aa r X i v : . [ c ond - m a t . s o f t ] J u l Flexoelectric blue phases
G. P. Alexander and J. M. Yeomans
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, England. (Dated: October 29, 2018)We describe the occurence and properties of liquid crystal phases showing two dimensional splay and benddistortions which are stabilised by flexoelectric interactions. These phases are characterised by regions of locallydouble splayed order separated by topological defects and are thus highly analogous to the blue phases ofcholesteric liquid crystals. We present a mean field analysis based upon the Landau–de Gennes Q -tensor theoryand construct a phase diagram for flexoelectric structures using analytic and numerical results. We stress thesimilarities and discrepancies between the cholesteric and flexoelectric cases. PACS numbers: 61.30.Gd, 64.70.Md, 61.30.Mp
Elastic distortions in the nematic phase of liquid crystalscan be categorised into three types: splay, twist and bend. Sta-ble phases with two dimensional twist distortions have beenwell characterised and observed experimentally. Here we pre-dict the occurence of structures with two dimensional splay–bend distortions.When nematics are doped with chiral molecules they canshow stable phases with a natural twist known as cholesterics,shown in Fig. 1(a). In general cholesteric phases comprise a one -dimensional helix. This is because topological constraintsmean that it is not possible to construct a state with helicalordering in two dimensions without introducing defects, ordisclination lines, into the structure. However local regions ofdouble twist are possible. Fig. 2(a) shows a simple, two di-mensional example of a director field where regions of doubletwist are separated by topological defects. If the free energyadvantage of the double twist regions offsets the disadvantageof the disclinations phases of this type will be stable.Such states, the so-called blue phases of cholesteric liq-uid crystals, have indeed been observed [1]. The disclinationstructure is, however, more complicated than that of the sim-ple example in Fig. 2(a). The disclinations form textures withcubic symmetry and the stable phases have space groups O − and O . The blue phases are stable at the isotropic–cholestericphase boundary as here the magnitude of the order is small andhence the free energy penalty associated with disclinations islow. They are stable only over a narrow temperature range ∼ K, although recently this has been extended to ∼ K byadding polymers or bimesogenic molecules [2, 3].One might therefore ask whether analogous behaviourcould be observed in a liquid crystal which shows splay andbend, rather than twist, distortions. In 1969 Meyer showed (a) (b)
FIG. 1: Schematic representation of elastic distortions of nematicliquid crystals, showing the director field projected into the plane ofthe page. (a) Twist in the cholesteric phase of chiral nematics. (b)Splay and bend in a nematic with flexoelectric interactions. (a) (b)
FIG. 2: Schematic examples of two dimensional director field con-figurations. (a) Structure displaying regions of local double twist,characteristic of cholesteric blue phases. (b) Analogous structurepossessing double splay distortions. that the one dimensional splay–bend distortion of the direc-tor field, shown in Fig. 1(b), could result from flexoelectriccoupling to an external field [4]. Here we extend these resultsto show that near the isotropic–nematic transition two dimen-sional splay–bend structures (e.g., Fig. 2(b)) can be stable.We shall term such phases flexoelectric blue phases. Similardirector field configurations, but stabilised by a saddle splayterm in the free energy, have been reported in [5].The coupling of a liquid crystal to an external field occursin two principal forms. The most usual is dielectric, wherethe field couples directly to the orientational order parame-ter. This coupling is quadratic in the field strength and themolecules tend to align parallel or perpendicular to the field,depending on the sign of the dielectric anisotropy. Flexoelec-tricity, which we shall deal with here, occurs because of theelastic properties of liquid crystals. If the liquid crystal iscomposed of pear-shaped molecules an elastic distortion canlead to a spontaneous polarisation. Conversely, if a polari-sation is induced by an external electric field then an elasticdistortion results. This is termed flexoelectric coupling, and itgives a response linear in the field strength.Our conclusions are based upon Landau–de Gennes mean-field theory. We present analytic and numerical results ex-ploring the phase diagram of the flexoelectric blue phases,stressing the analogy, and differences, between these and thecholesteric blue phases.At a macroscopic level, the order parameter Q may be takento be a traceless, symmetric, second rank tensor which is re-lated