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PHYSICAL REVIEW E 89, 030501(R) (2014)
Flexoelectricity in chiral nematic liquid crystals as a driving mechanism for the twist-bend and splay-bend modulated phases ˇ c,2,3 Mikhail A. Osipov,4 and Ewa Gorecka5 Nataˇsa Vaupotiˇc,1,2 Mojca Cepiˇ 1
Faculty of Natural Sciences and Mathematics, University of Maribor, Koroˇska 160, 2000 Maribor, Slovenia 2 Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia 3 Faculty of Education, University of Ljubljana, Kardeljeva ploˇscˇ ad 16, 1000 Ljubljana 4 Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond St., G1 Glasgow, United Kingdom 5 Department of Chemistry, University of Warsaw, Zwirki i Wigury 101, 02-089 Warsaw, Poland (Received 16 January 2014; published 26 March 2014) We present a continuum theoretical model describing the impact of chirality on nematic systems with large flexoelectricity. As opposed to achiral materials, where only one type of the modulated structure can exist in a given material, the model predicts that chirality can stabilize several modulated phases, which have already been observed experimentally [A. Zep et al., J. Mater. Chem. C 1, 46 (2013)]. DOI: 10.1103/PhysRevE.89.030501
PACS number(s): 61.30.Cz, 61.30.Eb, 64.70.M−
The flexoelectric effect can be a driving mechanism for the formation of the modulated nematic phases as already pointed out by Meyer [1]. Because very large flexoelectric coefficients are required, the idea of modulated nematic phases was practically forgotten until 2001 when Dozov [2] revived it by pointing out that bent-core liquid crystals could exhibit modulated nematic phases because of the specific shape of the constituent molecules. The flexoelectric coefficient can be very large for bent-core molecules [3] and as shown by [1,4], it can lead to the reduction of the bend elastic constant by which modulated nematic phases can become energetically favorable. There are, in general, two possible modulated nematic phases. In the twist-bend structure the nematic director, giving the average direction of the local direction of long molecular axes, bends and rotates on the cone. Such bend deformation is essentially coupled to the twist deformation of the director. In the splay-bend structure the direction of the nematic director exhibits an in-plane modulation. In achiral nematics one can observe only one of the modulated structures, depending on the ratio between the splay and twist elastic constant. The first observation of the nematic to modulated nematic phase transition was reported by Panov et al. [5] for mesogenic dimers, which have a bent shape when two monomers are linked by an odd-carbon aliphatic spacer [6,7]. Recently, the twist-bend modulated phases with helicoidal structures with an extremely short modulation period of only few molecular lengths were observed in achiral bent dimers [8–10]. Observation of only the twist-bend structures is not surprising, because the splay elastic constant in bent-core liquid crystal or bent dimers is very large [11–14]. On the other hand, chiral nematics made of chiral bent dimers show by far more complex phase behavior. As reported by Zep et al. [15] six nematic-type phases were observed in chiral dimers in a temperature range between the isotropic and crystal phase. Four of the observed phases have less than 1 K temperature range, which suggests that there are several strongly competing interactions in the system. The highest temperature phase is the conventional uniaxial cholesteric phase. The helix unwinds close to the transition to the first
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lower temperature phase. The lowest temperature nematic phase has a modulated structure. When the material is filled in thin cells, the modulated structure is identified by its stripe texture due to the focusing of light on the periodic changes of refractive index. Measurements also show that, at least in some temperature range, the modulations are one dimensional, because focusing can be observed only for the light polarized along the direction parallel to the rubbing direction of the cell surface. The pitch of modulation is in a micrometer range and it increases with increasing cell thickness, which is opposite to the effect observed in the SmC* phase, where the modulation periodicity increases with decreasing cell thickness as a result of helix unwinding caused by surface interactions [16]. In this Rapid Communication we report on the theoretical model of modulated structures in chiral nematics made of chiral bent dimers. The model predicts all the above-mentioned observed properties of nematics formed by chiral dimers, both in bulk and in thin cells. It predicts the existence of the twistbend and splay-bend structures in the same material and a phase transition between them. The key idea of the model is to take into account, that the modulated phase is essentially locally biaxial as shown in [4] using the Lebwohl-Lasher lattice model [17]. If the material is biaxial (but nonpolar), it can be described by the orientational tensor order parameter with three nonequivalent orthogonal primary axes. One of these axes specifies the primary director n which is the nonpolar direction of the average local orientation of the long molecular axis. The second axis corresponds to the unit vector l, perpendicular to n, which specifies the orientation of short molecular axes (in the direction of the molecular or dimer bend), while the third axis corresponds to the third director m = n × l (see Fig. 1). For the description of the elastic free energy density (fn ) of the biaxial nematic we use the expression derived by Stalinga and Vertogen [18]. The free energy includes 12 elastic constants plus three additional ones which are related to molecular chirality. The characteristic behavior of the system composed of chiral bent dimers can be explained already in the one constant approximation when all 12 elastic constants are
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assumed to be equal, although this is not a good approximation for real bent-core systems. In the one-constant approximation the elastic free energy density reads [18] fn = k1 l · (∇ × l) + k2 m · (∇ × m) + k3 n · (∇ × n) + 12 K[(∇ · l)2 + (∇ · m)2 + (∇ · n)2 ] + 12 K{[l · (∇ × l)]2 + [m · (∇ × m)]2 + [n · (∇ × n)]2 } + 12 K{[n · (∇ × m)]2 + [l · (∇ × n)]2 + [m · (∇ × l)]2 } + 12 K{[m · (∇ × n)]2 + [n · (∇ × l)]2 + [l · (∇ × m)]2 }.
