up being dominant while the remainingterms: the dipolar, Hd, the indirect dipolar interaction,Hj , the quadrupolar, Hq, and the chemical shift Hs inter-actions can be treated as a perturbation. In the contextof liquid crystals the molecules undergo rapid transla-tional and rotational diffusion motions which causes theperturbing Hamiltonian to be the result of an averag-ing process over these motions that causes interactionsbetween spin of different molecules to be averaged out.Additionally since some interactions are inversely propor-tional to the distance between the spins to the power ofthree which means nuclei that are far from each other

1 A more common definition of these angles would be φ, θ, ψ. Wechose to change notation to ensure that the parameters were notto be confused with the angular parameters that define the ori-entation of the molecule relative to the director.

have an almost neglectable interaction, allowing us toconsider small groups of spins and neglect interactionsbetween groups within each molecule. Considering theseapproximations and neglecting the indirect diplar inter-action term due to its small value for interacting protons,the Hamiltonian terms for a group of identical spins takethe form2:

Hz = −~N∑k=1

γkB0,z Ik,z (7)

Hs =∑k

γk~B0,z Ik,z[σisok +

⟨σsk,zz

⟩], (8)

Hq =∑k

3

4

eQk 〈Vk,zz〉Ik(2Ik − 1)

[(Ik,z)

2 − 1

3Ik(Ik + 1)

], (9)

Hd =

N∑k<l

−µ0

4πγkγl~2 〈Nkl,zz〉 ·

·[Ik,z Il,z −

1

4(Ik,+Il,− + Ik,−Il,+)

], (10)

where Ik is spin of the nucleus, Ik the spin angular mo-mentum operator, I+ = Ix + ıIy, I− = Ix − ıIy are theraising and lowering operators of angular momentum andeQk is the quadrupole moment of the nucleus k. In ad-

dition, 〈Nkl,zz〉 ≡⟨

3(nkl,z)2−1

r3kl

⟩, with nkl,z being the z

component of the unitary vector established between thenuclei k and l, and 〈Vk,zz〉 is the zz component of theelectric field gradient tensor at the nucleus site.

Considering that the director n is aligned along the zaxis of the laboratory frame, which is the case for LCsamples with positive magnetic suceptibility anisotropy,we can use the Saupe order matrix when spins k and lbelong to the same molecular segment, i.e.:

〈Nkl,zz〉 =∑α,β

(2

3Szzαβ +

1

3δαβ

)Nkl,αβ (11)

〈Vk,zz〉 =∑α,β

(2

3Szzαβ +

1

3δαβ

)Vk,αβ . (12)

If the spins k and l change their relative distance due to aconformation change, then expressions (11) and (12) canbe used in an approximate way by considering an aver-age molecular conformation to which a molecular frameis attached and the quantities Nkl,αβ and Vk,αβ in equa-tions (11) and (12) are replaced by their correspondingaverages over the spanned conformations. The quantitiesNkl,αβ and Vk,αβ can be related to their values in theirprincipal molecular frame through congruent transforma-tions. The quantity Vk,αβ has its principal axis definedalong the bond that connects an atom, in our case a car-bon atom C, to the deuterium, D. In this frame, Vk,A,B

2 For the simulation of the 1H NMR spectra we neglect the indirectdipolar interaction since its contribution is only of the order ofseveral hertz.

4

FIG. 2: Individual molecules make an angle θ with the directorn. βij defines the angle the inter-protonic axis makes with thethe most ordered axis, represented in red.

(A,B = x′, y′, z′) has the following form:

Vk,A,B = Vk,z′z′

η−12 0 00 −η+1

2 00 0 1

, (13)

η =Vk,x′x′ − Vk,y′y′

Vk,z′z′, (14)

with eQk

2π~ Vk,z′z′ = νq = 170kHz, which is a known molec-ular quantity. In the last of the equality we considereddeuterium nuclei in a sp3 orbital. Additionally, it is note-worthy to point out that η is not very large [16] and forthe CD bonds in the methylene group of our system weconsidered η = 0.

Finally, it is worth adding that if we are able to define amolecular frame as the principal frame of the Saupe ordertensor and since the molecular biaxiality D is, in general,much smaller than S it can, sometimes, be neglected weend up with the following equations for the line splittings:

∆νkl,d =3µ0

8π2γ2~

⟨1

r3kl

⟩S

[3

2cos2 βkl −

1

2

], (15)

∆νk,q =3

4π

e2qkQk~Ik(2Ik − 1)

S

[3

2cos2 βkl −

1

2

], (16)

with the angles βkl being the polar angles that defines theorientation of the inter-nuclear vector when the molecu-lar frame is written using spherical coordinates [17], seefigure 2.

