Director reorientation and order reconstruction: competing mechanisms in a nematic cell

Continuum Mech. Thermodyn. (2008) 20: 193–218DOI 10.1007/s00161-008-0077-x

ORIGINAL ARTICLE

Milan Ambrožic · Fulvio Bisi · Epifanio G. Virga

Director reorientation and order reconstruction: competingmechanisms in a nematic cell

Received: 5 December 2007 / Accepted: 9 June 2008 / Published online: 18 July 2008© Springer-Verlag 2008

Abstract We propose a model to explore the competition between two mechanisms possibly at work in anematic liquid crystal confined within a flat cell with strong uniaxial planar conditions on the bounding platesand subject to an external field. To obtain an electric field perpendicular to the plates, a voltage is imposedacross the cell; no further assumption is made on the electric potential within the cell, which is thereforecalculated together with the nematic texture. The Landau-de Gennes theory of liquid crystals is used to derivethe equilibrium nematic order tensor Q. When the voltage applied is low enough, the equilibrium texture isnearly homogeneous. Above a critical voltage, there exist two different possibilities for adjusting the ordertensor to the applied field within the cell: plain director reorientation, i.e., the classical Freedericksz transition,and order reconstruction. The former mechanism entails the rotation of the eigenvectors of Q and can bedescribed essentially by the orientation of the ordinary uniaxial nematic director, whilst the latter mechanismimplies a significant variation of the eigenvalues of Q within the cell, virtually without any rotation of itseigenvectors, but with the intervention of a wealth of biaxial states. Either mechanism can actually occur,which yields different nematic textures, depending on material parameters, temperature, cell thickness and theapplied potential. The equilibrium phase diagram illustrating the prevailing mechanism is constructed for asignificant set of parameters.

Keywords Order reconstruction · Nematic liquid crystals · Freedericksz transition · Biaxiality

PACS 61.30.Cz, 61.30.Dk, 61.30.Gd, 61.30.Pq

1 Introduction

In the last two decades, the role of order reconstruction in nematic liquid crystals (NLCs) has been assessedunder different conditions. Such a phenomenon was first retraced in defects [1]: within the core of a 1

2 -dis-clination, two uniaxial states, with orthogonal directors, are changed into one another through a plethora ofintervening biaxial states with no director rotation. Uniaxial states in liquid crystals are traditionally describedby a single director n, which represents the average molecular orientation, while biaxial states need to bedescribed by an order tensor Q with unequal eigenvalues, as molecules do not appear to be aligned in a singledirection. This description gives grounds for an early, perhaps more technical terminology, which also refers toorder reconstruction as eigenvalue exchange, as it occurs when two eigenvalues of Q gradually approach each

Communicated by S. Roux

M. AmbrožicEngineering Ceramics Department, Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia

F. Bisi (B) · E. G. VirgaDipartimento di Matematica and CNISM, Università di Pavia, via Ferrata 1, 27100 Pavia, ItalyE-mail: [email protected]

194 M. Ambrožic et al.

other and are eventually swapped, without this involving any rotation of the eigenframe of Q. More recently,order reconstruction has been studied in systems where antagonistic surface conditions affect the order in thebulk [2,3]; in these situations, it has been shown that two orthogonal directors prescribed on opposite boundingplates can be connected either by a distortion in the director field (a twist or a bend, according to the specificboundary condition), or through a reconstruction employing biaxial states.

At present, it is accepted that order reconstruction is not a mere theoretical speculation; indeed, there existsevidence supporting that a nematic order is first destroyed and then reconstructed upon applying a strongelectric field [4–6]. Moreover, it has been shown that mechanical instabilities observed in force measure-ments [7–11], previously also explained with the formation and interdigitation of smectic layers in the vicinityof the cell’s plates, can indeed be interpreted as a signature of nematic order reconstruction occurring beforeany smectic order establishes itself within the cell [12,13]. The mechanical signature of order reconstructionhas also been confirmed by a molecular dynamics simulation [14], an approach completely different from theone so far taken to illustrate order reconstruction. More recently, the interplay between order reconstructionsin a defect core and in bulk has been explored in greater detail; the presence of defects in a NLC cell exhibitingorder reconstruction may reduce the critical voltage at which this occurs [15]. Homogenous and inhomoge-neous order reconstructions, the latter involving both formation and migration of defects, have recently beencompared theoretically and observed experimentally [16].