Here the constants k1 , k2 , and k3 are chiral parameters which cause the twist distortion of the l, m, and n vector fields, respectively. The parameters can be either positive or negative, depending on molecular handedness. The elastic constant (K) is always positive. In the case of chiral biaxial nematics there are three possible helices with the axis along n, l, or m, respectively (Fig. 1). As an example, let us consider a twist distortion of n and l. The helical axis is along the m vector (let this be the z axis) and spatial variations of the n and l directors are expressed as n = {cos(qb z), sin(qb z),0}, l = {− sin(qb z), cos(qb z),0},
(3)
where qb is the modulation wave vector in the biaxial phase along the z axis. Minimizing the free energy fn dz we find k1 + k3 , 2K and the free energy per unit volume (Fb ) is
and (5) one can deduce the energy of the structure with the helical twist along the n or l axis. When the modulation wave vector for the helical twist along the m or l axis becomes small enough, a structure with a helicoidal modulation along, the n axis can be stabilized. The free energy density (f ) of the modulated nematic can be approximately expressed as a sum of the elastic energy, the leading flexoelectric term, and the term that describes the dielectric energy of the system [4]: f = fn + λP · [n × (∇ × n)] + μP2 ,
(2)
and
qb =
(4)
Fb = −Kqb2 .
(5)
In the high temperature uniaxial nematic phase the pitch is qu = kt /Kt , where Kt is the uniaxial twist elastic constant and kt describes the constant in front of the linear uniaxial twist term. If in the biaxial phase k1 and k3 are of the opposite sign, the modulation length (pitch) in the biaxial phase will be larger than in the uniaxial phase, and close to the phase transition temperature one should observe a pretransitional increase in the pitch due to the onset of the biaxial order. Such an increase was, indeed, observed in [15]. From Eqs. (4)
FIG. 1. (Color online) The three possible twist structures in orthorhombic nematic liquid crystals. The orientation of the blocks is defined by three perpendicular unit vectors: n (blue solid line), l (red dotted line), and m = n × l (green dashed line).
(1)
(6)
where λ is the bend flexoelectric coefficient, P is the local polarization, and the coefficient μ = 1/2χ⊥ is the inverse dielectric susceptibility of the nematic in the direction perpendicular to n. One notes that in the general case the free energy also contains the splay flexoelectric contribution λs P · n(∇ · n), and the dielectric energy term contains the dielectric susceptibility tensor χij . In typical bent-core nematics, however, the bend flexoelectric coefficient is much larger than the splay one, and thus the splay contribution can be neglected in the first approximation. In fact, the splay term vanishes in the case of perfect nematic order for banana-shaped molecules of C2v symmetry [19]. A general molecular theory of flexoelectricity in biaxial nematics has been developed in [20] and it has been shown that there are four additional flexocoefficients in the biaxial phase. Two of them are related to bend, but these “biaxial” coefficients are much smaller than the leading uniaxial one in Eq. (6) as they are proportional to small biaxial orientational order parameters. We assume that the nematic is not ferroelectric, that is, there is no spontaneous polarization (i.e., the parameter μ is positive), and therefore the local polarization is due to the flexoelectric effect. In this case the flexoelectric polarization is approximately perpendicular to n (the splay term is small), and thus the dielectric energy contains only the transverse susceptibility χ⊥ . In the case of bent-core molecules with large transverse dipoles the transverse susceptibility increases with the decreasing temperature due to the increase of the nematic order parameter. As a result the parameter μ decreases with decreasing temperature. To find the conditions for the stability of the splay-bend and twist-bend modulated structures in bulk, we use a harmonic ansatz, which includes the splay-bend and twist-bend modulation imposed on the chiral helical modulation along the n axis (Fig. 2). We assume that the local flexoelectric polarization is in the direction of the l vector, and is perpendicular to the n and m vectors. This is not a required condition, but we can expect that due to the bent shape of the molecules or dimers this component of local polarization will be the largest. In the splay-bend modulated chiral structure there is a helical twist
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FLEXOELECTRICITY IN CHIRAL NEMATIC LIQUID CRYSTALS . . .
FIG. 2. (Color online) The splay-bend (top) and twist-bend (bottom) modulation superposed over the chiral helix.
of l and m along n and the bend deformation of the n director in the yz plane superposed on the chiral modulation: n = {0, cos θ, sin θ },
(7)
l = {cos(qy), − sin θ sin(qy), cos θ sin(qy)},
(8)
P = p0 {cos(qy), − sin θ sin(qy), cos θ sin(qy)},
(9)
where θ = θ0 cos(qy). The phase of θ is chosen such that the structure is nonpolar over one period of modulation. A general phase can be used, providing that proper changes are made in the ansatz for l and P, as well. In the twist-bend structure the rotation of the n director on the cone is superposed on the chiral helix: n = {sin θ0 cos(qy), cos θ0 , sin θ0 sin(qy)},
(10)
l = {sin(qy),0, − cos(qy)},
(11)
P = p0 {sin(qy),0, − cos(qy)}.