Now that we have defined the high field Hamiltonianand how we relate the splittings with the orientationalorder of the system, it is possible to use first order timeindependent perturbation theory to calculate the NMRabsorption spectra3. The positions of the spectral linesin the spectra will correspond to transitions that obey theselection rule ∆m = 1, with m being the spin magnetic

3 The Fourier transform of the transverse complex magnetisationMx(t) + ıMy(t) following the application of a time oscillatingradio frequency field that causes the net magnetisation in thelaboratory frame to rotate π

2radians, gives the absorption spec-

tra of the nuclear spin Hamiltonian [18, p. 179].

quantum number. The area of each peak will be givenby:

Wmj→mj+1 =2π

~∣∣〈I, ...mj + 1...| I+

j |I, ...mj ..〉∣∣2

δ(∆E − ~ω1) (17)

provided that during the time scale τ of the perturbationthe spin populations change only a small amount and thepossible states between which absorption can occur mustbe spread over a range ∆E such that ∆E >> ~

τ [18].

III. EXPERIMENTAL INTERLUDE

The liquid crystal dimer α, ω − bis(4, 4′ −cyanobiphenyl) nonane), CBC9CB, studied in thiswork belongs to the CBCnCB series and is known topresent a periodically modulated nematic phase referredto as Ntb phase. It consists of two mesogenic groupslinked by a flexible chain. The mesogenic units arecyanobiphenyl (CB) groups and the flexible is odd alkylchain composed of nine carbon atoms [12]. The com-pound was synthesised in the Department of Chemistryat the University of Hull in the United Kingdom [6].

NMR spectroscopy studies were carried out in a BrukerAvance II 300 setup that generates a static magnetic fieldthrough a superconducting magnet. The setup is com-prised of an operator console, a console, a switcher/pre-amplifier and the magnet system that includes the super-conducting magnet, a probe head that houses the radio-frequency coil, the tunning and matching rf circuits andthe sample’s heating/cooling oven, and a shim systemlocated around probe head.

Lastly it is worth mentioning that all the experimentalwork was performed Laboratorio de Cristais Lıquidos eRessonancia Magnetica Nuclear of the Complex Fluids,NMR and Surfaces Group, Center of Physics and Engi-neering of Advanced Materials, CFNMRS-CeFEMA, atIST.

IV. EXPERIMENTAL RESULTS

A. Preliminary NMR Study

As a first approach to study the orientational order inboth nematic phases with 1H NMR we carried out ananalysis which improves on the one performed in [12].The acquired spectra, in a magnetic field of T = 7.04T,are represented in figure 3. The isotropic to uniaxial ne-matic phase transition occurred at 399.1 ± 0.8K4 whichis higher than the temperatures recorded in the Polar-ising Optical Microscopy (POM) studies (395 ∼ 396K)and those reported in [5]. This seems to indicates that

4 This error was estimated through a linear calibration with a ther-mocouple that was previously calibrated with the boiling andfreezing point of distilled water.

5

FIG. 3: Colour map representation of the results with in-terpolation between temperatures. Ntb-Nu phase transitionoccurred at T = (TC − 19.0) ± 0.1K, (TC = 399.1 ± 0.8K)according to the POM studies.

the magnetic field may induce some order in the systemwhich increases the phase transition temperature5.

Looking at the spectra in the figure 3 we can see thatthe overall shape does not change significantly and, there-fore, to study the evolution of the order parameter we fol-lowed a procedure identical to the one in [12] with someimprovements. The procedure consists of choosing a areference spectrum at a specific temperature and proceedto fit the other temperatures with a function G′(ω) de-fined as the pieciwise linear interpolation of the referencespectrum (G(ω)) but with its argument shifted by a fac-tor ω0, scaled by a factor A and its amplitude scaledby a factor B, that is G′(ω) = B · G(ω−ω0

A ). The fitswere carried out using the Levenberg-Marquardt mini-mization method [20] (implemented in [21]), disregardinga percentage of the lowest data, in order to suppress thecontribution of the tails and the central region of thespectra. Our interest lies in the parameter A, since thishypothesis assumes the spectra evolution with tempera-ture is due to the evolution of the product of the orderparameter S by a molecular dependent parameter withtemperature. A plot that depicts the evolution of this pa-rameter with temperature is represented in 4a. We findindications that the transition Nu-Ntb occurs in betweenT = (TC − 20.7) ± 0.8K and T = (TC − 16.3) ± 0.8Kwhere we noticed a discontinuity in the parameter evolu-tion. This transition has an intersection with the regionwhere our POM studies indicate that this transition oc-curs, represented in figure 4. Additionally this regionalso covers the transition temperatures indicated in [5],represented also in figure 4.

