Mechanisms to Splay-Bend Nematic Phases
MMechanisms to Splay-Bend Nematic Phases
N. Chaturvedi ∗ and Randall D. Kamien † Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, 19104-6396, USA
While twist-bend nematic phases have been extensively studied, the experimental observation oftwo dimensional, oscillating splay-bend phases is recent. We consider two theoretical models thathave been used to explain the formation of twist-bend phases – flexoelectricity and bond orientationalorder – as mechanisms to induce splay-bend phases. Flexoelectricity is a viable mechanism, andsplay and bend flexoelectric couplings can lead to splay-bend phases with different modulations.We show that while bond orientational order circumvents the need for higher order terms in thefree energy, the important role of nematic symmetry and phase chirality rules it out as a basicmechanism.
I. INTRODUCTION
Liquid crystalline materials show a rich variety ofstructures and phases. Indeed even if we focus on thesmectic or cholesteric mesophases, there are a nearly un-limited variety of structures and motifs. On the otherhand, achiral nematic phases, the backbone of the dis-play industry, the workhorse of experiment, and the mostwell understood have only a few variants (it has not es-caped our attention that their simplicity is the key totheir value as devices). Indeed, only a handful of dis-tinct nematic phases have been found, and the spaceof possible configurations is highly restricted for achiralmolecules. It is well known that achiral rod-like and dis-cotic molecules form uniaxial nematics, and also biaxialnematics [1–3]. Over the past few decades, the study ofbent core molecules has led to the discovery of a nematicphase in which the director field of achiral molecules fol-lows an oblique helicoid, maintaining a constant obliqueangle with a helical axis [4–8]. The texture is splay-free, having only twist and bend distortions. This newphase, the twist-bend phase, has attracted attention dueto its unusual properties – a spontaneously chiral phaseis formed out of achiral molecules [9, 10]. Additionally,experiments show three times larger bend flexoelectriccoefficients in bent core molecules than the typical valuein rod-like liquid crystals [11, 12]. A schematic of thisphase is shown in Fig. 1.With this phase as the backdrop, it is natural to con-template additional nematic phases that show only twistand splay, or only splay and bend deformations. Inthis note we consider both bond orientational order andflexoelectricity as effects that can stabilize “splay-bend”phases, also shown in the schematic in Fig. 1. Althoughflexoelectricity has been considered before, we show thatdifferent forms of the splay and bend couplings can giveus two distinct splay-bend phases with different modu-lations [13, 14]. The paper is organized as follows. Insection II, we consider bond orientational order and findthat nematic symmetry and phase chirality make bond ∗ [email protected] † [email protected] (a) Twist-Bend (b) Splay-Bend FIG. 1: The figure shows a twist-bend and splay-bendstructure. the twist-bend structure has molecules rotatingabout the z direction while maintaining a constant anglewith it. In the splay-bend texture, molecules oscillate alongthe z direction in two dimensions. order an unlikely mechanism for splay-bend. Next, insection III we consider flexoelectric effects and look atthe splay and bend flexoelectric couplings that could giverise to splay-bend phases with different modulations. Insection IV, we look at the two different ‘splay’ phasesthat have been addressed in the literature, splay-bend[13] and splay nematic phases [14], and show that theseare related to each other by an exchange of the bend andsplay deformations.The mechanism behind the emergence of the twist-bend and splay-bend phases remains debated. Initialwork argued that a purely elastic instability, resultingfrom negative bend elastic constants, could explain the a r X i v : . [ c ond - m a t . s o f t ] A p r emergence of both these phases [13, 15, 16]. However,this leads to a free energy unbounded from below – higherorder and degree terms are necessary to find stable ex-trema. More recent theoretical work shows that a linearcoupling between polar order and the deformations ofthe nematic director can give effective elastic constants,which can then be driven negative with changing tem-perature [17, 18]. Bend flexoelectric couplings have beenused to explain twist-bend phases, and a combinationof both bend and splay flexoelectricity to explain splay-bend phases. Recent work shows that combinations offlexoelectricity and intrinsic chirality also predict yet un-seen, but related, modulated phases [19].Another mechanism that does not require higher or-der terms does exist for the twist-bend texture [20] butrequires chiral bond order: upon cooling, nematic liq-uid crystals can give rise to a liquid crystalline phasewith nematic order and hexatic order in the plane per-pendicular to it [21]. If the hexatic order is itself chiral,then the twist-bend texture is stable. Such a mechanismwould predict the emergence of twist-bend and splay-bend phases without the need for stabilizing arbitraryhigher order terms but pushes the problem on to find amechanism for spontaneous achiral symmetry breakingin the case of achiral molecules. II. BOND ORIENTATIONAL ORDER