to the anisotropic part of the dielectric tensor [6]. Usinga tensor allows both the magnitude and the direction of theorder to be recorded. The direction of the order, describedby a vector n called the director, is defined as the eigenvectorcorresponding to the maximal eigenvalue of Q . We considera bulk sample with periodic boundary conditions and assumethat the liquid crystal has zero dielectric anisotropy and theelectric field is uniform throughout the sample.The equilibrium thermodynamics of a nematic liquid crys-tal can be described by the Landau–de Gennes free energy [6],which we supplement by a familiar term for chiral couplingand an additional flexoelectric coupling F = V Z Ω d r (cid:26) τ tr (cid:0) Q (cid:1) − √ tr (cid:0) Q (cid:1) + (cid:16) tr (cid:0) Q (cid:1)(cid:17) + L (cid:0) ∇ a Q bc (cid:1) + 2 q L Q · ∇ × Q + ε f Q ab (cid:0) E a ∇ c − E c ∇ a (cid:1) Q bc (cid:27) . (1)Here, τ is a reduced temperature, L is an elastic constant, q determines the pitch in the cholesteric phase, ε f is a flexoelec-tric coupling constant, E is the electric field and V is the vol-ume of the domain, Ω . There are four flexoelectric couplingsup to second order in Q , three of which may be written as atotal divergence and converted to a surface term using Stokes’theorem. Since we neglect surface terms in this work only theterm quoted need be retained. For simplicity, we adopt the oneelastic constant approximation where the magnitude of twist,splay and bend are taken to be equal.Since chirality and flexoelectricity induce structure in thefluid it is useful to analyse potential stable states by introduc-ing a Fourier decomposition of the Q -tensor [7, 8] Q ( r ) = X k N − / k X m = − Q m ( k ) e iψ m ( k ) M m (ˆ k ) e − i k . r , (2)where N k is a normalising factor counting the number ofwavevectors with the same magnitude, Q m ( k ) , ψ m ( k ) are theamplitude and phase of the Fourier component and M m (ˆ k ) are a set of orthonormal basis tensors. Reality of Q is ensuredby requiring ψ m ( − k ) = − ψ m ( k ) and M m ( − ˆ k ) = M † m (ˆ k ) .With these definitions the quadratic part of the free energytakes the form F (2) = X k N − k X m,m ′ Q m ( k ) Q m ′ ( k ) e i ( ψ m ( k ) − ψ m ′ ( k )) × M † m ′ (cid:0) ˆ k (cid:1) ab (cid:26)(cid:16) τ + L k (cid:17) δ ac − i q L kǫ adc ˆ k d − iε f Ek (cid:0) ˆ E a ˆ k c − ˆ k a ˆ E c (cid:1)(cid:27) M m (cid:0) ˆ k (cid:1) bc . (3)The basis tensors M m (ˆ k ) should be chosen so as to diago-nalise F (2) . It is convenient to introduce the quantity ∆ k := q − (cid:0) ˆ E . ˆ k (cid:1) and use the direction of the electric field andof the wavevector for a given Fourier mode to define a lo-cal, right-handed, orthonormal frame field { ˆ k , ˆ v , ˆ w } such that ˆ E = (cid:0) ˆ E . ˆ k (cid:1) ˆ k + ∆ k ˆ w . We then make use of the observationthat the flexoelectric coupling is antisymmetric to rewrite itas + iε f E ∆ k kǫ adc ˆ v d , revealing a formal mathematical simi-larity between flexoelectric and chiral couplings. Combiningthese two coupling terms motivates a transformation to a newset of basis vectors, obtained by means of a rotation about ˆ w : a := µ k (cid:0) q L ˆ k − ε f E ∆ k ˆ v (cid:1) , b := µ k (cid:0) ε f E ∆ k ˆ k + 2 q L ˆ v (cid:1) , c := ˆ w , (4)where µ k = (cid:0) (2 q L ) + ( ε f E ∆ k ) (cid:1) / is a normalisationfactor. In this rotated local frame, the choice of basis tensors M ± = (cid:0) b ± i c (cid:1) ⊗ (cid:0) b ± i c (cid:1) , M ± = h a ⊗ (cid:0) b ± i c (cid:1) + (cid:0) b ± i c (cid:1) ⊗ a i , M = √ (cid:0) a ⊗ a − I (cid:1) , (5)results in the desired diagonalisation of F (2) , F (2) = X k N − k X m = − Q m ( k ) n τ + L k − m µ k k o . (6)We see from Eqs. (4) that there is a strong sense in whichthe transition from purely chiral to purely flexoelectric cou-pling may be viewed geometrically as a π/ rotation. Indeed,this is already evident in the one dimensional cholesteric andsplay–bend states of Fig. 1, which are related by a π/ rota-tion about the vertical direction. This rotation is none otherthan the well known flexoelectro-optic effect [9].Given this formal similarity it is natural to ask whethera precise correspondence can be made between the stablephases, and their properties, in the purely chiral and purelyflexoelectric limits. To this end we now focus our attention onthe purely flexoelectric sector and analyse a number of struc-tures motivated by the analogy with cholesterics.The simplest flexoelectric structure is a one dimensionalphase that was described in Meyer’s original paper [4], whichwe shall call splay–bend, see Fig. 1(b). The Q -tensor appro-priate to splay–bend is the analogue of that for the cholesterichelix in chiral liquid crystals: Q = Q √ cos( kx ) 0 sin( kx )0 0 0sin( kx ) 0 − cos( kx ) − Q h √ − − (7)where we have used the electric field to define the z -axis ofa Cartesian coordinate system