In the high temperature uniaxial nematic phase,theparameter A increases monotonously as temperature de-creases and its temperature dependence fits nicely to anHaller-type law, for S(T ), approximating T† = TC

6, see

5 High magnetic fields are known to have effects in the phase tran-sition of compounds that exhibit the Ntb phase [19].

6 We initially considered T† as a free parameter but it yielded atemperature smaller than TC , which is physically impossible. In

equation (2), with A(T ) given by:

A(T ) = K · S(T ) · P2[cosβ(T )]

P2(cosβr), (18)

where K is a proportionality factor, P2 is the second de-gree Legendre polynomial and β(T ) is the the angle rep-resented in 2 and βr is the value of β(T ) in the Nu phase.We assume that βr is temperature independent and,

therefore, the P2[cos β(T )]P2(cos βr) parcel can be dropped in the Nu

phase. In the Ntb phase, defining ∆β(T ) = β(T ) − βrto account for a temperature dependent additional tilt-ing beyond βr and using a zeroth order approximationwe end up with:

A(T ) ≈ K · S(T ) ·[

3

2cos(∆β)− 1

2

]. (19)

The paramter ∆β translates the occurrence of a signifi-cant change in the orientation of the most ordered molec-ular axis molecules which is associated with the onset ofthe Ntb phase [12]. Using an Haller type law to fit thehigh temperature region of the Nu phase we are able toobtain ∆β as a function of temperature. The results arerepresented in 4b. By further examination of figure 4b wecan conclude that this orientational change starts occur-ring before the Nu-Ntb phase transition, which explains adecrease in A(T ) before the transitions as one approachesthe Nu-Ntb from above. When the phase transition oc-curs the rate of change of this angle it varies discontinu-ously. This variation could be associated with a change inthe molecular packing that possibly occurs at the phasetransition.

V. MODEL AND NUMERICALIMPLEMENTATION

We next built a model and its numerical implementa-tion from more fundamental principles able to describethe 1H NMR spectra and, along with deuterium NMRdata available in the literature on the same compound,characterise the orientational order in the system. Themain provision of this model was to simulate the 1H NMRspectrum along with the quadrupolar splittings from aoriented monodomain liquid crystal sample, consideringthe molecular structure conformation and orientationalorder. In order to simulate the spectrum we must firstconstruct the Hamiltonian. Since we are in high fieldNMR the Zeeman term is dominant and the remain-ing terms can be treated as a perturbation. It is alsonoteworthy to point out that the quadruploar interac-tion has a null contribution for the proton absorptionspectra. The summations in the NMR Hamiltonian in-clude all the spins in all the molecules, however, due tofast self-diffusion the interaction between spins of differ-ent molecules are averaged out and only contribute tothe broadening of the resonance lines. Additionally the

this way we chose this approximation since TC is the lower boundof T†.

6

-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0

T-TC [K]

0.60

0.64

0.68

0.72

0.76

0.80

0.84

0.88

0.92

0.96

1.00

A

Nx

Nu

(a) Parameter A as function of temperature.

-56.0 -48.0 -40.0 -32.0 -24.0 -16.0

T-TC [K]

0.071

0.142

0.213

0.284

0.355

0.426

0.497

0.568

0.639

∆β

[rad]

(b) ∆β(T ), black dashed line corresponds to thelinear interpolation of the data.

FIG. 4: The blue dotted lines indicate the temperature atwhich Nu-Ntb occurred according to our POM studies. Theviolet dot dashed line indicates the last temperature in whichthe deuterium splittings published in [22] were equal and thered dashed lines indicate the highest and lowest transitiontemperature reported in [5], respectively.

state space for a spin group with N spins has dimensionof D = (2I + 1)N , with I = 1

2 . In this way, a scalabil-ity problem in the simulation also becomes obvious, theHamiltonian matrices have size 2N × 2N . To handle thisproblem we considered that our system is composed ofthree separate molecular groups (see figure 5) the firstone, represented in blue (B), corresponding to the meso-genic group defined by the aromatic rings and the firstmethylene group, the second one, represented in red (R),is the majority of the linking chain composed of the re-maining methylene groups expect for the central methy-lene which by itself defines a third molecular group, rep-resented in green (G). This assumption is not unmeritedbecause we do not expect that the linking chain shouldhave as much orientational order as the mesogenic body.Finally the central group also had to be differentiatedfrom the rest of the chain, since the CH bonds in thisgroup are NMR equivalent in both Nu and Ntb phases[7]. No interaction between protons of different groupswill be considered. It is also noteworthy to point out thatwe are only effectively simulating half of the molecule,because we assume the linking chain has enough internalmobility to ensure that the two halves of the moleculeare equivalent to NMR, as considered also by other au-thors studying dimer systems with NMR [4] [8] [22] andis suggested by the proton and deuterium data on these

