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1 Flexoelectric effect in biaxial nematics

A. Kapanowski

Institute of Physics, Jagiellonian University,

ulica Reymonta 4, 30-059 Cracow, Poland

February 10, 2011

Abstract

The flexoelectric (FE) effect provides a linear coupling betweenelectric polarization and orientational deformation in liquid crystals.It influences many electrooptical phenomena and it is used in somebistable nematic devices. A statistical theory of dipole FE polariza-tion in biaxial nematic liquid crystals is used to calculate temperaturedependence of order parameters, elastic constants, and FE coefficients.The splitting of the two Meyer FE coefficients and the appearance ofnew FE coefficients is obtained at the uniaxial-biaxial nematic tran-sition. The ordering of the splited FE coefficients corresponds to theordering of the splited elastic constants.

Keywords: biaxial nematic liquid crystals, flexoelectric effect, elasticconstants.

1 Introduction

The biaxial nematic liquid crystals were first predicted by Freiser [1], whoshowed that molecules with shapes that deviate from cylindrical symmetrycould possess a nematic phase with three distinct optical axes. A biaxialnematic phase was first observed in a lyotropic mixture by Yu and Saupe [2]in 1980 but the existence of a thermotropic biaxial system was not certain formany years. Several reports appeared in 2004 on thermotropic liquid crystalsformed by side-chain polymers [3], bent-core molecules [4, 5], and tetrapodes[6]. The number of new biaxial systems is constantly growing.

1

Many theoretical papers [7] and computer simulations [8] show that themolecular shape and pair interaction biaxiality are important for the biaxialphase to exist. However, very often real systems favour packing in the smec-tic or crystalline biaxial phases. It is a challenge for the theory to find factorsresponsible for absolute stability of the biaxial nematic phase. It was shownthat fluctuations in molecular shape can influence the biaxial nematic phasestability [9]. The motivation for this search ranges from purely academicinterest to the potential usage of these materials in faster displays, wherein principle the commutation of the secondary director should give lower re-sponse times compared to the conventional twisted nematic and ferroelectricsmectic devices.

A static electric field imposed on a nematic liquid crystal have many phys-ical effects, but the most important are two of them. One is connected withthe anisotropy of the dielectric constant. The second effect is the appearanceof the spontaneous polarization in a deformed liquid crystal; this is calledthe flexoelectric effect. Conversely, an electric field may induce distortionsin the bulk. In 1969 Meyer showed that it is a steric effect due to the shapeasymmetry of polar molecules [10]. In case of nonpolar molecules the FEeffect originates from a gradient of quadrupole moment density [11].

The two FE coefficients were introduced by Meyer for splay and benddistortions of the uniaxial nematic phase [10]. Recently, a statistical theoryfor the dipole FE polarization was derived in the case of the biaxial nematicphase composed of the C2v molecules [12]. There are six splay-bend defor-mations of the biaxial nematic phase and thus six FE coefficients are defined,but only five of them are independent. General microscopic expressions forthe FE coefficients involve the one-particle distribution function and the po-tential energy of two-body short-ranged interactions.

The FE effect has a large influence on many phenomena in liquid crys-tals: electrooptical phenomena and defect formation, for instance. It playsa key role in some device applications. Flexoelectric switching is importantin bistable displays [13]-[15]. Flexoelectric coupling in chiral and twisted ne-matic crystals [16] leads to a linear rotation of the optic axis and also leadsto device applications [17]. Flexoelectric coupling in smectic liquid crystalshas been shown to stabilize helical structures [18].

The purpose of this study is to calculate the values of the FE coefficientsand other material parameters for model systems. The proper form of theinteraction potential energy allows us to calculate the temperature depen-dence of the order parameters, the FE coefficients, and the elastic constants.

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The uniaxial and biaxial nematic phases are considered.