(12)
The modulation wave vector along the y direction (q) can be expressed as q = q0 (1 + q ),
(13)
where q0 = −(k1 + k2 )/(2K) is the modulation wave vector along the n axis due to the chiral twist. The dimensionless parameter q gives the relative change of the wave-vector magnitude in the splay-bend or the twist-bend modulated phase with respect to the phase with only pure chiral twist. Similarly, the free energy F = f dy is expressed as F = F0 + F,
(14)
where F0 = −(k1 + k2 ) /(4K) is the free energy of the phase with only chiral twist and F is the change in energy due to the splay-bend or twist-bend modulation. Let us consider the splay-bend structure first. We expand the free energy up to the fourth order in θ0 and minimize the free energy over p0 , θ0 , and q. Complex expressions are obtained and they can be significantly simplified, if (without losing the generality of the result) we set k2 = 0. Minimization shows that the modulated structure is stable if μ < μ(sb) 0 : 2
λ2 , (15) 6K where the index (sb) stands for the splay-bend structure. The temperature dependent parameter μ is then expressed as = μ(sb) 0
μ = μ(sb) μ), 0 (1 −
(16)
where μ is proportional to T = T0 − T , and T0 is the temperature at which the modulated structure becomes stable.
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Then we assume that below the critical value of μ0 the μ)k to quantities p0 , θ0 , q, and F are proportional to ( the lowest order where the exponent k is determined from the corresponding equations. As a result one finds the following expressions: √ 2 9 2 k1 (sb) (sb) θ0 = 6 μ, p0 = − √ μ, (17) 31 31 λ 9 27 F (sb) = μ, ( μ)2 . 31 F0 31 For the twist-bend (tb) structure one finds q (sb) =
(18)
λ2 , (19) 4k31 K where the ratio between the chiral twist parameters k31 = k3 /k1 was introduced. If the helix along the m axis unwinds upon the transition to the biaxial phase, we expect k3 and k1 to be of the opposite sign [see Eq. (4)], so μ(tb) 0 is positive. To the lowest order term in μ, we find the following expressions for the model parameters and the energy: μ μ k3 (tb) (tb) θ0 = , p0 = 2 , (20) 2 + k31 λ 2 + k31 μ(tb) 0 =−
q (tb) = −
k31 μ, 2 + k31
F (tb) k31 =− ( μ)2 . F0 2 + k31 (21)
Comparing Eqs. (15) and (19) we see that the signs and the magnitudes of the chiral parameters define which modulated structure is stable and in which temperature range. By taking k31 = −1, one finds μ(sb) = 2/3μ(tb) so the twist0 0 bend structure occurs at higher temperatures. Using μ = μ and a similar expression for the twist-bend case, it μ(sb) 0 is straightforward to find F (sb) /F0 = −31.4(Kμ/λ2 )2 and F (tb) /F0 = −16(Kμ/λ2 )2 from which it follows that the splay-bend structure has lower energy. One can thus envisage the following phase sequence upon the reduction of temperature: uniaxial nematic (chiral twist along the z axis) → biaxial nematic (three possibilities for the chiral twist direction) → modulated twist-bend nematic → modulated splay-bend nematic. This simple model can thus explain the occurrence of six nematic phases. Additional, three-dimensional structures are also possible. One notes that the predicted phase sequence is specific for the chosen set of parameters. For example, if k31 > 0, then μ(tb) 0 < 0 [see Eq. (19)], so in this case only the splay-bend structure is stable. When the chiral terms vanish (achiral material), there is no reason to take the corresponding limit in the above expressions, because they represent only one specific solution of the set of equations, obtained by the free energy minimization under the assumption that k1 and k3 are nonzero. One solution, obtained by the minimization of the free energy, is always a uniform structure with θ0 , p0 , and q being equal to zero. The solution for achiral nematics cannot be directly obtained from this model, because in achiral systems the modulated structures can be obtained only if the coupling between the magnitude of polarization and its spatial variation is taken into consideration [4]. Comparing the temperature dependence of the reduced