and taken the wavevector to beorthogonal to the field so as to maximise ∆ k and hence min-imise the free energy. The free energy is readily shown to be F = τ (cid:0) Q + Q h (cid:1) − Q Q h + Q h + (cid:0) Q + Q h (cid:1) + Q (cid:0) L k − ε f Ek (cid:1) . (8)It is formally identical to that of the cholesteric phase with ε f E/L playing the role of the chiral parameter, q . There-fore the solutions for Q and Q h as a function of the reducedtemperature, τ , and field strength, E , obtained by minimis-ing Eq. (8), are the same as those obtained for the cholestericphase [10], provided one substitutes q with ε f E/L .Extending the analogy with cholesterics, we now considerflexoelectric phases with a higher dimensional structure; flex-oelectric analogues of the cholesteric blue phases. Based onthe heuristic observations that the field picks out a preferreddirection in space, and that the flexoelectric coupling vanishesfor wavevectors parallel to the field, it seems likely that thefree energy will be minimised by two dimensional structures,containing only wavevectors orthogonal to the field. There-fore we consider first a two dimensional phase possessinghexagonal symmetry which may be obtained by choosing thefundamental set of Fourier modes to be along the directions {± e x , ± ( − e x + √ e y ) / , ± ( − e x − √ e y ) / } . An approxi-mate Q -tensor, comprising only this fundamental set, all with m = 2 , and a homogeneous component, is given by Q = Q √ (cid:26) (cid:16) e x + i e z (cid:17) ⊗ S e − i ( kx − ψ ) + (cid:16) − e x + √ e y + i e z (cid:17) ⊗ S e − i ( k ( − x + √ y ) / − ψ ) + (cid:16) − e x − √ e y + i e z (cid:17) ⊗ S e − i ( k ( − x −√ y ) / − ψ ) + H.c. (cid:27) + Q h e iδ √ (cid:16) e z ⊗ e z − I (cid:17) (9)where H.c. stands for Hermitian conjugate, ψ , ψ , ψ ∈ [0 , π ) , δ ∈ { , π } and the short-hand notation ⊗ S denotesa symmetrised tensor product, i.e., u ⊗ S := u ⊗ u . The direc-tor field of this hexagonal flexoelectric blue phase is shownin Fig. 3. The structure consists of an hexagonal lattice ofstrength − / disclinations separating regions in which thedistortion is one of pure splay along any straight line pass-ing through the centre of the hexagon. Indeed, expanding incylindrical polars ( ρ, φ, z ) , the director field in a local neigh-bourhood of the centre of each hexagon is n = cos (cid:16) Q kρ Q +2 Q h (cid:17) e z − sin (cid:16) Q kρ Q +2 Q h (cid:17) e ρ , (10)and therefore the structure may be aptly referred to as a doublesplay cylinder since it is the clear analogue of double twistcylinders in the cholesteric blue phases.The free energy of the hexagonal flexoelectric blue phase is F = τ (cid:16) Q + Q h (cid:17) −
14 2 ε f E L Q − e iδ Q Q h − e iδ Q h + cos( ψ + ψ + ψ ) Q + Q + Q Q h + Q h − cos( ψ + ψ + ψ + δ ) Q Q h . (11)This is minimised by choosing the phases to satisfy ψ + ψ + ψ = π and δ = 0 . As in the case of the splay–bend phase, thisis identical to the free energy of the cholesteric analogue, theplanar hexagonal blue phase [7, 8]. Consequently, we may FIG. 3: Director field of the hexagonal flexoelectric blue phase de-scribed in the main text obtained from a numerical minimisation atparamter values κ E = 1 , τ − κ E = 0 where the phase is stable, seeFig. 4. The structure is periodic in both directions. draw on this previous work for cholesterics to conclude thatthe hexagonal flexoelectric phase has lower free energy thanthe one dimensional splay–bend phase at sufficiently largefield strength. However, in the chiral case the hexagonal bluephase is not stable since one of the cubic blue phases alwayshas lower free energy. It is therefore necessary to see whethera similar result also holds in the flexoelectric case.Although one may reasonably expect that the structure ofthe cubic blue phases provides a good starting point for aconsideration of flexoelectric blue phases, the latter differ ina number of respects. In particular, since the flexoelectriccoupling is vectorial, it defines a preferred direction in spacewhich will act to lower the symmetry from cubic to tetrag-onal. We therefore considered the set of possible structuresobtained by selecting the Fourier modes of the Q -tensor to be n , ± πL x e x , ± πL x e y , ± πL x (cid:0) e x + e y (cid:1) , ± πL x (cid:0) − e x + e y (cid:1) , ± πL z e z , ± πL x (cid:0) e x + L x L z e z (cid:1) , ± πL x (cid:0) e y + L x L z e z (cid:1) , ± πL x (cid:0) − e x + L x L z e z (cid:1) , ± πL x (cid:0) − e y + L x L z e z (cid:1)o , (12)where L x , L z are the lattice parameters along the x − and z − axes of the conventional unit cell respectively. This set ofstructures allows for considerable scope, containing as spe-cial cases Q -tensors that are exact analogues of those repre-senting the cholesteric cubic blue phases as well as that of thetwo dimensional square structure shown in Fig. 2(b). We haveinvestigated