2 Description of the system

Let us consider a set of N molecules contained in a volume V , at the tem-perature T . The molecules are rigid blocks (C2v symmetry) with three trans-lational and three rotational degrees of freedom. It is assumed that themolecules interact via two-body short-range forces that depend on the dis-tance between the molecules (~u = ~r2 − ~r1 = u~∆) and their orientations de-scribed by the three Euler angles R = (φ, θ, ψ) or three orthonormal vectors

(~l, ~m,~n).The microscopic free energy F for the system is given by

F = Fent + Fint, (1)

βFent =∫

d~rdRG(~r, R){ln[G(~r, R)Λ]− 1}, (2)

βFint = −1

2

∫

d~r1dR1d~r2dR2G(~r1, R1)G(~r2, R2)f12. (3)

Here f12 = exp(−βΦ12) − 1 is the Mayer function, Φ12 the potential en-ergy of interactions, dR = dφdθ sin θdψ, β = 1/(kBT ), and Λ is related tothe ideal gas properties. The one-particle distribution function G has thenormalization

∫

d~rdRG(~r, R) = N. (4)

The equilibrium distribution G minimizing the free energy (1) satisfies theequation

ln[G(~r1, R1)Λ]−∫

d~r2dR2G(~r2, R2)f12 = const. (5)

For the homogeneous phase G = G0 does not depend on the position and ithas the form [19]

G0(R) = G0(~l · ~L,~l · ~N,~n · ~L, ~n · ~N), (6)

were the orthonormal vectors (~L, ~M, ~N) define the biaxial nematic phaseaxes. In practice we characterize the alignment not through the full functionG, but by some numerical parameters - order parameters. In the case of

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the biaxial nematic phase the main order parameters are the orientationaldistribution averages of the following four functions [20]: F

(2)00 , F

(2)02 , F

(2)20 , and

F(2)22 . We note that there are other notations [22]. In the uniaxial nematic

phase the functions F(2)00 and F

(2)02 have nonzero averages only. The molecule

alignment can be also described by order tensors Q which are often calculatedfor computer simulations [23] where the molecular and laboratory axes mustbe distinguished. The order tensors are defined as

Qllαβ = (3lαlβ − δαβ)/2, (7)

Qmmαβ = (3mαmβ − δαβ)/2, (8)

Qnnαβ = (3nαnβ − δαβ)/2. (9)

3 Elastic deformations of the phase

Orientational ordering of biaxial nematics is usually described by the theeorthonormal vectors

~L = R1α~eα, ~M = R2α~eα, ~N = R3α~eα. (10)

In the homogeneous phase the vectors (~L, ~M, ~N) are constant in space, butin a deformed phase they depend on the position in space. In a continuumapproach the distortion free-energy density fd is obtained as an expansionabout an undistorted reference state with respect to gradients of the vectors(~L, ~M, ~N). The form of the fd can be derived in many alternative ways butwe use the form presented by Stallinga and Vertogen [21] (the surface termsare neglected)

fd =1

2K1111(D11)

2 +1

2K1212(D12)

2 +1

2K1313(D13)

2

+1

2K2121(D21)

2 +1

2K2222(D22)

2 +1

2K2323(D23)

2

+1

2K3131(D31)

2 +1

2K3232(D32)

2 +1

2K3333(D33)

2

+K1122D11D22 +K1133D11D33 +K2233D22D33

+K1221D12D21 +K1331D13D31 +K2332D23D32. (11)

Dij =1

2ǫjklRiαRkβ∂αRlβ. (12)

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Microscopic expressions for the elastic constants Kijkl were derived in [19]and it was shown that there are 12 independent bulk constants because

K1221 = K1122, K1331 = K1133, K2332 = K2233. (13)

4 Flexoelectric polarization

Liquid crystalline phases often consist of polar molecules but in homoge-neous nematic phases the average polarization is zero. On the other hand, aphase distortion can produce a polarization and this is called the FE effect.In a continuum approach the FE polarization of the biaxial nematic phasedepends on the spatial derivatives of the vectors (~L, ~M, ~N) [12]

Pα =∑

i

(siiRiα∂βRiβ + biiRiβ∂βRiα). (14)

The parameters sii and bii, (i = 1, 2, 3) are not unique because if we addany constant to all of them, the polarization will not change. The physicalFE coefficients ai (i = 4, . . . , 9) are

a4 = s33 − b11, a5 = s22 − b11, a6 = s33 − b22,

a7 = s11 − b22, a8 = s22 − b33, a9 = s11 − b33. (15)

The coefficients satisfy the identity

a4 − a5 − a6 + a7 + a8 − a9 = 0. (16)

Deformations of the biaxial nematic phase connected with the FE effect aregiven in Table 1. In the case of the uniaxial nematic phase the FE polarizationhas the form

Pα = e1Nα∂βNβ + e3Nβ∂βNα. (17)