Order reconstruction has been widely studied in recent years, especially to explain the molecular reorga-nisation occurring in a NLC cell subject to an external field. However, this is not the only mechanism possiblyat work; indeed, director reorientation, as in the classical Freedericksz transition, is a phenomenon known fora long time. At first, in the early decades of the past century, it was observed and described for systems subjectto a magnetic field [17–19]; later, its electric analogue was also analysed [20], alongside with the effects ofcombining both magnetic and electric fields [21]. The theoretical analysis of systems like this is often carriedout for a magnetic field, as the electromagnetic equations can then be easily decoupled from those governingthe elastic director distortions; however, most applications are driven by an electric field. There, if the voltageexceeds a critical value, the elastic curvature energy attains its minimum not for a homogeneous texture butfor a distorted one, tending to align the nematic director along the electric field; in such a case, the thresholdvoltage is independent of the separation between the two plates in the cell. Mathematically, this is a typicalbifurcation scenario. In practice, Freedericksz transition also allows measuring the elastic constants of liquidcrystals [22,23].

To sum up, Freedericksz transition has so far been interpreted in terms of an elastic distortion in whichthe molecular reorganisation occurs through a reorientation of the nematic director, but order reconstruction isan alternative mechanism, capable in principle of also explaining Freedericksz transition without any directorreorientation. Thus, a nematic cell appears as an ideal case study, where both order reconstruction and directorreorientation can be contrasted. It should be interesting to see which of these interpretation models is moreadequate, and—more importantly—under what circumstances. It is not just a matter of interpretation, as thesetwo models bear distinctive features, subjectable to experimental scrutiny. In fact, for pure director reorienta-tion we would expect a critical voltage independent of the dimension of the cell; for pure order reconstruction,this should not be the case.

Imagining a kind of coexistence between these pure models, in this paper we explore how the criticalthreshold depends on the distance between the plates of the cell. The paper is organised as follows. In Sect. 2,within the Landau-de Gennes theory of liquid crystals, we study the equilibrium configurations of a nematiccell with strong-anchoring uniaxial boundary conditions; furthermore, in the presence of an electric field, weprescribe the electric potential on the plates, but we make no further assumption on the behaviour of the fieldinside the cell. The liquid crystal ordering is described in terms of the symmetric, traceless tensor Q, for whicha useful representation is introduced in Sect. 2.3; the calculation of the free energy allows introducing twonormalisation constants, see Sect. 2.4, which give intrinsic scale factors to measure the thickness of the celland the voltage applied to it. The differential equations governing the equilibrium states of the nematic cell arederived in Sect. 2.5, and the boundary conditions appropriate for this representation are presented in Sect. 2.6.In Sect. 3, we discuss how different classes of solutions are obtained. In one class, the nematic director tendsto align itself parallel to the electric field; here the director reorientation is dominant. In the other class, theeigenframe of Q is nearly immobile; here the order reconstruction is dominant. We discuss how these typesof solution compete and how they can be transformed into one another as a function of both the potentialdifference between the plates delimiting the cell and their separation. Additional mathematical details aregiven in the closing Appendices.

Director reorientation and order reconstruction 195

Fig. 1 A sketch of the plates and the chosen reference frame {ex , ey, ez}. The orthogonal unit vectors ex and ey are in the planecommon to the parallel plates; ez is orthogonal to both ex and ey . The rotated frame {e1, e2, e3} has e1 = ex ; ϕ is the angle thate2 makes with ey . B is the region occupied by the nematic liquid crystal; ∂B is its boundary, comprised of two plates delimitingthe cell, to which the voltage generator is connected

2 The model

2.1 Geometric setting

In our case study, we will consider a cell delimited by parallel rectangular plates; the thickness of the cell willbe taken to be 2d . We introduce a frame {ex , ey, ez}, in which the ez direction is orthogonal to the plates; thesize of the cell in both directions x and y is supposed to be much larger than the thickness of the cell, so thatwe can treat the plates as infinitely extended in both directions (see Fig. 1). For symmetry reasons, the originof the frame is placed at mid-distance between the plates, so that they are situated at z = ±d .

The symmetry of the system in Fig. 1, in the case of infinite plates, suggests searching for solutions thatretain only the variable z along ez in the equilibrium equations (see Eqs. (22) below). Furthermore, it is assumedthat the eigenframe of Q, denoted by {e1, e2, e3}, has e1 = ex , uniformly across the cell, and it is determinedby the rotation angle ϕ that e2 makes with ey . This choice is consistent with the classical description of theFreedericksz transition in the director formalism, in which the instability emerges as a rotation of the nematicdirector n about an axis parallel to the plates. Accordingly, strong anchoring boundary conditions for thenematic director field will be assumed on both plates along the y direction. The external electric field will beapplied along the z axis and described by a potential field U = U (z), which attains prescribed values on thecell plates:

U (±d) = ±U0, (1)

so that the total voltage across the cell is 2U0.