FIG. 5: Ball and stick representation of the simulated sys-tem. The pink (P), blue (B), and green (G), correspond to themolecular groups defined in the simulation. The blue (xyz)and red (x′y′z′) corresponds do selected and principal molec-ular frames. In a orange shade we highlighted the deuteriumnuclei 1 and 2. Henceforth we will use this notation as asubscript to distinguish the different assignments. The rep-resented yOz plane defines the plane of symmetry of the CHbonds of the flexible chain.

systems. The molecular frame we considered is also rep-resented in figure 5 in which the z axis is aligned with thelongest molecular axis and the y axis is defined so thatthe yOz plane defines a symmetry of the carbon hydrogen(CH) bonds of the flexible chain. The x axis was simplydefined as to be perpendicular to the yOz plane.

In order to calculate the Hamiltonians we need to sup-ply the quantities

⟨V izz⟩

for the quadupolar, σisok and⟨σsk,zz

⟩for the chemical shift and

⟨3(nkl,z)2−1

r3kl

⟩for the

dipolar Hamiltonians, for each spin and spin-spin interac-tion. Regarding the chemical shift term, it is noteworthyto point out that this simulation was computed consid-ering the rotating frame which means, quantum mechan-ically, that the contribution of the Zeeman Hamiltonianin the Eigenvalue calculation is suppressed and thus thespectra will be centered around zero Hz instead of theLarmor frequency. However we have to consider a contri-bution of offset to the resonance. In this model, we con-sidered two different chemical shift coefficients as fittingparameters, one for the protons in aromatic groups, thatusually have a characteristic chemical shift contribution,and one for the protons in the methyl chain, which allhave the same hybridisation and hence we predict thattheir contribution to the chemical shift will be similar.This consideration was due to the fact that the chem-ical shift is associated with the movement of the elec-tronic cloud around the nuclei which induces local mag-netic fields that oppose the external magnetic field andare usually characteristic of certain functional groups ina molecule. By doing this approximation, we effectivelyinclude off resonance effect in these coefficients and con-

sider both terms σisok and⟨σsk,zz

⟩to be indistinguishable.

Regarding the quadrupolar interaction, we will use thequadrupolar splittings reported in [22] of a deuteratedmolecule (the protons in the first methylene group of thelinking chain were substituted by deuterons, see figure5). In order to calculate the splittings, we need to use acongruent transformation to change the principal frameof the electric field gradient tensor to molecular frame.

7

Subsequently we can calculate the splitting using the fol-lowing equation:

∆νk =3

2

∑k,αβ

(2

3Szzαβ +

1

3δαβ

)(eQ

2π~V mk,αβ

). (20)

Since the Saupe order tensor is dependent on S, D andthe angular parameters α, β, δ and we have two deuter-ated nuclei, we can establish a system of two equationsfor five unknowns:

∆ν1 = a11(α, β, δ, νq)S + a12(α, β, δ, νq)D, (21)

∆ν2 = a21(α, β, δ, νq)S + a22(α, β, δ, νq)D. (22)

We have chosen to solve for S and D since they are linearin these parameters. In this way, S and D can be de-termined from the quadrupolar splitting and the anglesα, β, δ. However we also expect that the biaxiliaty param-eter D will be much smaller than the S [23]. As a start-ing point we considered the angle δ that minimises themolecular biaxiality paramater D using the Brent routineexplained with detail in [20]. The remaining parametersα and β will be considered as fitting parameters. In thisway we are able to obtain all the the molecular parame-ters of the mesogenic molecular group. We will considerthat D = 0 and, for simplicity, δ = 0 (since they are cou-pled) for the remaining groups. Additionally, we assumeβ is the same for all molecular groups and that α is thesame for the flexible chain and mesogenic body while itsfixed at π

2 for the central CH2 group. This ensures thatthe CH bonds of this group never loose their NMR equiv-alence. The flexible chain and central CH2 will both haveindependent order parameters left as a fitting parameter.

Now that we have established how we take into ac-count, in our simulation, the quadrupolar splitting we aregoing to discuss the generation of the coefficient 〈Nkl,zz〉in order to calculate the dipolar Hamiltonian. The quan-

tity⟨

1r3kl

(3nkl ⊗ nkl − δαβ)⟩

can be easily calculated for

each pair of interacting spins in each molecular groupsimply using each protons position, determining the interproton normalised vector, nkl and computing the tensorproduct. In our model we only consider one average con-formation, however, we considered fast π flips of bothbenzene rings and also considered the orientation of eachone of them in the molecular planes is defined by two an-gles that are fitting parameters. The coupled π rotationswere obtained simply rotating both benzene rings alongthe axis defined by its bonds to the rest of the molecule byan angle θ. Taking into account spin flips means that wehave to consider two possible conformations of the aro-

matic rings, calculate the⟨

1r3kl

(3nkl ⊗ nkl − δαβ)⟩

tensor

for all pairs of each interacting spin and then calculatethe average tensor. Finally, similarly to the quadrupolarinteraction we have to change from the molecular frameto the laboratory frame by contracting this tensor withthe Saupe order tensor,⟨