Let us define the molecule electric dipole moment as

µα = µ1lα + µ2mα + µ3nα. (18)

In the case of the molecular iteractions described below, the FE coefficientscan be expressed as follows

a4 =∫

d~udR1dR2f12G0(R1)µ3n1z(−ux)(U2z −W2x), (19)

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a5 =∫

d~udR1dR2f12G0(R1)µ3n1y(−ux)U2y, (20)

a6 =∫

d~udR1dR2f12G0(R1)µ3n1zuyW2y, (21)

a7 =∫

d~udR1dR2f12G0(R1)µ3n1xuyU2y, (22)

a8 =∫

d~udR1dR2f12G0(R1)µ3n1y(−uz)W2y, (23)

a9 =∫

d~udR1dR2f12G0(R1)µ3n1xuz(U2z −W2x), (24)

where it is assumed that ~n defines the molecule C2 axis and

Uα = ∂1G0lα + ∂3G0nα, Wα = ∂2G0lα + ∂4G0nα. (25)

5 Results

We performed our calculations for the square-well potential energy of theform

Φ12(u/σ) =

+∞ for (u/σ) < 1,−ǫ for 1 < (u/σ) < 2,0 for (u/σ) > 2,

(26)

were ǫ is the depth of the well and σ depends on the molecule orientationsand on the vector ~∆

σ = σ0 + σ1(~∆ · ~n1 − ~∆ · ~n2) + σ2[

(~∆ · ~n1)2 + (~∆ · ~n2)

2]

+ σ3[

(~∆ ·~l1)2 + (~∆ ·~l2)

2]

(27)

The parameter σ0 defines the length scale, σ1 defines the FE term, σ2 andσ3 define biaxial nematic terms. We used the density NVmol/V = 0.1, themolecule volume Vmol was estimated from the mutually excluded volume.The FE coefficients were expressed in µi/σ

20 , the elastic constants in ǫ/σ0, and

the temperature in ǫ/kB. The parameters σi are given in Table 2. The twophysical systems are considered that consist of wedge-shaped and banana-shaped molecules.

5.1 Wedge-shaped biaxial molecules

In the system of wedge-shaped biaxial molecules, on decreasing the temper-ature we meet the first order transition from the isotropic to the uniaxial

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nematic phase at TIN = 0.618 and the second order transition to the biaxialnematic phase at TNB = 0.401. The temperature dependence of the ordertensors is presented in Fig. 1. The temperature dependence of the elasticconstants and the FE coefficients are presented in Figs. 2 and 3, respectively.

The values of Qnnzz show that long molecule axes are directed along the

Z axis in the whole nematic region, whereas the values of Qllxx reveal the

alignment of short molecule axes along the X axis and it is enhanced inthe biaxial nematic phase. The splay elastic constant K1 splits into K1212

and K2121. The bend elastic constant K3 splits into K3232 and K3131. Notethat the equality K1 = K3 is accidental and results from neglecting orderparameters F (j)

µν with j > 2. The splay FE coefficient e1 splits into a4 and a6(a4 > a6 > 0). The bend FE coefficient −e3 splits into a8 and a9 (0 > a8 >a9). The coefficients a5 and a7 are small and almost always negative.

5.2 Banana-shaped biaxial molecules

In the system of banana-shaped biaxial molecules, on decreasing the tem-perature we meet the first order transition from the isotropic to the uniaxialnematic phase at TIN = 0.595 and the second order transition to the biaxialnematic phase at TNB = 0.382. The temperature dependence of the ordertensors is presented in Fig. 4. The temperature dependence of the elasticconstants and the FE coefficients are presented in Figs. 5 and 6, respectively.

According to the values of Qllzz long molecule axes are directed along the

Z axis in the whole nematic region, whereas the values of Qnnxx show the

alignment of short molecule axes along the X axis and it is enhanced in thebiaxial nematic phase. The behaviour of the elastic constants is similar tothe case of the wedge-shaped molecules because the FE term is small in bothcases. The bend FE coefficient −e3 splits into a8 and a9 (0 > a8 > a9). Thesplay FE coefficient e1 is smaller then e3 and it splits into a4 and a6. Thecoefficients a5 and a7 are again small but comparable with a4 and a6. Thesign of some coefficient can change on changing the temperature.