Previous work shows how hexatic bond order in a chiralliquid crystal can give rise to a twist-bend phase, whilecircumventing the need for higher order terms [20]. Weconsider now whether this is a viable mechanism to in-duce the splay-bend phase. Consider a nematic systemwith bond orientational order in the plane perpendicularto the nematic director. For our purposes, it is sufficientto consider the general case without specifying the num-ber of nearest neighbors.The fluctuations in the nematic director, n , are givenby the Frank free energy density, f n = K n ( ∇ · n )] + K n · ( ∇ × n )] + K n · ∇ ) n ] (1)where K , K and K are the splay, twist and bend elasticconstants, respectively. Here and throughout we requirethat these elastic constants are positive. Apart from thecontributions to the free energy from modulations in thedirector field, we want to account for interactions be-tween the director and the bond angle. The bond angle,Φ, quantifies the bond order in the system. The defini-tion of Φ depends on the definition of the nematic directorfield [20]. In particular, it follows the nematic symmetry,and Φ → − Φ under the transformation n → − n . We ex-pect that the bond order contribution to the free energydensity has a term that penalizes any sharp changes in Φ, and a term that captures the interaction between Φand n .Since we require that the overall nematic symmetryis preserved in the free energy density, any term thatrepresents the interaction between the bond angle andthe nematic director must have an even power of Φ and n together. This means, for a term linear in ∇ Φ, theinteraction term must have an odd power in n .The twist-bend phase has a chiral structure, and so achiral interaction term is expected. In order to constructthe interaction term then, we want a vector with an oddnumber of derivatives to account for chirality, and anodd power of n to preserve nematic symmetry. The low-est order term that satisfies these constraints is n · ∇ Φ.Considering this term, f Φ = K A ∇ Φ) − K A q ( n · ∇ Φ) (2)where the full free energy density is f = f n + f Φ and f Φ is the contribution to the free energy density frombond orientational order. The total free energy can beminimized to determine the parameters of the phase andthe bond angle, Φ, as a function of the Frank constants,the pitch ( q ), and the bond-angle stiffness ( K A ). Sincethe interaction term is of lower order than the termsin the Frank elastic energy, the total free energy re-mains bounded from below. Indeed, extremizing overΦ we have ∇ Φ = q ∇ · n . For the twist-bend texture n tb = [cos( qz ) cos θ, sin( qz ) cos θ, sin θ ] and we can onlyhave ∇ Φ = v ˆ z , a constant vector along the z -axis. Min-imizing over the value of v and integrating over a periodgenerates a term [20] f Φ = − K A q θ (3)and the bond order acts as a magnetic aligning field asstudied half a century ago by R.B. Meyer [22], stabilizingthe texture.Following a similar argument, we might consider thepossibility that bond orientational order is also a mech-anism for the formation of splay-bend phases. For splay-bend, we use the ansatz , n = (cid:110) sin [ θ sin( qz )] , , cos [ θ sin( qz )] (cid:111) (4)This describes an oscillating, two-dimensional structurethat alternates between regions of splay and bend defor-mations. Here, q is the pitch of the phase and θ is themaximum angle to which the molecules tilt. The direc-tion of modulation here is parallel to the average nematicdirector field as shown in Fig. 2.Using this to calculate the splay and bend free energydensity contributions, and averaging over a period π/q ,we find a free energy density¯ f n = K q (cid:104) θ − θJ (2 θ ) (cid:105) + K q (cid:104) θ + θJ (2 θ ) (cid:105) (5) (a) Splay-bend phase inEq. (4) (b) Splay-bend phase inEq. (7) FIG. 2: Schematics of the splay-bend phase in the ansatz inEqs. (4) and (7) showing the maximum angle θ . Themodulations are parallel to the average nematic directordirection in the left schematic, and perpendicular to it in theright. Here, J ν ( z ) are Bessel functions of the first kind. Usingthe properties of Bessel functions it is straightforward tocheck that when both K and K are