the free energy minima of these structures bothusing approximate analytic calculations and with an exact nu-merical minimisation of the free energy for a range of initialconditions, varying the ratio of the lattice parameters L x /L z and the relative amplitudes and phases of the Fourier compo-nents. The numerical minimisation was performed using a lat-tice Boltzmann algorithm with an identical technique to thatdescribed in [10]. In all instances a single minimum was ob-tained which corresponds to the two dimensional square flex-oelectric blue phase of Fig. 2(b).These results were combined with the analytic expressionsfor the free energy of the splay–bend and hexagonal struc-tures, Eqs. (8) and (11), to construct an approximate phase κ E τ − κ E Isotropic Splay−−bend Hexagonal
FIG. 4: Phase diagram of the flexoelectric blue phases. The circlesand solid lines show the numerical phase boundaries, while the ana-lytic results are given by the dashed lines. diagram. We find that although the two dimensional squarephase is a local minimum of the free energy within the set ofstructures defined by (12), it never has lower free energy thanthe hexagonal phase. The same result is obtained numerically,leading to the phase diagram shown in Fig. 4, which has beenplotted in the κ E , ( τ − κ E ) plane, where κ E := q ε f E /L ,in accordance with the conventions adopted in cholesterics[10]. As the electric field strength is increased the hexagonalflexoelectric blue phase becomes stabilised over an increas-ing temperature interval between the isotropic and splay–bendphases. The discrepancy between the numerical and analyticphase boundaries is similar to that found in cholesterics andarises from neglecting higher order wavevector harmonics inanalytic calculations. In the cholesteric blue phases the lat-tice periodicity is not identical to the cholesteric pitch, but isgenerally somewhat larger. From the numerical minimisationwe find the same feature in the hexagonal flexoelectric bluephase, with the lattice parameter being approximately larger than the ‘pitch’ ( πL /ε f E ) of the splay–bend phase atthe triple point.The phase diagram bears a qualitative resemblance to thatof cholesterics [10], but the details reveal differences betweenchiral and flexoelectric couplings. In particular, it seems thatthe suppression of wavevectors parallel to the field introducedby the quantity ∆ k is sufficient to convert the fully three di-mensional states which are stable at large chiral coupling intotwo dimensional states under flexoelectric coupling.We comment that the local director field in the double splayregions of the flexoelectric blue phases is the same as that inthe escape configuration lattice phase recently suggested byChakrabarti et. al. [5]. However, the mechanism by which thestructure occurs, and is stabilised, is different. Their proposedstructure arises without the application of an electric field, be-ing instead stabilised by the saddle splay elastic constant andincluding weak anchoring of the director field and surface ten-sion at the interface between the nematic and isotropic regions of the fluid. Note also that the hexagonal symmetry of thephase we obtain is different to the square symmetry predictedin that work.It is important for potential experimental work to give an es-timate of the field strength required to stabilise the two dimen-sional hexagonal flexoelectric blue phase. A good estimate isthat the strength of the flexoelectric coupling, ε f E , should beas large as the chiral coupling, q L , in a compound whichpossesses cholesteric blue phases. This gives a field strength E ≈ q L /ε f ≈ πK/ep , where we have replaced theLandau–de Gennes parameters with their director field equiv-alents; an average Frank elastic constant, K , and the averageof the flexoelectric coupling constants, e := ( e s + e b ) / . p isthe pitch in the cholesteric phase of a compound which alsodisplays blue phases, typically ∼ . µm . Recently materi-als have been developed for flexoelectric purposes and foundto have a large flexoelastic ratio e/K ∼ V − [11]. Thusin these materials the required field strength is expected to be E ∼ V µm − , which is within the experimentally accessiblerange. In addition, for the proposed flexoelectric blue phasesto be seen, it will be helpful to use a material with as close tozero dielectric anisotropy as possible and to take care to en-sure that surface effects are negligible compared to the bulk.The analogy with cholesterics further suggests that flexoelec-tric blue phases are only likely to be found in a very narrowtemperature range ( ∼ K) just below the isotropic–nematictransition temperature, although this is predicted to expandwith increasing field strength.We thank Davide Marenduzzo for useful discussions. [1] D. C. Wright and N. D. Mermin,
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