6 Conclusions

In this paper, the statistical theory was used to study the temperature de-pendence of the order parameters, elastic constants, and FE coefficients ofbiaxial nematic liquid crystals. In order to calculate these macroscopic pa-

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rameters one needs the one-particle distribution function and the potentialenergy of molecular interactions. The two physical systems were considered.The splittings of the FE coefficients and the elastic constants were obtained atthe uniaxial-biaxial nematic transition. New small FE coefficients appearedat the transition. The ordering of the splited FE coefficients corresponds tothe ordering of the splited elastic constants.

The FE coefficients were proportional to the dipole moment componentparallel to the molecule C2v symmetry axis. This was the result of the inter-actions potential symmetry. The beaviour of the main FE coefficients, e1 forthe wegde-shaped molecules and e3 for the banana-shaped molecules, is clearand it is in the agreement with previous studies [24]. On the other hand,it seems that other FE coefficients should be interpreted with caution. It ispossible that higher order parameters can have a significant contribution.

At present stage, the direct comparison between the theory and the ex-periment in not possible because to our knowledge the FE coefficients havenot been measured for the biaxial nematic phase. What is more, even forthe uniaxial nematic phase the experimental data are still scarce and some-times contradictory [25]. However, when biaxial nematic phases become morewidespread, the presented theory will be helpful in practical applications.

Acknowledgements

The author is grateful to J. Spałek for his support and discussions.

References

[1] M. J. Freiser, "Ordered states of a nematic liquid", Phys. Rev. Lett. 24,1041-1043 (1970).

[2] L. J. Yu and A. Saupe, "Observation of a biaxial nematic phase inpotassium-1-decanol-water mixtures", Phys. Rev. Lett. 45, 1000-1003(1980).

[3] K. Severing and K. Saalwachter, "Biaxial nematic phase in a ther-motropic liquid crystalline side-chain polymer", Phys. Rev. Lett. 92,125501 (2004).

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[4] L. A. Madsen, T. J. Dingemans, M. Nakata, and E. T. Samulski, "Ther-motropic biaxial nematic liquid crystals", Phys. Rev. Lett. 92, 145505(2004).

[5] B. R. Acharya, A. Primak, and S. Kumar, "Biaxial nematic phase inbent-core thermotropic mesogens", Phys. Rev. Lett. 92, 145506 (2004).

[6] K. Merkel, A. Kocot, J. K. Vij, R. Korlacki, G. H. Mehl, andT. Meyer, "Thermotropic biaxial nematic phase in liquid crystallineorgano-siloxane tetrapodes", Phys. Rev. Lett. 93, 237801 (2004).

[7] L. Longa, P. Grzybowski, S. Romano, and E. Virga, "Minimal couplingmodel of the biaxial nematic phase", Phys. Rev. E 71, 051714 (2005).

[8] R. Berardi, L. Muccioli, S. Orlandi, M. Ricci, and C. Zannoni, "Com-puter simulations of biaxial nematics", J. Phys.: Condens. Matter 20,463101 (2008).

[9] L. Longa, G. Pajśk, and T. Wydro, "Stability of biaxial nematic phasefor systems with variable molecular shape anisotropy", Phys. Rev. E 76,011703 (2007).

[10] R. B. Meyer, "Piezoelectric effects in liquid crystals", Phys. Rev. Lett.22, 918-921 (1969).

[11] J. Prost and J. P. Marcerou, "On the microscopic interpretation of flex-oelectricity", J. Phys. (Paris) 38, 315-324 (1977).

[12] A. Kapanowski, "Flexoelectric polarization in the biaxial nematicphase", Phys. Rev. E 77, 052702 (2008).

[13] C. Denniston and J. M. Yeomans, "Flexoelectric surface switching ofbistable nematic devices", Phys. Rev. Lett. 87, 275505 (2001).

[14] A. J. Davidson and N. J. Mottram, "Flexoelectric switching in a bistablenematic device", Phys. Rev. E 65, 051710 (2002).

[15] L. A. Parry-Jones, R. B. Meyer, and S. J. Elston, "Mechanisms of flexo-electric switching in a zenithally bistable nematic device", J. Appl. Phys.106, 014510 (2009).

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[16] J. S. Patel and R. B. Meyer, "Flexoelectric electro-optics of a cholestericliquid crystal", Phys. Rev. Lett. 58, 1538-1540 (1987).