positive, ¯ f n ≥ θ = 0, the uniaxial na-matic. Since the splay-bend phase is achiral an achiralcoupling is necessary, ˜ f Φ . The symmetries that the newterm must have are as follows: continuing to require thatthe nematic symmetry, n → − n , is preserved, the inter-action term must have an even power of Φ and n together.Further, since the texture is achiral, we assume that theinteraction term must also be achiral and thus even inderivatives of fields. Thus a term linear in ∇ Φ, requiresa vector with an odd order of derivatives, and an oddpower of n .We may then list our the possibilities for the lowestorder term: one could consider interactions that involvethe splay vector, n ( ∇ · n ), but these do not follow thenematic symmetry. The same is true for interactions thatinvolve the bend vector, n × ( ∇ × n ).One possibility that has the required symmetriesis ∇ Φ · ( ∇ × n ). In this case, the extremal equa-tion for Φ is again ∇ Φ = 0. Since ∇ × n =[ qθ cos( qz ) sin ( θ sin( qz )) , , ∇ Φ · ∇ × n (cid:54) = 0 then Φ must depend on x . There isthe solution linear in x which, when inserted and aver-aged over a z period results in no coupling between thebond order and the director. Other solutions are of theform cosh( αx i ) cos( αx j ) where i (cid:54) = j and α is a constant.Since we would need ∂ x Φ (cid:54) = 0, the only possible termthat would not vanish upon spatial averaging would beof the form Φ = cosh( αx ) cos( αz ) (up to translations).Unfortunately, while surviving the z -averaging, a solu-tion like this would lead to an unbounded free energydensity. Whether it is possible to have defect walls be-tween regions of bounded ∇ Φ is the topic of future work.Finally, were we to consider an interaction higher or- der in derivatives than either the bend or splay vectors,we would generate an odd power of q higher than 2 inthe free energy integrated over one pitch, requiring evenhigher order terms to assure stability. Since that wasthe raison d’ˆetre for considering this mechanism, we con-clude that there are then no interaction terms that havethe appropriate symmetries, and a low enough order togive a non-trivial minimum for q and θ .We conclude then, that bond orientational order is nota simple mechanism that can give splay-bend phases. Inorder to get a splay-bend phase, a vector field, like thepolarization vector, P , is required [17]. Such a field playsthe part of a vector that need not follow the nematicsymmetry. Several of the interaction terms that are notavailable to us with the bond angle are then permittedby symmetry. III. FLEXOELECTRICITY
Recall that the flexoelectric effect is a linear couplingbetween a polarization vector and director deformations.A coupling may be constructed with either the splay orbend vectors that, in turn, gives rise to an effective nega-tive K or K , respectively [17]. Such a coupling inducesspontaneous splay or bend in the system. Previous workhas shown how a negative effective K can lead to boththe twist-bend and splay-bend phases [13]. Similarly, anegative effective value of K has been used to explainthe observation of the splay nematic phase [14].We look at both of these couplings independently. Weconsider the following ansatz for the polarization vector, P and nematic director field n [14], n = (cid:110) sin [ θ sin( qx )] , , cos [ θ sin( qx )] (cid:111) (6) P = n p cos qx (7)This ansatz is different from the one in Eq. (4), andthe direction of modulation is perpendicular to the av-erage direction of the nematic director field, as is shownin Fig. 2. We will show in the next section how thetwo splay-bend systems can be mapped on to each other.This form for the polarization, P , breaks the nematicup/down symmetry. When averaging over the sample,modulations that are at different wavelengths than 2 π/q will vanish and so we pick the dipole modulation accord-ingly. For a splay flexoelectric coupling, the free energydensity is f splay = f n − γ P · [ n ( ∇ · n )] + b ∇ P ) + t P (8)where γ , b and t are Landau coefficients. The value of t changes with temperature and drives the transition to aspontaneously polarized state [14]. Since the free energyis second order in n , the effective period of its variationis π/q . Inserting the ansatz into the free energy density,and integrating over a period π/q , we find an average freeenergy density of¯ f splay = K q (cid:2) θ + θJ (2 θ ) (cid:3) + K q (cid:2) θ − θJ (2 θ ) (cid:3) − γpq J ( θ ) θ + tp b p q (cid:0) θ + 4 (cid:1) (9)This free energy can then be minimized with respect to p and q to obtain the following expressions at the freeenergy minimum, q splay = − √ tb (3 θ + 4) × (10) (cid:40) √ t + 2 √ γ | J ( | θ | ) | (cid:112) ( K + K ) θ + θ ( K − K ) J (2 θ ) (cid:41) p splay = 8 θγJ ( | θ | ) q splay | θ | (cid:16) b (3 θ + 4) q splay + 4 t (cid:17) (11)Note that the radicand in (10) is non-negative ( ¯ f n ≥ θ ∼