2.2 General equilibrium equations

In general, let B be the region in space occupied by the NLC, subject to an electric field produced by anappropriate voltage applied to the boundary ∂B of the region B. The nematic texture in B is described by thesecond-rank, symmetric, and traceless order tensor Q, which measures the amount to which the probabilitydistribution of the molecular long axis of the nematic molecules differs from being isotropic (see [24, pp.56–57]). Being Q a symmetric tensor, it can be represented in the orthonormal basis of its eigenvectors{e1, e2, e3} as

Q =3∑

i=1

λi ei ⊗ ei ,

and the eigenvalues λi must obey the constraint

λ1 + λ2 + λ3 = 0,


as tr Q = 0. A uniaxial state is described by the condition that two eigenvalues coincide, and, in that case, wecan write

Q = S

(n ⊗ n − 1

3I)

, (2)

where the S ∈ [− 12 , 1] is the scalar order parameter, n is the nematic director, and I is the identity tensor. It

is worth noting that the upper bound of S corresponds to a configuration in which the molecules are orientedalong n, whilst the lower bound describes the case in which the molecules are on average distributed in theplane orthogonal to n. On the other hand, a biaxial state is characterised by the condition that all eigenvaluesof Q be distinct; an index of how far a biaxial state is from a uniaxial one is the degree of biaxiality [25]

β2 := 1 − 6(tr Q3)2

(tr Q2)3 , (3)

which ranges in the interval [0, 1]. In all uniaxial states β2 = 0; conversely, states with maximal biaxialitycorrespond to β2 = 1. Since tr Q3 = 3 det Q, the states with maximal biaxiality are precisely those wheredet Q = 0, that is, where at least one eigenvalue of Q vanishes. For any state, we define the reduced scalarorder parameter

s :=√

3

2tr Q2. (4)

For perfectly uniaxial states (β2 = 0), s coincides with the absolute value |S| of the scalar order parameterintroduced in Eq. (2).

The energy density f has three different contributions:

f = fe + fb + fel , (5)

where fe is the elastic energy, fb is the Landau-de Gennes bulk energy, and fel is the electric field energy.More precisely,

fe := L

2|∇Q|2, (6a)

fb := A

2tr Q2 − B

3tr Q3 + C

4(tr Q2)2, (6b)

fel := −1

2E · D . (6c)

In Eq. (6a) we adopt the one-constant approximation. Furthermore, in Eq. (6b), B and C are positive constitutiveconstants, whereas A depends on the absolute temperature T through the equation

A = a(T − T ∗), (7)

where a > 0 and T ∗, the temperature at which A vanishes, is called the super-cooling temperature. Finally, inEq. (6c) we use the SI form of the equations for the electric field E and the electric displacement vector D.

The order tensor Q can also be interpreted as the ensemble average of the molecular uniaxial tensor

q := m ⊗ m − 1

3I, (8)

where m is the unit vector along the symmetry axis of the nematic molecule; thus,

Q = 〈m ⊗ m〉 − 1

3I, (9)

where 〈·〉 denotes the ensemble average.Similarly, in the uniaxial symmetry, the molecular dielectric tensor εm has the form

εm = ε‖ m ⊗ m + ε⊥ (I − m ⊗ m), (10)


where ε‖ and ε⊥ are the molecular dielectric permittivities for applied electric fields parallel and orthogonalto m, respectively. Thus, by Eq. (9), the macroscopic dielectric tensor ε can be expressed as

ε := 〈εm〉 = ε⊥I + (ε‖ − ε⊥)

(Q + 1

3I)

. (11)

Equation (11) can be recast in the simpler form

ε = εi I + εaQ, (12)

where εi and εa are the isotropic and anisotropic dielectric constants, defined by

εi := 2

3ε⊥ + 1

3ε‖ (13)

and

εa := ε‖ − ε⊥ . (14)

The tensor ε in both Eqs. (11) and (12) is dimensionless; within the SI standards, we write the linear constitutivelaw for the electric displacement vector D as

D = ε0εE = ε0(εi E + εaQE), (15)

where the permittivity ε0 of free space is given by ε0 = 8.85 pF/m. As no free charge is present, the corre-sponding Maxwell equation for D is

div D = 0 . (16)

Moreover, the electric field E can be derived from the electric potential U through the formula

E(p) = −∇U (p) ∀ p ∈ B . (17)

By using Eqs. (15) and (17) in Eq. (6c) we finally obtain

fel = −ε0

2(εi∇U · ∇U + εa∇U · Q∇U ) . (18)

Thus, the total free energy F becomes a functional of the fields Q and U , subject to the differential constraint(16):

F [Q, U ] :=∫

B

( fe + fb + fel)dV

=∫

B

[L

2|∇Q|2 + A

2tr Q2 − B

3tr Q3 + C

4(tr Q2)2 − ε0

2(εi∇U · ∇U + εa∇U · Q∇U )

]dV .