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energy of the modulated structure in the achiral and chiral nematics, one concludes that the tendency to form modulated structures is much stronger in chiral nematics, because the reduced energy in the chiral nematics is of the lower order [it is proportional to (T )2 , while in achiral nematics it is proportional to (T )3 ; see [4]]. Finally, let us consider the modulated structures in confined geometry, in planar thin cells of thickness L with the z axis being perpendicular to the cell surface and the y axis along the cell surface. We neglect all the effects that surfaces impose on the chiral helix and assume only that the splay-bend or twistbend modulation reduces to zero at the surface. The director modulation is along the y axis, which is along the surfaces. The ansatz given by Eqs. (7)–(9) for the splay-bend structure and Eqs. (10)–(12) for the twist-bend structure are modified to include the effect of the surfaces, such that θ0 → θ0 sin(π z/L) and p0 → p0 sin(π z/L). Minimizing the free energy, we find x 2 λ2 , 1 + 6x 2 K
(22)
2x 2 λ2 , K(1 − 8k31 x 2 )
(23)
= μ(sb) 0 μ(tb) 0 =
where x is defined as a ratio between the cell thickness and the modulation length x = Lq0 /(2π ). The temperature of the transition to the modulated phase depends on the cell thickness, as expected, and in the limit of large L expressions (22) and (23) reduce to Eqs. (15) and (19), respectively. For the change in the wave-vector magnitude and energy one obtains (1 + x 2 )(1 + 6x 2 )2 K μ, 2Ax 2 λ2 (1 − 8k31 x 2 )2 (1 − 4k31 x 2 ) K q (tb) = μ, 2Bx 2 λ2 2 (1 + 6x 2 )4 K (sb) μ , F /F0 = − 4Ax 4 λ2 2 (1 − 8k31 x 2 )4 K (tb) μ , F /F0 = − 16Bx 4 λ2
q (sb) =
FIG. 3. (Color online) (a) The relative change in the wave-vector magnitude ( q ) in the twist-bend (dashed blue line) and splay-bend (solid red line) modulated structures as a function of the cell thickness L given in units of the modulation length 2π/q0 [x = Lq0 /(2π )]. = F /F0 ) as a function of (b) The relative reduction in energy (F (tb) , blue dashed line) and the cell thickness for the twist-bend (F (sb) , red solid line) modulated structures. Parameter splay-bend (F values: k31 = −1, μK/λ2 = 10−3 . Inset: the value of μ at which the modulated structure becomes stable. Blue (dashed) line: μ(tb) 0 . Red solid line: μ(sb) 0 .
(24)
(26)
The results for the change in the wave-vector magnitude and energy reduction as a function of cell thickness are given in Fig. 3 for both the splay-bend and twist-bend structures. At a chosen set of parameters the model predicts that the modulation wave vector decreases with increasing cell thickness [see Fig. 3(a)], meaning that the modulation wavelength increases with increasing cell thickness, as observed experimentally. The splay-bend structure has lower free energy than the twist-bend structure [Fig. 3(b)]. However, the splay-bend structure becomes stable at a lower temperature [see the inset to Fig. 3(b)]. To conclude, we have proposed a theoretical model, which describes the impact of strong flexoelectricity on chiral nematic systems. The chirality stabilizes both the splay-bend and twist-bend structures in the same material, while in achiral materials only one of the modulated structures can exist.
(27)
ˇ acknowledge the financial support of the N.V. and M.C. ARRS research program P1-0055, and E.G. acknowledges the financial support of the project “Mistrz 2013”.
[1] R. B. Meyer, in Structural Problems in Liquid Crystal Physics, edited by R. Balian and G. Weil, Les Houches Summer School in Theoretical Physics, 1973. Molecular Fluids (Gordon and Breach, New York, 1976), pp. 273–373.
[2] I. Dozov, Europhys. Lett. 56, 247 (2001). [3] J. Harden, B. Mbanga, N. Eber, K. Fodor-Csorba, S. Sprunt, J. T. Gleeson, and A. Jakli, Phys. Rev. Lett. 97, 157802 (2006).
(25)
where A = −1 + x 2 + 16x 4 , 2 4 B = −16k31 x + 8k31 x 2 (1 − 6x 2 ) − 1 + 8x 2 .
The modulated structure is stable if A and/or B are positive, which gives us the critical value of the cell thickness (Lcr ) below which the modulated structures are expelled: L(sb) cr = 0.47 × 2π/q0 , L(tb) cr =
1 2
√ 1 + k31 − 1 − k31 2π/q0 . k31 (3 + k31 )
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FLEXOELECTRICITY IN CHIRAL NEMATIC LIQUID CRYSTALS . . . [4] S. M. Shamid, S. Dhakal, and J. V. Selinger, Phys. Rev. E 87, 052503 (2013). [5] V. P. Panov, M. Nagaraj, J. K. Vij, Y. P. Panarin, A. Kohlmeier, M. G. Tamba, R. A. Lewis, and G. H. Mehl, Phys. Rev. Lett. 105, 167801 (2010). [6] P. J. Barnes, A. G. Douglass, S. K. Heeks, and G. R. Luckhurst, Liq. Cryst. 13, 603 (1993). [7] C. T. Imrie and P. A. Hendersson, Chem. Soc. Rev. 36, 2096 (2007). [8] C. Meyer, G. R. Luckhurst, and I. Dozov, Phys. Rev. Lett. 111, 067801 (2013). [9] 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, and O. D. Lavrentovich, Nat. Commun. 4, 2635 (2013). [10] D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, Proc. Natl. Acad. Sci. USA 110, 15931 (2013).