1. Using K = 10 pN , K = 1 pN , γ = 10 − V , b =2 × − V m / (A s) and t = 8 × − V m/(A s), weobtain q = 0 . − and p = 10 (A s) / m . This is con-sistent with experiments where a nanometer range forpitch is observed [14]. Further, using the typical densityof 1 g / cm , the value of the polarization density, p trans-lates to a molecular polarization of 10 Debye, which is ap-proximately the same as that of the molecule of RM734seen to form splay-bend phases [14].We substitute these expressions for p splay and q splay into the free energy and plot it as a function of the maxi-mum angle θ in Fig. 3. As can be seen, there is a nontriv-ial minimum at a non-zero value of θ , so the splay-bendphase is stable in the case of a splay flexoelectric cou-pling.In the case of a bend flexoelectric coupling, the onlyterm that changes is the interaction term with cou-pling γ . A bend flexoelectric coupling is of the form P × [ n × ( ∇ × n )]. However, this is a vector. If, how-ever, the material were sandwiched between two dif-ferent plates separated in the direction perpendicularto director (the y -axis), then a coupling of the formˆ y · ( P × [ n × ( ∇ × n )]) is allowed. In this case the av-erage free energy density is¯ f bend = K q (cid:2) θ + θJ (2 θ ) (cid:3) + K q (cid:2) θ − θJ (2 θ ) (cid:3) − γpqHHH ( θ ) + tp b p q (cid:0) θ + 4 (cid:1) (12)Here, HHH ν ( z ) is the Struve function of order ν . Repeatingthe same procedure as earlier, we plot the average freeenergy density in Fig. 3. As can be seen, a nontrivialminimum exists at a higher value of θ than for splay f splay f bend - π - π π π θ - - - - f FIG. 3: Plot of ¯ f splay and ¯ f bend as a function of θ at the freeenergy minimizing values of q and p . The plots clearly showthat both free energies have a minimum at a non trivialvalue of θ , implying that the splay-bend phase is apossibility with both couplings. The parameter values are K = 1 . , K = 2 , γ = 40 , b = 2 and t = 10. flexoelectricity. Thus, we conclude that the splay-bendphase given by the ansatz in Eq. (7) can be obtained byeither splay flexoelectric coupling or a bend flexoelectriccoupling along with a sample asymmetry, providing thedirection ˆ y . IV. SPLAY-BEND AND SPLAY NEMATICPHASES
Previous work on nematic phases with splay and bendmodulations makes a distinction between the ansatz inEq. (7), a ‘splay nematic phase’, and the ‘splay-bendphase’ in Eq. (4) [14]. In particular, the direction of themodulation is perpendicular to the director in Eq. (7),as opposed to along the director, as in Eq. (4). In the‘splay-bend phase’, the splay and bend contributions tothe free energy density, integrated over a period π/q , arethen,¯ f n = K q (cid:104) θ + θJ (2 θ ) (cid:105) + K q (cid:104) θ − θJ (2 θ ) (cid:105) (13)As can be seen from a comparison of the above equa-tion with Eq. (5), the splay and bend contributions havebeen interchanged. The two systems can be mapped onto each other by exchanging K with K . The ‘splaynematic phase’ and the ‘splay-bend phase’ are closely re-lated phases. This is expected since a rotation of thenematic director field by π/
2, as would be required toturn n into n , would turn splay deformations into benddeformations and bend into splay.We could have begun by using the ansatz in Eq. (4),and repeated the process outlined in section III by in-serting the new ansatz into the free energies with thetwo different flexoelectric couplings. Minimizing with re-spect to q and p , we would find that the results in SectionIII are reversed, and the curves for ¯ f splay and ¯ f bend inter-changed in Fig. 3. Thus, both the splay-bend and ‘splaynematic’ phases can be obtained with splay and bendflexoelectric couplings, and can be related to each otherby an exhange of the bend and splay elastic constants. V. CONCLUSIONS
We have demonstrated that while bond orientationalorder is a mechanism that could circumpass the prob-lem of an unbounded free energy, nematic symmetry andachirality of the splay-bend phase prevent it from ex-plaining the formation of the splay-bend phase. Flexo-electricity provides a viable mechanism for introducingsplay and bend modulations in nematic systems. Bend and splay flexoelectric couplings lead to effective elas-tic constants that stabilize splay and bend modulations.Both flexoelectric couplings can give rise to splay-bendphases with modulation in the average direction of the di-rector field or modulations perpendicular to the averagedirection of the nematic director. These two modula-tions, treated previously in the literature as ‘splay-bend’and ‘splay-nematic’ phases, are related to each other byan exchange of the splay and bend elastic constants.