(19)

In the scenario we envisage, both fields Q and U are subject to Dirichlet boundary conditions; thus, the valuesof both the order tensor and the electric potential will be prescribed on ∂B:

Q|∂B = Q� U |∂B = U�, (20)

with Q� and U� assigned function on ∂B.The constraint (16) is treated by introducing a Lagrange multiplier λ; the functional F is replaced by the

modified functional F ∗, whose stationary points are free from any constraint:

F ∗ = F −∫

B

λ div D dV = F +∫

B

λ div [ε0(εi∇U + εaQ(∇U ))] dV

= F −∫

B

∇λ · [ε0(εi∇U + εaQ(∇U ))] dV −∫

∂B

λD · ν dA, (21)


where use has been made of the divergence theorem and ν denotes the outer unit vector normal to ∂B. Thestationarity conditions for F ∗, derived in Appendix A, read in B as

L∇2Q − ∂ fb

∂Q+ ε0

2εa ∇U ⊗ ∇U + ε0εa ∇λ ⊗ ∇U = 0, (22a)

div ((εi I + εaQ)∇U ) = 0, (22b)

div ((εi I + εaQ)∇λ) = 0, (22c)

where . . . denotes the symmetric traceless part of a tensor; they are subject to (20) and to (see Appendix A)

λ = 0 on ∂B . (23)

2.3 Order representation

For our case study, we now introduce a representation for the order tensor Q similar to that employed in [3];the Cartesian components of Q in the frame {ex , ey, ez} are given the following form:

[Q] =⎡

⎣−2q1 0 0

0 q1 − q2 q3

0 q3 q1 + q2

⎤

⎦ , (24)

where all qi ’s are functions of z only. The representation (24) automatically obeys the constraint tr Q = 0 andembodies the assumption on Q made in the preceding subsection. The meaning of the parameters qi can beappreciated more conveniently by considering the eigenvalues and eigenvectors of Q. When ϕ = 0, so thatthe eigenframe {e1, e2, e3} coincides with the reference frame {ex , ey, ez}, a uniaxial calamitic state with Qas in (2), n = ey , and S > 0 is represented by q1 > 0, q2 = −3q1, q3 = 0, and the corresponding eigenvaluesof Q are λx = λz = −2q1, λy = 4q1. In general, ϕ is implicitly defined by

tan(2ϕ) = −q3

q2, (25)

and the eigenvalues of Q corresponding to the eigenvectors

e1 = ex , (26a)

e2 = cos ϕ ey + sin ϕ ez, (26b)

e3 = − sin ϕ ey + cos ϕ ez, (26c)

are

λ1 = −2q1, (27a)

λ2 = q1 +√

q22 + q2

3 , (27b)

λ3 = q1 −√

q22 + q2

3 . (27c)

Further properties of Q follow from the representation (24). First,

[Q2] =⎡

⎢⎣4q2

1 0 0

0 q23 + (q1 − q2)

2 2q1q3

0 2q1q3 q23 + (q1 + q2)

2

⎤

⎥⎦ , (28)

so that

tr Q2 = 2(3q21 + q2

2 + q23 ), (29)


and

Q2 =⎡

⎢⎣4q2

1 − 13 tr Q2 0 0

0 q23 + (q1 − q2)

2 − 13 tr Q2 2q1q3

0 2q1q3 q23 + (q1 + q2)

2 − 13 tr Q2

⎤

⎥⎦

=⎡

⎢⎣2q2

1 − 23 (q2

2 + q23 ) 0 0

0 −q21 + 1

3 (q22 + q2

3 ) − 2q1q2 2q1q3

0 2q1q3 −q21 + 1

3 (q22 + q2

3 ) + 2q1q2

⎤

⎥⎦ . (30)

Similarly, we obtain

tr Q3 = −6q1(q21 − q2

2 − q23 ) . (31)

We heed here for later use that, for Q, U , and λ depending only on the variable z, Eqs. (22b) and (22c)become