3(nkl,z)2 − 1

r3kl

⟩=∑αβ

(2

3Szzαβ +

1

3δαβ

)1

r3kl

(3nkl ⊗ nkl − δαβ), (23)

the over bar indicates an average over the π flips of therings. Since in this analysis, we are considering that thismesogenic group adopts one preferred conformation, theSaupe order matrix relates to that conformation.

Now that all the necessary quantities for the differentparcels of the Hamiltonian were discussed, we are ableto build it. Subsequently, we have to compute its eigen-values and eigenvectors. In order to do so, we used theJacobi algorithm, described in [20].

After calculating the eigenvalues Ei and eigenvectors|ψ〉 of the Hamiltonian, proceed to the calculation of theabsorption spectra taking into account equation (17) in-dicating that the area of the peak corresponding to eachtransition is proportional to

Wij ∝ κ∣∣〈ψj | I+ |ψi〉

∣∣2 δ(∆ω − ~ω1), (24)

with κ being a proportionality constant and ∆ω =Ej−Ei

~with Ej and Ei being eigenvalues of the final and initialstate, respectively. Due to spin-spin relaxations each al-lowed transition in the spin system will contribute to theabsortion spectra with a shape function centered at thetransition frequency ω = ∆E

~ , with an area given by (24).The width of the shape function is parametrized by a linebroadening parameter LBij and the shape function is as-sumed to be Lorentzian. In this way, our simulated freeinduction decay is given by:

φ(t) = κ

N∑i=1

N∑i<j

Wije−[ı(

Ei−Ej~

)+LBij

]t, (25)

with κ being, once again, a proportionality constant andLBij the broadening coefficients.

Most of the LBij coefficients should be correlated and,therefore, we divided the spins in four different groupsand considered that each group has its broadening coef-ficient Ki

7. The groups considered were: one for thespins in the aromatic rings, one for the spins in the firstmethylene group, which in spite of having a chemical en-vironment different from the aromatic ring should haveits orientation strongly coupled with them, one for theremaining flexible chain and, finally, one for the centralmethylene group, whose CD bonds, as we previously men-tioned, do not loose their NMR equivalence which willcauses the orientational order of this group to be differ-ent than the rest of the chain. Each eigenstate is a linearcombination of produced states of up and down spins, inthis way to estimate the contribution of each spin flip to agiven transition we may calculate the following quantity:

Xkij =∣∣∣⟨ψj ∣∣∣I+

k

∣∣∣ψi⟩∣∣∣2 , (26)

where I+k = 11 ⊗ ...⊗ I+

n ⊗ ...⊗ 1N . Finally LBij simplybecomes

LBij =

N∑k=1

XkijKk∑Nk=1Xkij

, (27)

7 The dipolar interaction is dominant in the transverse relaxationof the signal and consequently the broadening of the transitions.

8

with Ki being the previously defined spin dependentbroadening coefficients.

After computing the signal, in order to produce thespectra we calculated the Fourier transform of the signal.We chose to use the Fast Fourier Transform algorithm,explained in detail in [20].

Lastly, regarding the fitting algorithm, the Leastsquare minimisation we used to run these simulations wasthe Levenberg-Marquardt, adapted to use the Eigen li-brary [24], discussed in [20]. It is also worth noting thatto match the intensities of both simulated and experi-mental spectra we multiplied our simulation by a linearfunction and left both slope and constant offset as fittingparameters. In this way, our model considered fourteendifferent fitting parameters.

To define the error of each data point in the spectra, weconsidered the standard deviation of the data correspond-ing to high absolute values of frequency in which thespectra has already decayed to zero. The error of the fit-ting parameters is estimated through the covariance ma-trix computed in the Levenberg-Marquardt method [25].However, since some parameters are obtained through thesplitting equations (21) and (22) (S and D) and througha minimisation routine (δ). The error of these parame-ters is obtained by calculating the standard deviation ofdistribution of solutions computed by generating Gaus-sian distributions of the necessary parameters requiredto compute them. Finally, regarding the splittings tran-scribed from [8], the error these parameters was approx-imated by the maximal reasonable deviation when digi-tising the data of the aforementioned reference.