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[11] S. Kaur, J. Addis, C. Greco, A. Ferrarini, V. G¨ortz, J. W. Goodby, and H. F. Gleeson, Phys. Rev. E 86, 041703 (2012). [12] S. Kaur, L. Tian, H. Liu, C. Greco, A. Ferrarini, J. Seltmann, M. Lehmann, and H. F. Gleeson, J. Mater. Chem. C 1, 2416 (2013). [13] M. Majumdar, P. Salamon, A. J´akli, J. T. Gleeson, and S. Sprunt, Phys. Rev. E 83, 031701 (2011). ´ [14] P. Salamon, N. Eber, J. Seltmann, M. Lehmann, J. T. Gleeson, S. Sprunt, and A. J´akli, Phys. Rev. E 85, 061704 (2012). [15] A. Zep, S. Aya, K. Aihara, K. Ema, D. Pociecha, K. Madrak, P. Bernatowicz, H. Takezoe, and E. Gorecka, J. Mater. Chem. C 1, 46 (2013). [16] M. Brunet and N. Isaert, Ferroelectrics 84, 25 (1988). [17] P. Lebwohl and G. Lasher, Phys. Rev. A 6, 426 (1972). [18] S. Stallinga and G. Vertogen, Phys. Rev. E 49, 1483 (1994). [19] M. A. Osipov, in Flexoelectricity in Liquid Crystals: Theory, Experiments and Applications, edited by A. Buka and N. Eber (Imperial College Press, London, 2013). [20] A. Kapanowski, Phys. Rev. E 77, 052702 (2008).
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Flexoelectricity in chiral nematic liquid crystals as a driving mechanism for the twist-bend and splay-bend modulated phases ˇ c,2,3 Mikhail A. Osipov,4 and Ewa Gorecka5 Nataˇsa Vaupotiˇc,1,2 Mojca Cepiˇ 1
Faculty of Natural Sciences and Mathematics, University of Maribor, Koroˇska 160, 2000 Maribor, Slovenia 2 Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia 3 Faculty of Education, University of Ljubljana, Kardeljeva ploˇscˇ ad 16, 1000 Ljubljana 4 Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond St., G1 Glasgow, United Kingdom 5 Department of Chemistry, University of Warsaw, Zwirki i Wigury 101, 02-089 Warsaw, Poland (Received 16 January 2014; published 26 March 2014) We present a continuum theoretical model describing the impact of chirality on nematic systems with large flexoelectricity. As opposed to achiral materials, where only one type of the modulated structure can exist in a given material, the model predicts that chirality can stabilize several modulated phases, which have already been observed experimentally [A. Zep et al., J. Mater. Chem. C 1, 46 (2013)]. DOI: 10.1103/PhysRevE.89.030501
PACS number(s): 61.30.Cz, 61.30.Eb, 64.70.M−
The flexoelectric effect can be a driving mechanism for the formation of the modulated nematic phases as already pointed out by Meyer [1]. Because very large flexoelectric coefficients are required, the idea of modulated nematic phases was practically forgotten until 2001 when Dozov [2] revived it by pointing out that bent-core liquid crystals could exhibit modulated nematic phases because of the specific shape of the constituent molecules. The flexoelectric coefficient can be very large for bent-core molecules [3] and as shown by [1,4], it can lead to the reduction of the bend elastic constant by which modulated nematic phases can become energetically favorable. There are, in general, two possible modulated nematic phases. In the twist-bend structure the nematic director, giving the average direction of the local direction of long molecular axes, bends and rotates on the cone. Such bend deformation is essentially coupled to the twist deformation of the director. In the splay-bend structure the direction of the nematic director exhibits an in-plane modulation. In achiral nematics one can observe only one of the modulated structures, depending on the ratio between the splay and twist elastic constant. The first observation of the nematic to modulated nematic phase transition was reported by Panov et al. [5] for mesogenic dimers, which have a bent shape when two monomers are linked by an odd-carbon aliphatic spacer [6,7]. Recently, the twist-bend modulated phases with helicoidal structures with an extremely short modulation period of only few molecular lengths were observed in achiral bent dimers [8–10]. Observation of only the twist-bend structures is not surprising, because the splay elastic constant in bent-core liquid crystal or bent dimers is very large [11–14]. On the other hand, chiral nematics made of chiral bent dimers show by far more complex phase behavior. As reported by Zep et al. [15] six nematic-type phases were observed in chiral dimers in a temperature range between the isotropic and crystal phase. Four of the observed phases have less than 1 K temperature range, which suggests that there are several strongly competing interactions in the system. The highest temperature phase is the conventional uniaxial cholesteric phase. The helix unwinds close to the transition to the first
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lower temperature phase. The lowest temperature nematic phase has a modulated structure. When the material is filled in thin cells, the modulated structure is identified by its stripe texture due to the focusing of light on the periodic changes of refractive index. Measurements also show that, at least in some temperature range, the modulations are one dimensional, because focusing can be observed only for the light polarized along the direction parallel to the rubbing direction of the cell surface. The pitch of modulation is in a micrometer range and it increases with increasing cell thickness, which is opposite to the effect observed in the SmC* phase, where the modulation periodicity increases with decreasing cell thickness as a result of helix unwinding caused by surface interactions [16]. In this Rapid Communication we report on the theoretical model of modulated structures in chiral nematics made of chiral bent dimers. The model predicts all the above-mentioned observed properties of nematics formed by chiral dimers, both in bulk and in thin cells. It predicts the existence of the twistbend and splay-bend structures in the same material and a phase transition between them. The key idea of the model is to take into account, that the modulated phase is essentially locally biaxial as shown in [4] using the Lebwohl-Lasher lattice model [17]. If the material is biaxial (but nonpolar), it can be described by the orientational tensor order parameter with three nonequivalent orthogonal primary axes. One of these axes specifies the primary director n which is the nonpolar direction of the average local orientation of the long molecular axis. The second axis corresponds to the unit vector l, perpendicular to n, which specifies the orientation of short molecular axes (in the direction of the molecular or dimer bend), while the third axis corresponds to the third director m = n × l (see Fig. 1). For the description of the elastic free energy density (fn ) of the biaxial nematic we use the expression derived by Stalinga and Vertogen [18]. The free energy includes 12 elastic constants plus three additional ones which are related to molecular chirality. The characteristic behavior of the system composed of chiral bent dimers can be explained already in the one constant approximation when all 12 elastic constants are
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assumed to be equal, although this is not a good approximation for real bent-core systems. In the one-constant approximation the elastic free energy density reads [18] fn = k1 l · (∇ × l) + k2 m · (∇ × m) + k3 n · (∇ × n) + 12 K[(∇ · l)2 + (∇ · m)2 + (∇ · n)2 ] + 12 K{[l · (∇ × l)]2 + [m · (∇ × m)]2 + [n · (∇ × n)]2 } + 12 K{[n · (∇ × m)]2 + [l · (∇ × n)]2 + [m · (∇ × l)]2 } + 12 K{[m · (∇ × n)]2 + [n · (∇ × l)]2 + [l · (∇ × m)]2 }.