{[εi + εa(q1 + q2)] U ′}′ = 0 (32)

and{[εi + εa(q1 + q2)] λ′}′ = 0, (33)

respectively. Moreover, in Eq. (22a),

∇U ⊗ ∇U = U ′2(

ez ⊗ ez − 1

3I)

, (34)

and

∇λ ⊗ ∇U = λ′U ′(

ez ⊗ ez − 1

3I)

, (35)

where a prime ′ denotes differentiation with respect to z. Finally, by Eqs. (24), (28), and (29), a Cartesianrepresentation in the frame {ex , ey, ez} can also be given for

∂ fb

∂Q= AQ − B Q2 + C tr Q2 Q . (36)

2.4 Free energy and normalisation constants

We are now in a position to calculate the free-energy integral in Eq. (19) for the nematic cell in Fig. 1. Denotingby F the free energy per unit area of the cell’s plates, we arrive at

F[Q, U ] =d∫

−d

( fe + fb + fel) dz

=d∫

−d

{L

[3q ′2

1 + q ′22 + q ′2

3

] + fb − ε0

2(εi + εa(q1 + q2)) U ′2} dz, (37)

where

fb = A

2tr Q2 − B

3tr Q3 + C

4(tr Q2)2

= A(3q21 + q2

2 + q23 ) + 2Bq1(q

21 − q2

2 − q23 ) + C(3q2

1 + q22 + q2

3 )2 . (38)


It is easily seen [3] that, for A ≥ A∗ := B2/24C , the potential fb in Eq. (38) admits Q = 0 as uniqueminimiser, corresponding to the isotropic phase. According to Eq. (7), A = A∗ for T = T ∗∗, where

T ∗∗ := T ∗ + B2

24aC(39)

is the super-heating temperature. We define the reduced temperature θ through the equation

θ := A

A∗= T − T ∗

T ∗∗ − T ∗ ; (40)

we point out that θ < 1 for T < T ∗∗, i.e., in the absolute temperature range for which a nematic phase is pres-ent. We further scale the energy F in Eq. (19) to a characteristic energy F0, independent of the temperature,but depending on the the material and on the cell’s thickness,

F0 := 3A∗B2d3

8C2 . (41)

For θ > 0, the potential fb attains a local minimum for Q �= 0 (see [3] for further details); such a localminimiser is uniaxial as in Eq. (2) with n however oriented in space and

S = sb := B + √B2 − 24AC

4C. (42)

It readily follows from Eq. (42) and the definition of A∗ that, for T = T ∗∗, sb = s∗ := B/4C . We normaliseeach qi , as well as sb, to s∗. Moreover, we scale the space variable z to the half-thickness d of the cell. Moreprecisely, we write:

Q := 1

s∗Q, sb := sb

s∗= √

1 − θ + 1, z := z

d. (43)

Accordingly, we normalise the reduced scalar order parameter as

s := s

s∗, (44)

with s defined as in Eq. (4). We also introduce the bare biaxial coherence length ξ0 as

ξ0 :=√

L

Bs∗=

[4LC

B2

]1/2

. (45)

This is an absolute parameter characterising the material. As in [3], a temperature-dependent biaxial coherencelength ξb is often used instead; such a length measures the distance over which a biaxial perturbation of thenematic order would persist in space. The lengths ξ0 and ξb are easily converted into each other by the followingequation

(√1 − θ + 1

)ξ2

b := sbξ2b = ξ2

0 . (46)

Upon using the second of Eqs. (43), we also define a coherence length ξ0 normalised to the half-thickness d:

ξ0 := ξ0

d. (47)

Similarly, we introduce a normalisation for the electric potential independent of the temperature: we define ascaling voltage U∗ as

U∗ :=√

3A∗Bξ20 s∗

εiε0C= B

C

√L

8εiε0, (48)


and consequently we shall write

U = U

U∗. (49)

Furthermore, in the spirit of Eqs. (43), we also define a normalised relative anisotropy dielectric constant εa as

εa := εas∗εi

. (50)

It is instructive to estimate the values of the normalising constants we have defined in a typical situation.Using the values of the constants obtained for 5CB in [5], we set a = 0.2 × 106 J/K m3, B = 7.2 × 106 J/m3,and C = 8.8 × 106 J/m3 [26], L � 5 × 10−12 N [27], ε⊥ = 5, ε‖ = 20 [28]; with a reduced temperatureθ = −8, we have εa = 0.307, and U∗ = 69 mV.