VI. FITTING RESULTS

After introducing the numerical model that is able toexplain both the proton and deuterium NMR data, wewill focus on presenting and discussing the results we ob-tained. The model we developed has several fitting pa-rameters increasing the chances that the topology of χ2

as a function of the free parameters will include severallocal minimums. Since both the proton and deuteriumNMR results vary monotonously with temperature, forresults that had similar χ2 we favoured results that werelocally monotonous. Additionally, we do not have infor-mation regarding which CD bond corresponds to whichquadupolar splitting and so we obtained fits for both pos-sible assignments. In figure 6 we represented the dif-ference between the simulation and experimental resultsalong with a representative fit for the final solution ofthe assignment |νq,1| ≥ |νq,2| (see figure 5 for subscriptnotation). In figures 7a, 7b and 7c we represented theevolution of the molecular parameters with temperaturefor both assignments, |νq,1| ≥ |νq,2| and |νq,1| ≤ |νq,2|.Regarding the order parameter of the mesogenic group,it fluctuates between ∼ 0.44 and ∼ 0.47 with a maximumdeviation from its mean of ∼ 3% if |νq,1| ≥ |νq,2| and∼ 1.8% if |νq,1| ≤ |νq,2|. This means that in this temper-ature region, the evolution of this order parameter is notresponsible for the change of the profile of the 1H and 2HNMR results. If we compare our results with the resultsof the 5CB liquid crystal, which has a chemical formula

that is identical to the molecular segment we simulated,we see that the order parameter is relatively higher afterit stabilises in the Nu phase (∼ 0.5% [26]). This indicatesthat the shape of the CBC9CB induces packing difficul-ties that end up reducing the order. The order param-eter of the flexible chain has a profile which is also verysimilar between assignments and it is also relatively con-stant throughout the temperature range. This behaviouris similar to the order parameter of the mesogenic group,even though it is always lower. This decrease is to beexpected since the flexible chain should have more con-formation freedom but since the difference is so small, wehave reason to believe that this molecular group is stillsignificantly correlated with the mesogenic group. Finallythe order parameter of the central CH2 is also extremelysimilar between assignments and it decreases its order asthe temperature decreases.

The parameter D is also non null in the Ntb phasefor both assignments, which indicates that the moleculepresents some biaxiality. This result is concordant withthe hypothesis that the molecules pack themselves intwisted and bent domains. This parameter is null inthe Nu phase and then with decreasing temperature itincreases if |νq,1| ≥ |νq,2| and decreases if |νq,1| ≤ |νq,2|until it stabilises at approximately T = 363.2±0.8K. It isnoteworthy to point out that its evolution is symmetricwith respect to the axis defined by D = 0 which meansthat if we consider |νq,1| ≥ |νq,2| the director distributionwill more concentrated in the vicinity of the xOz planeof the principal frame while if νq,1| ≤ |νq,2| it would bemore concentrated in the vicinity of the yOz. Regard-ing the angular parameters, β exhibits an approximatelylinearly increase with decreasing temperature for both as-signments, while α and δ exhibit a symmetric evolutionbetween assignments. It is also worth mentioning thatthese parameters stabilise at about T = 365K remain-ing approximately constant at lower temperatures whileβ continues to increase as temperature decreases. Theseangular parameters together with D are the main mech-anism that causes the change of profile in the spectra.

In the previous chapter we mentioned the parameterδ was obtained minimising D. In order to ensure thatthis condition was not being too restrictive we relaxed itand fitted the experimental data in some representativetemperatures. The results we obtained matched the min-imums we had obtained before and did not resulted in asignificant decrease in the χ2/n.d.f. (n.d.f=number of ex-perimental points - number of fitting parameters), whichmeans our results withe D minimized are compatible withthe best χ2/n.d.f..

VII. CONCLUSIONS

In this work we presented a proton and deuteriumNMR study of the CBC9CB liquid crystal dimer system.We focused on the describing the molecular orientationalorder of this system in the Nu and Ntb phase.

In our preliminary analysis of the proton NMR spectrawe concluded that in the high temperature region of theNu phase, the horizontal scaling of the spectra fits wellan Haller-type law. However before the onset of the Ntb

9

(a) Colour map representation of difference betweenthe simulation results and experimental data.

Frequency [Hz] ×104

-2 -1 0 1 2 3

Inte

nsi

ty [

a.u

.]

×10-5

0

1

2

3

4

5

6

7

8

Final Result

Mesogenic Group

Flexible Chain

Central CH2

Experimental Data

(b) Representative fit at temperatureT = 358.8 ± 0.8K, in the Ntb phase.

FIG. 6: Simulation results for the assignment νq,1| ≥ |νq,2|.

phase the spectra is no longer modelled solely by this lawand we have to take into account the variation of angleβ, see figure 2. In a zeroth order order approximation weconcluded that the difference between β in the Nu phaseand at a given temperature in the Ntb phase increaseswith decreasing temperature. After the onset of the Ntbphase, this difference increases almost linearly with de-creasing temperature.