Here the constants k1 , k2 , and k3 are chiral parameters which cause the twist distortion of the l, m, and n vector fields, respectively. The parameters can be either positive or negative, depending on molecular handedness. The elastic constant (K) is always positive. In the case of chiral biaxial nematics there are three possible helices with the axis along n, l, or m, respectively (Fig. 1). As an example, let us consider a twist distortion of n and l. The helical axis is along the m vector (let this be the z axis) and spatial variations of the n and l directors are expressed as n = {cos(qb z), sin(qb z),0}, l = {− sin(qb z), cos(qb z),0},
(3)
where qb is the modulation wave vector in the biaxial phase along the z axis. Minimizing the free energy fn dz we find k1 + k3 , 2K and the free energy per unit volume (Fb ) is
and (5) one can deduce the energy of the structure with the helical twist along the n or l axis. When the modulation wave vector for the helical twist along the m or l axis becomes small enough, a structure with a helicoidal modulation along, the n axis can be stabilized. The free energy density (f ) of the modulated nematic can be approximately expressed as a sum of the elastic energy, the leading flexoelectric term, and the term that describes the dielectric energy of the system [4]: f = fn + λP · [n × (∇ × n)] + μP2 ,
(2)
and
qb =
(4)
Fb = −Kqb2 .
(5)
In the high temperature uniaxial nematic phase the pitch is qu = kt /Kt , where Kt is the uniaxial twist elastic constant and kt describes the constant in front of the linear uniaxial twist term. If in the biaxial phase k1 and k3 are of the opposite sign, the modulation length (pitch) in the biaxial phase will be larger than in the uniaxial phase, and close to the phase transition temperature one should observe a pretransitional increase in the pitch due to the onset of the biaxial order. Such an increase was, indeed, observed in [15]. From Eqs. (4)
FIG. 1. (Color online) The three possible twist structures in orthorhombic nematic liquid crystals. The orientation of the blocks is defined by three perpendicular unit vectors: n (blue solid line), l (red dotted line), and m = n × l (green dashed line).
(1)
(6)
where λ is the bend flexoelectric coefficient, P is the local polarization, and the coefficient μ = 1/2χ⊥ is the inverse dielectric susceptibility of the nematic in the direction perpendicular to n. One notes that in the general case the free energy also contains the splay flexoelectric contribution λs P · n(∇ · n), and the dielectric energy term contains the dielectric susceptibility tensor χij . In typical bent-core nematics, however, the bend flexoelectric coefficient is much larger than the splay one, and thus the splay contribution can be neglected in the first approximation. In fact, the splay term vanishes in the case of perfect nematic order for banana-shaped molecules of C2v symmetry [19]. A general molecular theory of flexoelectricity in biaxial nematics has been developed in [20] and it has been shown that there are four additional flexocoefficients in the biaxial phase. Two of them are related to bend, but these “biaxial” coefficients are much smaller than the leading uniaxial one in Eq. (6) as they are proportional to small biaxial orientational order parameters. We assume that the nematic is not ferroelectric, that is, there is no spontaneous polarization (i.e., the parameter μ is positive), and therefore the local polarization is due to the flexoelectric effect. In this case the flexoelectric polarization is approximately perpendicular to n (the splay term is small), and thus the dielectric energy contains only the transverse susceptibility χ⊥ . In the case of bent-core molecules with large transverse dipoles the transverse susceptibility increases with the decreasing temperature due to the increase of the nematic order parameter. As a result the parameter μ decreases with decreasing temperature. To find the conditions for the stability of the splay-bend and twist-bend modulated structures in bulk, we use a harmonic ansatz, which includes the splay-bend and twist-bend modulation imposed on the chiral helical modulation along the n axis (Fig. 2). We assume that the local flexoelectric polarization is in the direction of the l vector, and is perpendicular to the n and m vectors. This is not a required condition, but we can expect that due to the bent shape of the molecules or dimers this component of local polarization will be the largest. In the splay-bend modulated chiral structure there is a helical twist
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FIG. 2. (Color online) The splay-bend (top) and twist-bend (bottom) modulation superposed over the chiral helix.