To sum up, in the scenario here adopted we rewrite the free energy functional in Eq. (37) as

F = F0

d2 (Φ1 + Φ2 + Φ3), (51)

where

Φ1 :=1∫

−1

ξ02 [

3q ′21 + q ′2

2 + q ′23

]dz, (52a)

Φ2 :=1∫

−1

[θ

6(3q2

1 + q22 + q2

3) + 2q1(q21 − q2

2 − q23) + 1

4(3q2

1 + q22 + q2

3)2]

dz, (52b)

Φ3 := −1∫

−1

ξ02 [

1 + εa(q1 + q2)]

U ′2 dz . (52c)

To avoid clutter, in Eqs. (52) and in what follows, whenever normalised quantities are involved, a prime ′denotes differentiation with respect to the variable z.

2.5 Equilibrium equations

In the representation introduced in (24), the equilibrium equations (22) become

Lq ′′1 = Aq1 − B

3(q2

2 + q23 − 3q2

1 ) + 2C(3q21 + q2

2 + q23 )q1 − ε0εa

12

[U ′2 + 2λ′U ′] , (53a)

Lq ′′2 = Aq2 − 2Bq1q2 + 2C(3q2

1 + q22 + q2

3 )q2 − ε0εa

4

[U ′2 + 2λ′U ′] , (53b)

Lq ′′3 = Aq3 − 2Bq1q3 + 2C(3q2

1 + q22 + q2

3 )q3, (53c)

[εi + εa(q1 + q2)] U ′′ = −εa(q ′1 + q ′

2)U, (53d)

[εi + εa(q1 + q2)] λ′′ = −εa(q ′1 + q ′

2)λ′ . (53e)

Equation (53b) is obtained by linear combination of the second and third diagonal elements in the matrixrepresentation of Eq. (22a). Equations (53d) and (53e) are the expanded versions of Eqs. (32) and (33). It canbe shown that λ ≡ 0 is the only solution to Eq. (53e) compatible with our assumptions. Details of the proofare given in Appendix B. By use of this latter result, the scaled equilibrium equations within the cell read as

ξ02q ′′

1 = θ

6q1 − 1

3(q2

2 + q23 − 3q2

1) + 1

2(3q2

1 + q22 + q2

3)q1 − 1

6ξ0

2εaU

′2, (54a)

ξ02q ′′

2 = θ

6q2 − 2q1q2 + 1

2(3q2

1 + q22 + q2

3)q2 − 1

2ξ0

2εaU

′2, (54b)

ξ02q ′′

3 = θ

6q3 − 2q1q3 + 1

2(3q2

1 + q22 + q2

3)q3, (54c)


{[1 + εa(q1 + q2)

]U

′}′ = 0, (54d)

λ(z) = 0 ∀ z ∈ [−1, 1], (54e)

It ought to be noticed that with our choice of dimensionless variables, the explicit dependence on the tem-perature in Eqs. (54a)-(54c) is concentrated in the parameter θ . Moreover, it is worth pointing out that λ ≡ 0means that the constraint expressed by Eq. (16) is ineffective in the present setting. Equivalently, Eq. (53d)can be seen as the equilibrium equation for the unconstrained potential U , derived directly from Eq. (51).

2.6 Boundary conditions

We now describe how to impose on the equilibrium equations the boundary conditions appropriate to ourproblem; we shall also comment on how they are sensitive to the parametrisation chosen for the order tensor.The order tensor Q is prescribed to be Q� on both plates, with Q� uniaxial as in Eq. (2), S equal to the bulkequilibrium value sb, and director n = ey :

Q� = sb

(ey ⊗ ey − 1

3I)

. (55)

This reflects on the continuum scale a strong anchoring for the nematic molecules, imagined to be kept onaverage aligned along ey by an appropriate surface potential that enforces the same degree of order as wouldfb in the undistorted bulk.

Comparing Eq. (55) with Eq. (24), we readily arrive at the corresponding boundary conditions for the qi ’s:

q1(−d) = q1(+d) = sb

6, (56a)

q2(−d) = q2(+d) = − sb

2, (56b)

q3(−d) = q3(+d) = 0 . (56c)

Correspondingly, the angle ϕ defined in Eq. (25) vanishes at both plates. Intuitively, this angle is more tellingfor—as in the director analogue—it describes the distortion induced by the electric field in a better way: whenϕ grows inside the cell, though being bound to vanish at the plates, the field starts being effective, and its effectsaturates when ϕ approaches π

2 . Here ϕ retains much of its intuitive meaning, although it describes properlythe rotation of the eigenframe of Q, a tensor which is expected to lose its uniaxial character in the bulk. InAppendix C we show how our equilibrium problem can equivalently be phrased in terms of the variable ϕ.However, in the body of the paper it will remain a derived parameter. Moreover, the boundary conditions inEqs. (56) must be supplemented by (1).