[17] A. E. Blatch, M. J. Coles, B. Musgrave, and H. J. Coles, "Flexoelectricliquid crystal bimesogens", Mol. Cryst. Liq. Cryst. 401, 161-169 (2003).

[18] M. Cepic and B. Zeks, "Flexoelectricity and Piezoelectricity: The Rea-son for the Rich Variety of Phases in Antiferroelectric Smectic LiquidCrystals", Phys. Rev. Lett. 87, 085501 (2001).

[19] A. Kapanowski, "Statistical theory of elastic constants of biaxial nematicliquid crystals", Phys. Rev. E 55, 7090-7104 (1997).

[20] M. Fialkowski, A. Kapanowski, K. Sokalski, "Microscopic approach totheory of biaxial nematic liquid crystals", Mol. Cryst. Liq. Cryst. 265,371-385 (1995).

[21] S. Stallinga, G. Vertogen, "Theory of orientational elasticity", Phys.Rev. E 49, 1483-1495 (1994).

[22] R. Rosso, "Orientational order parameters in biaxial nematics: Poly-morfic notation", Liq. Cryst. 34, 737-748 (2007).

[23] P. J. Camp and M. P. Allen, "Phase diagram of the hard biaxial ellipsoidfluid", J. Chem. Phys. 106, 6681 (1997).

[24] A. Kapanowski, "Flexoelectric effect modelling", Opto-Electron. Rev.16, 9-12 (2008).

[25] A. G. Petrov, Measurements and interpretation of flexoelectricity", inPhysical properties of liquid crystals: nematics, pp. 251-264, edited byD. Dunmur, A. Fukuda, and G. L. Luckhurst, INSPEC, The Institutionof Electrical Engineers, London, 2001.

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Table 1: Deformations of the biaxial nematic phase connected with the FEeffect. The corresponding elastic constants and the FE coefficients are given,the values for the uniaxial nematic phase are in parentheses.

Deformation Elastic constant FE coefficient~N splay, ~L bend K1212 (K1) a4 (e1)~M splay, ~L bend K1313 (0) a5 (0)~N splay, ~M bend K2121 (K1) a6 (e1)~L splay, ~M bend K2323 (0) a7 (0)~M splay, ~N bend K3131 (K3) a8 (−e3)~L splay, ~N bend K3232 (K3) a9 (−e3)

Table 2: Parameters σi used in calculations.

Molecules σ1/σ0 σ2/σ0 σ3/σ0 Long axis Short axis C2 axis

wedge-like 0.2 0.5 −0.4 ~n ~l ~n

banana-like 0.2 −0.4 0.5 ~l ~n ~n

11

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Ord

er te

nsor

s

T

QllxxQllyyQllzz

QnnxxQnnyyQnnzz

Figure 1: The temperature dependence of the order tensors for wedge-shapedbiaxial molecules.

12

0

2

4

6

8

10

12

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Ela

stic

con

stan

ts

T

K1212K1313K2121K2323K3131K3232

Figure 2: The temperature dependence of the elastic constants for wedge-shaped biaxial molecules. The squares, triangles, and circles denote defor-mations with ~N splay, ~N bend, and ~N constant, respectively. The empty(filled) symbols indicate the larger (smaller) parameter.

13

-1

0

1

2

3

4

5

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Fle

xoel

ectr

ic c

oeffi

cien

ts

T

a4a5a6a7a8a9

Figure 3: The temperature dependence of the flexoelectric coefficients forwedge-shaped biaxial molecules. Symbols have the same meaming as inFig. 2.

14

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Ord

er te

nsor

s

T

QllxxQllyyQllzz

QnnxxQnnyyQnnzz

Figure 4: The temperature dependence of the order tensors for banana-shaped biaxial molecules.

15

0

2

4

6

8

10

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Ela

stic

con

stan

ts

T

K1212K1313K2121K2323K3131K3232

Figure 5: The temperature dependence of the elastic constants for banana-shaped biaxial molecules. Symbols have the same meaming as in Fig. 2.

16

-2

-1.5

-1

-0.5

0

0.5

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Fle

xoel

ectr

ic c

oeffi

cien

ts

T

a4a5a6a7a8a9

Figure 6: The temperature dependence of the flexoelectric coefficients forbanana-shaped biaxial molecules. Symbols have the same meaming as inFig. 2.

17