Using our numerical model we were able draw someconclusions regarding the molecular orientational order inthe Ntb phase and the Nu-Ntb phase transition. In thistemperature range we concluded that the order param-eter of the mesogenic group remained almost constant.This result is not contradictory with our previous inter-pretation since far away from the Nu to isotropic liquidphase transition the order parameter, according to anHaller type law, should exhibit little variation. The or-der parameter S of the flexible chain, without the centralCH2, which defines the second most important contri-bution to the spectra, has a very similar behaviour tothe order parameter of the mesogenic group, in spite ofbeing slightly lower. This indicates that these two molec-ular groups are still significantly correlated. We also con-cluded that the main mechanism that induces the changeof 1H NMR spectrum and the quadrupolar splittings isthe change in the orientation of the principal molecularframe, along with molecular biaxility parameter ceasing

Temperature [K]

340 345 350 355 360 365 370 375 380

Mole

cula

r P

aram

eter

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Szz - |νq,1| ≥ |νq,2|Szz - |νq,1| ≤ |νq,2|D - |νq,1| ≥ |νq,2|D - |νq,1| ≤ |νq,2|

(a) Molecular order parameters of the mesogenicgroup.

Temperature [K]

340 350 360 370 380 390

Angula

r P

aram

eter

[R

ad]

0

0.5

1

1.5

2

2.5

3

3.5 α - |νq,1| > |νq,2|α - |νq,1| < |νq,2|β - |νq,1| > |νq,2|β - |νq,1| < |νq,2|δ - |νq,1| > |νq,2|δ - |νq,1| < |νq,2|

(b) Order parameters of the different moleculargroups.

Temperature [K]

340 350 360 370 380

Ord

er P

aram

eter

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Szz,rij. - |νq,1| ≥ |νq,2|Szz,rij. - |νq,1| ≤ |νq,2|Szz,ch. - |νq,1| ≥ |νq,2|Szz,ch. - |νq,1| ≤ |νq,2|Szz,cent. - |νq,1| ≥ |νq,2|Szz,cent. - |νq,1| ≤ |νq,2|

(c) Angular parameters that define the principalmolecular frame.

FIG. 7: Representation of the simulation parameters, as afunction of the temperature, for each assignment.

to be null in the Ntb to Nu phase transition. The angu-lar parameter β, which describes the angle between theprincipal molecular frame’s z axis, to which the directoris, on average, parallel to, and the z axis of the molec-ular frame we defined, which is parallel to the para-axisof the benzene rings, increases almost linearly with de-creasing temperature. This behaviour matches the re-sults we obtained in our preliminary study, with an ap-proximate description of β. It is also noteworthy to pointout that in the Ntb phase the z axis of the principal

10

molecular frame leaves the plane of symmetry definedby the two CD bonds of the deuterated CH2 (see figure5) We believe that these results are in agreement witha collective molecular arrangement in twisted and bentchiral domains, in the Ntb phase, because this packingsuggests that the mesogenic groups are be packed in atilted-twisted configuration. This configuration is coher-ent with our results since the most ordered molecular axis,associated with the mesogenic group, establishes an an-gle β with the para-axis of the benzene rings and leavesthe plane of symmetry defined by the CD bonds. Ad-ditionally, this packing is compatible with the moleculesacquiring some biaxilaty (D <> 0) which is also in agree-ment with our results.

When comparing the results of both assignments weconclude that they were mutually compatible in almostall relevant parameters except in the angular parame-ters α and δ that define the orientation of the principalmolecular frame and in the molecular biaxiality param-eter D. These results, however, seem to exhibit a sym-metry between assignments, arising probably from thesimple quadrupolar splitting exchange between both as-signments.

As future work, it would be interesting to carry out

additional NMR studies that could provide uncorrelatedinformation to restrict the free parameters of our modelto further validate our results. Studies that could pro-vide this type of information would be double quantum1H experiments and the carbon 13 absorption spectra.In addition, it would also be interesting to use our modeland study the evolution of the orientational order in thelower temperatures of the Nu phase, since our preliminarystudy indicates that the orientation changes of the prin-cipal molecular frame start occurring in those tempera-tures, this provided we also could obtain the evolution ofthe quadrupolar splitting with temperature in that regionor other independent NMR data. Other issues still persistin this phase, since in the literature the elastic constantshave yet to be measured in the Ntb phase, a study ofthe magnetically induced Freedericksz transitions, in thisphase, would be very important. Additionally, it wouldalso be relevant to measure the dielectric permittivity inthese conditions where we can ensure that the directoris aligned with the magnetic field, as the measurementsreported in literature seem to overlook the director align-ment in the Ntb phase leading to incorrect values beingreported for the dielectric permitivities in this phase.