Finally, the conditions expressed by Eqs. (56) and (1) are readily given the dimensionless form adoptedhere:

q1(−1) = q1(1) = 1

6(√

1 − θ + 1), (57a)

q2(−1) = q2(1) = −1

2(√

1 − θ + 1), (57b)

q3(−1) = q3(1) = 0, (57c)

U (−1) = −U 0 := −U0

U∗, U (1) = U 0 = U0

U∗. (57d)

3 Results

To capture the physical picture, potentially obscured by a plethora of values for the many parameters involved,we perform our calculations for a fixed value of the dimensionless temperature θ (or, equivalently, of theparameter sb). As we are interested in the order textures in the deep nematic phase, the typical value θ = −8(or sb = 4) has been used throughout the paper. Furthermore, a definite value of reduced anisotropy has also


been chosen, precisely εa = 0.307, corresponding to one of the most common nematic compounds in liquidcrystal studies and applications, i.e., 5CB. In this case, only two dimensionless parameters remain to be varied:d/ξ0 and U 0.

For different values of both parameters, different equilibrium order textures are obtained by solvingEqs. (54) numerically. This has been accomplished by using two independent codes specifically designed.The first code employed the over-relaxation method (see, for example, [29]) to solve the finite-differencediscretised version of the equilibrium equations (54) with a number of mesh points ranging from 100 to 200.The second code employed the matcont [30] package, which integrates into matlab [31] software package.A finite difference method has been applied to discretise equations (54). All of the plots reported here havebeen obtained with the matlab/matcont code, validated by the first code outcomes. The number of equallyspaced mesh points used in this case is N + 2, with N = 2n + 1; N is the number of internal points in thegrid, where z ranges in ] − 1, 1[, and the two additional points correspond to z = ±1, where the boundaryconditions are imposed. The choice of an odd number of points allows computing the values of all variablesat z = 0, which turns out to be useful when assessing several features of the equilibrium order texture. Forbasic explorations, n was chosen to be 5; this value allows building the phase diagram with sufficient accuracy.Mesh grids have been refined for the production of smoother plots of the order parameters q1, q2, q3, and thequantities derived from them, such as the eigenvalues of the order tensor Q, the rotation angle ϕ, the degree ofbiaxiality β2 and the reduced order parameter s. In doing this, we have checked that the values for the criticalfields described below were scarcely influenced by the number of grid points, for n > 5. The graphs havebeen drawn from data obtained with n = 8, except for Figs. 10 and 11, for which choosing n = 7 provedsufficient to produce plots reasonably smooth. No data interpolation has been performed, and all of the plotsare simply obtained by joining with straight line segments the data points. Although below we mostly presentthe outcomes of the second code, a close agreement between these and the ones arrived at by the first codehas systematically been achieved. To avoid cumbersome notations, in the following we use dimensionless orrescaled quantities only whenever necessary (e.g. when discussing numerical values or plots); in any case, abar over a symbol will denote the corresponding scaled quantity.

It is of great importance understanding the behaviour of the nematic texture as a function of the appliedvoltage, while keeping the cell’s thickness constant. When the applied voltage is zero (U0 = 0), the texture ishomogeneous throughout the cell and it is determined uniquely by the boundary condition, i.e., q1(z) = sb/6,q2(z) = −sb/2, q3(z) = 0 ∀ z ∈ [−d, d]. When U0 is increased, the nematic texture in the cell becomes moreand more distorted. The equilibrium texture depends on the thickness 2d of the cell. There exist essentiallytwo regimes, corresponding to two ranges of d .

3.1 Thick cells

To analyse the different textures that can be observed and how they depend on the applied voltage, we imagineto increase such a voltage gradually. For thick cells (d � ξb) we find the following results. For U0 smallenough, the parameter q3 remains uniformly zero, while the deviations of q1 and q2 from their boundaryvalues are very small within the cell (for θ = −8, they stay close to q1 = sb/6 and q2 = −sb/2). Thereis also no noticeable deviation of the potential within the cell from the linear dependence U (z) = U0 z/d ,which corresponds to a homogeneous electric field inside the cell. Thus, the bulk nematic texture is essentiallyundistorted, with the nematic director field almost completely aligned in the y direction. We will refer to thisas the homogeneous texture, and we shall denote it by H, even if it exhibits a distortion of the eigenvaluesmore pronounced for smaller d/ξb. Upon increasing U0 from zero, at a critical boundary potential U (cr)

0,1 there

is a continuous transition into a rotated state (q3 �= 0). For large values of d/ξb, the critical value U (cr)0,1 is

independent of the thickness of the cell; for d = 5 ξb, its scaled value is U(cr)

0,1 = 5.66, coinciding virtually withits limiting value for large d/ξb, which matches the critical threshold voltage for the classical Freedericksztransition (see Appendix D), fully described by the classical nematic director n, with a constant scalar orderparameter.