of l and m along n and the bend deformation of the n director in the yz plane superposed on the chiral modulation: n = {0, cos θ, sin θ },
(7)
l = {cos(qy), − sin θ sin(qy), cos θ sin(qy)},
(8)
P = p0 {cos(qy), − sin θ sin(qy), cos θ sin(qy)},
(9)
where θ = θ0 cos(qy). The phase of θ is chosen such that the structure is nonpolar over one period of modulation. A general phase can be used, providing that proper changes are made in the ansatz for l and P, as well. In the twist-bend structure the rotation of the n director on the cone is superposed on the chiral helix: n = {sin θ0 cos(qy), cos θ0 , sin θ0 sin(qy)},
(10)
l = {sin(qy),0, − cos(qy)},
(11)
P = p0 {sin(qy),0, − cos(qy)}.
(12)
The modulation wave vector along the y direction (q) can be expressed as q = q0 (1 + q ),
(13)
where q0 = −(k1 + k2 )/(2K) is the modulation wave vector along the n axis due to the chiral twist. The dimensionless parameter q gives the relative change of the wave-vector magnitude in the splay-bend or the twist-bend modulated phase with respect to the phase with only pure chiral twist. Similarly, the free energy F = f dy is expressed as F = F0 + F,
(14)
where F0 = −(k1 + k2 ) /(4K) is the free energy of the phase with only chiral twist and F is the change in energy due to the splay-bend or twist-bend modulation. Let us consider the splay-bend structure first. We expand the free energy up to the fourth order in θ0 and minimize the free energy over p0 , θ0 , and q. Complex expressions are obtained and they can be significantly simplified, if (without losing the generality of the result) we set k2 = 0. Minimization shows that the modulated structure is stable if μ < μ(sb) 0 : 2
λ2 , (15) 6K where the index (sb) stands for the splay-bend structure. The temperature dependent parameter μ is then expressed as = μ(sb) 0
μ = μ(sb) μ), 0 (1 −
(16)
where μ is proportional to T = T0 − T , and T0 is the temperature at which the modulated structure becomes stable.
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Then we assume that below the critical value of μ0 the μ)k to quantities p0 , θ0 , q, and F are proportional to ( the lowest order where the exponent k is determined from the corresponding equations. As a result one finds the following expressions: √ 2 9 2 k1 (sb) (sb) θ0 = 6 μ, p0 = − √ μ, (17) 31 31 λ 9 27 F (sb) = μ, ( μ)2 . 31 F0 31 For the twist-bend (tb) structure one finds q (sb) =
(18)
λ2 , (19) 4k31 K where the ratio between the chiral twist parameters k31 = k3 /k1 was introduced. If the helix along the m axis unwinds upon the transition to the biaxial phase, we expect k3 and k1 to be of the opposite sign [see Eq. (4)], so μ(tb) 0 is positive. To the lowest order term in μ, we find the following expressions for the model parameters and the energy: μ μ k3 (tb) (tb) θ0 = , p0 = 2 , (20) 2 + k31 λ 2 + k31 μ(tb) 0 =−
q (tb) = −
k31 μ, 2 + k31
F (tb) k31 =− ( μ)2 . F0 2 + k31 (21)
Comparing Eqs. (15) and (19) we see that the signs and the magnitudes of the chiral parameters define which modulated structure is stable and in which temperature range. By taking k31 = −1, one finds μ(sb) = 2/3μ(tb) so the twist0 0 bend structure occurs at higher temperatures. Using μ = μ and a similar expression for the twist-bend case, it μ(sb) 0 is straightforward to find F (sb) /F0 = −31.4(Kμ/λ2 )2 and F (tb) /F0 = −16(Kμ/λ2 )2 from which it follows that the splay-bend structure has lower energy. One can thus envisage the following phase sequence upon the reduction of temperature: uniaxial nematic (chiral twist along the z axis) → biaxial nematic (three possibilities for the chiral twist direction) → modulated twist-bend nematic → modulated splay-bend nematic. This simple model can thus explain the occurrence of six nematic phases. Additional, three-dimensional structures are also possible. One notes that the predicted phase sequence is specific for the chosen set of parameters. For example, if k31 > 0, then μ(tb) 0 < 0 [see Eq. (19)], so in this case only the splay-bend structure is stable. When the chiral terms vanish (achiral material), there is no reason to take the corresponding limit in the above expressions, because they represent only one specific solution of the set of equations, obtained by the free energy minimization under the assumption that k1 and k3 are nonzero. One solution, obtained by the minimization of the free energy, is always a uniform structure with θ0 , p0 , and q being equal to zero. The solution for achiral nematics cannot be directly obtained from this model, because in achiral systems the modulated structures can be obtained only if the coupling between the magnitude of polarization and its spatial variation is taken into consideration [4]. Comparing the temperature dependence of the reduced