ACKNOWLEDGMENTS
We thank A. Mertelj for helpful discussions. This workwas supported by a Simons Investigator grant from theSimons Foundation to R.D.K. and NSF DMR12-62047. [1] L. A. Madsen, T. J. Dingemans, M. Nakata, and E. T.Samulski, Phys. Rev. Lett. , 145505 (2004).[2] G. R. Luckhurst and T. J. Sluckin, Biaxial Nematic Liq-uid Crystals: theory, simulation and experiment (JohnWiley & Sons, 2015).[3] J. P. Straley, Phys. Rev. A , 1881 (1974).[4] V. Borshch, Y.-K. Kim, J. Xiang, M. Gao, A. J´akli, V. P.Panov, J. K. Vij, C. T. Imrie, M.-G. Tamba, G. H. Mehl, et al. , Nature Commun. , 3635 (2013).[5] P. A. Henderson and C. T. Imrie, Liq. Cryst. , 1407(2011).[6] M. Cestari, S. Diez-Berart, D. Dunmur, A. Ferrarini,M. de La Fuente, D. Jackson, D. Lopez, G. Luckhurst,M. Perez-Jubindo, R. Richardson, et al. , Phys. Rev. E , 031704 (2011).[7] D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai,U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser,J. E. Maclennan, et al. , Phys. Rev. E , 022506 (2014).[8] R. J. Mandle, E. J. Davis, C. T. Archbold, S. J. Cowling,and J. W. Goodby, J. Materials Chem. C , 556 (2014).[9] C. Pr¨asang, A. C. Whitwood, and D. W. Bruce, Chem.Commun. , 2137 (2008).[10] C. Meyer, G. Luckhurst, and I. Dozov, J. of MaterialsChem. C , 318 (2015). [11] J. Harden, B. Mbanga, N. ´Eber, K. Fodor-Csorba,S. Sprunt, J. T. Gleeson, and A. Jakli, Phys. Rev. Lett. , 157802 (2006).[12] J. Harden, M. Chambers, R. Verduzco, P. Luchette, J. T.Gleeson, S. Sprunt, and A. J´akli, Appl. Phys. Lett. ,102907 (2010).[13] I. Dozov, EPL (Europhys. Lett.) , 247 (2001).[14] A. Mertelj, L. Cmok, N. Sebasti´an, R. J. Mandle, R. R.Parker, A. C. Whitwood, J. W. Goodby, and M. ˇCopiˇc,Phys. Rev. X , 041025 (2018).[15] R. Memmer, Liq. Cryst. , 483 (2002).[16] C. Meyer and I. Dozov, Soft Matter , 574 (2016).[17] S. M. Shamid, S. Dhakal, and J. V. Selinger, Phys. Rev.E , 052503 (2013).[18] C. Meyer, G. Luckhurst, and I. Dozov, Phys. Rev. Lett. , 067801 (2013).[19] L. Longa and G. Paj¸ak, Phys. Rev. E , 040701 (2016).[20] R. D. Kamien, Journal de Physique II , 461 (1996).[21] J. Toner, Phys. Rev. A , 1157 (1983).[22] R. B. Meyer, Appl. Phys. Lett.14