[1] D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai,U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser,J. E. Maclennan, and . Noel A. Clark, PHYSICAL RE-VIEW E 89, 022506 (2014).

[2] V. Borshch, Y.-K. Kim, J. Xiang, M. Gao, A. Jakli,V. Panov, J. Vij, C. Imrie, M. G. Tamba, G. H. Mehl,and O. D. Lavrentocich, Nature Communications (2013),10.1038/ncomms3635.

[3] P. J. Barnes, A. G. Douglass, S. K. Heeks, andG. R. Luckhurst, Liquid Crystals 13, 603 (1993),http://dx.doi.org/10.1080/02678299308026332 .

[4] M. Cestari, S. Diez-Berart, D. A. Dunmur, A. Ferrarini,M. R. de la Fuente, D. J. B. Jackson, D. O. Lopez,G. R. Luckhurst, M. A. Perez-Jubindo, R. M. Richardson,J. Salud, and H. Zimmermann, PHYSICAL REVIEW E84, 031704 (2011).

[5] B. Robles-Hernandez, N. Sebastian, M. R. de la Fuente,D. O. Lopez, S. Diez-Berart, J. Salud, M. B. Ros, D. A.Dunmur, G. R. Luckhurst, and B. A. Timimi, Phys. Rev.E 92, 062505 (2015).

[6] C. S. P. Tripathi, P. Losada-Perez, C. Glorieux,A. Kohlmeier, M.-G. Tamba, G. H. Mehl, and J. Leys,Phys. Rev. E 84, 041707 (2011).

[7] L. Beguin, J. W. Emsley, M. Lelli, A. Lesage, G. R.Luckhurst, B. A. Timimi, and H. Zimmermann, J. Phys.Chem. B, 116, 7940-7951 (2012), 10.1021/jp302705n.

[8] A. Hoffmann, A. G. Vanakaras, A. Kohlmeier, G. H.Mehl, and D. J. Photinos, Soft Matter 11, 850 (2015).

[9] J. W. Emsley, M. Lelli, H. Joy, M.-G. Tamba, and G. H.Mehl, Phys. Chem. Chem. Phys. 18, 9419 (2016).

[10] C. Meyer, G. R. Luckhurst, and I. Dozov, PHYSICALREVIEW LETTERS 111,067801 (2013).

[11] V. P. Panov, R. Balachandran, M. Nagaraj, J. K. Vij,and M. G. Tamba, APPLIED PHYSICS LETTERS 99,261903 (2011).

[12] H. M. M. Cachitas, Visco-Elastic Parameters of a Liq-uid Crystal with a Nematic-Nematic Transition, Master’sthesis, Instituto Superior Tecnico (2013).

[13] P. J. Collings, Liquid Crystals Nature’s Delicate Phase ofMatter (IOP Publishing Ltd, 1990).

[14] I. Haller, Progress in Solid State Chemistry 10, Part 2,103 (1975).

[15] D. Brink and G. Satchler, Angular Momentum, Oxfordscience publications (Clarendon Press, 1993).

[16] J. Emsley, Nuclear Magnetic Resonance of Liquid Crys-tals, Nato Science Series C: (Springer Netherlands, 2012).

[17] J. P. de Almeida Martins, Molecular dynamics study ofchromonics doped with salts, Master’s thesis, Instituto Su-perio Tecnico (2014).

[18] C. Slichter, Principles of Magnetic Resonance, SpringerSeries in Solid-State Sciences (Springer Berlin Heidelberg,1996).

[19] P. K. Challa, V. B. B. Parri, C. Imrie, S. N. Sprunt,J. T. Gleeson, O. D. Lavrentocich, and A. Jakli, PhysicalReview (2014).

[20] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery, Numerical Recipes in C The Art of ScientificComputing, edited by C. U. Press (1997).

[21] M. Galassi et al., “Gnu scientific library reference manual(3rd ed.),” ISBN 0954612078.

[22] A. G. Vanakaras and D. J. Photinos, Soft Matter 12, 2208(2016).

[23] H. Cachitas, A. Aluculesei, C. Cruz, J. L. Figueirinhas,F. Chavez, P. J. Sebastiao, M. G. Tamba, A. Kohlmeier,and G. H. Mehl, in ILCC (2014).

[24] G. Guennebaud, B. Jacob, et al., “Eigen v3,”http://eigen.tuxfamily.org (2010).

[25] F. James and M. Roos, Comput. Phys. Commun. 10, 343(1975).

[26] O. Mensio, R. C. Zamar, E. Anoardo, R. H. Acosta,and R. Y. Dong, The Journal of Chemical Physics 123,204911 (2005), http://dx.doi.org/10.1063/1.2121650 .