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ˇ CEPI ˇ ˇ OSIPOV, AND GORECKA VAUPOTIC, C,
PHYSICAL REVIEW E 89, 030501(R) (2014)
energy of the modulated structure in the achiral and chiral nematics, one concludes that the tendency to form modulated structures is much stronger in chiral nematics, because the reduced energy in the chiral nematics is of the lower order [it is proportional to (T )2 , while in achiral nematics it is proportional to (T )3 ; see [4]]. Finally, let us consider the modulated structures in confined geometry, in planar thin cells of thickness L with the z axis being perpendicular to the cell surface and the y axis along the cell surface. We neglect all the effects that surfaces impose on the chiral helix and assume only that the splay-bend or twistbend modulation reduces to zero at the surface. The director modulation is along the y axis, which is along the surfaces. The ansatz given by Eqs. (7)–(9) for the splay-bend structure and Eqs. (10)–(12) for the twist-bend structure are modified to include the effect of the surfaces, such that θ0 → θ0 sin(π z/L) and p0 → p0 sin(π z/L). Minimizing the free energy, we find x 2 λ2 , 1 + 6x 2 K
(22)
2x 2 λ2 , K(1 − 8k31 x 2 )
(23)
= μ(sb) 0 μ(tb) 0 =
where x is defined as a ratio between the cell thickness and the modulation length x = Lq0 /(2π ). The temperature of the transition to the modulated phase depends on the cell thickness, as expected, and in the limit of large L expressions (22) and (23) reduce to Eqs. (15) and (19), respectively. For the change in the wave-vector magnitude and energy one obtains (1 + x 2 )(1 + 6x 2 )2 K μ, 2Ax 2 λ2 (1 − 8k31 x 2 )2 (1 − 4k31 x 2 ) K q (tb) = μ, 2Bx 2 λ2 2 (1 + 6x 2 )4 K (sb) μ , F /F0 = − 4Ax 4 λ2 2 (1 − 8k31 x 2 )4 K (tb) μ , F /F0 = − 16Bx 4 λ2
q (sb) =
FIG. 3. (Color online) (a) The relative change in the wave-vector magnitude ( q ) in the twist-bend (dashed blue line) and splay-bend (solid red line) modulated structures as a function of the cell thickness L given in units of the modulation length 2π/q0 [x = Lq0 /(2π )]. = F /F0 ) as a function of (b) The relative reduction in energy (F (tb) , blue dashed line) and the cell thickness for the twist-bend (F (sb) , red solid line) modulated structures. Parameter splay-bend (F values: k31 = −1, μK/λ2 = 10−3 . Inset: the value of μ at which the modulated structure becomes stable. Blue (dashed) line: μ(tb) 0 . Red solid line: μ(sb) 0 .
(24)
(26)
The results for the change in the wave-vector magnitude and energy reduction as a function of cell thickness are given in Fig. 3 for both the splay-bend and twist-bend structures. At a chosen set of parameters the model predicts that the modulation wave vector decreases with increasing cell thickness [see Fig. 3(a)], meaning that the modulation wavelength increases with increasing cell thickness, as observed experimentally. The splay-bend structure has lower free energy than the twist-bend structure [Fig. 3(b)]. However, the splay-bend structure becomes stable at a lower temperature [see the inset to Fig. 3(b)]. To conclude, we have proposed a theoretical model, which describes the impact of strong flexoelectricity on chiral nematic systems. The chirality stabilizes both the splay-bend and twist-bend structures in the same material, while in achiral materials only one of the modulated structures can exist.
(27)
ˇ acknowledge the financial support of the N.V. and M.C. ARRS research program P1-0055, and E.G. acknowledges the financial support of the project “Mistrz 2013”.
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(25)
where A = −1 + x 2 + 16x 4 , 2 4 B = −16k31 x + 8k31 x 2 (1 − 6x 2 ) − 1 + 8x 2 .
The modulated structure is stable if A and/or B are positive, which gives us the critical value of the cell thickness (Lcr ) below which the modulated structures are expelled: L(sb) cr = 0.47 × 2π/q0 , L(tb) cr =
1 2
√ 1 + k31 − 1 − k31 2π/q0 . k31 (3 + k31 )
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FLEXOELECTRICITY IN CHIRAL NEMATIC LIQUID CRYSTALS . . . [4] S. M. Shamid, S. Dhakal, and J. V. Selinger, Phys. Rev. E 87, 052503 (2013). [5] V. P. Panov, M. Nagaraj, J. K. Vij, Y. P. Panarin, A. Kohlmeier, M. G. Tamba, R. A. Lewis, and G. H. Mehl, Phys. Rev. Lett. 105, 167801 (2010). [6] P. J. Barnes, A. G. Douglass, S. K. Heeks, and G. R. Luckhurst, Liq. Cryst. 13, 603 (1993). [7] C. T. Imrie and P. A. Hendersson, Chem. Soc. Rev. 36, 2096 (2007). [8] C. Meyer, G. R. Luckhurst, and I. Dozov, Phys. Rev. Lett. 111, 067801 (2013). [9] 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, and O. D. Lavrentovich, Nat. Commun. 4, 2635 (2013). [10] D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, Proc. Natl. Acad. Sci. USA 110, 15931 (2013).
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