As soon as U0 is increased above U (cr)0,1 , the parameter q3 starts increasing and reaches its maximum value

in the middle of the cell, for z = 0; similarly, the parameter q2 changes in a conspicuous way, whilst q1 isnot much affected. This means that not only do the eigenvectors of Q rotate almost to match the nematictexture to the electric field inside the cell, but also the eigenvalues λ2 and λ3 change significantly. Inspiredby Eq. (2), we can also identify the effective scalar order parameter S and the effective nematic director n: we


set S := 32λM , where λM is the largest eigenvalue of Q, and identify n with the corresponding eigenvector. It

is clear from Eqs. (27) that, for q1 > 0, n = e2. Thus, we can say that a rotation of the nematic director inthe (y, z) plane coexists with a partial order reconstruction. Hereafter, we shall refer to this as the distortedFreedericksz texture, and denote it by DF, as it is endowed with a structure similar to the one expected in theclassical Freedericksz scenario.

Upon further increasing U0 above the critical value U (cr)0,1 (for fixed d/ξb), the nematic director is almost

oriented in the z direction in most of the cell, except near the plates. For fixed d/ξb, at some value of thepotential U0, q3(0) reaches its maximum and then it is again depressed for increasing values of U0. On theother hand, q2(0) steadily increases with increasing potential and becomes even positive. For large enoughU0, there are two planes at a distance z0 from the middle of the cell where q2 is zero, i.e., q2(±z0) = 0; at thesame time, q3 is very small everywhere in the cell. Thus, according to Eqs. (27), near z = ±z0 the eigenvaluesλ2 and λ3 come very close together. Finally, at a second critical potential U (cr)

0,2 , q3 becomes identically zerothroughout the cell, and so the eigenvalues λ2 and λ3 meet at the planes corresponding to z = ±z0 within thecell. Thus, for U0 > U (cr)

0,2 , we can formally write the eigenvalues λ2 and λ3 as λ2 = q1 +q2 and λ3 = q1 −q2.Where the parameter q2 changes its sign within the cell, the two eigenvalues are exchanged. There, the nematicstate is uniaxial along n = ex with negative order parameter, S = −3q1. While the eigenframe of Q is fixedin space, the effective nematic director n changes from ey , for z0 < |z| ≤ d , to ez , for |z| < z0. In brief,above the second critical potential, the order texture shows an exchange of eigenvalues and no nematic directorreorientation. As this is the pattern characteristic of the order reconstruction, we denote such an equilibriumtexture OR. The transition between different textures at U (cr)

0,2 is continuous, like the one at U (cr)0,1 ; however,

unlike U (cr)0,1 , U (cr)

0,2 increases with the thickness of the cell. For potentials not much larger than U (cr)0,1 , DF is

the prevailing texture. In such a case, the nematic director is oriented by the electric field; in particular, inthe middle of the cell, the angle ϕ between the nematic director and the y axis, given by Eq. (25), steadilyapproaches the value π/2, as the potential U0 is increased.

3.2 Thin cells

For thin cells (d ∼ ξb) the equilibrium picture changes considerably; the DF texture simply disappears, asdoes any trace of nematic director rotation. There is, instead, a third critical potential U (cr)

0,3 , in correspondenceof which a direct continuous transition from the H-texture to the OR-texture takes place; the critical potentialU (cr)

0,3 is obtained as the smallest boundary potential at which q2 crosses zero, which, by symmetry, occurs at

z = 0. Mathematically, U (cr)0,3 is defined like U (cr)

0,2 , as the new texture developing there, upon increasing U0, isOR in both cases. What distinguishes them is the pre-existing texture: H for the former, and DF for the latter.

3.3 Graphical synopsis

The analysis summarised above is detailed in the following set of figures, which contain plots for significant,exemplary cases. The figures are organised so as to ease comparison between the quantities relative to fixedvalues of the half-thickness d of the cell. As we need discuss numerical values, explicit reference will bemade to scaled quantities. Figure 2 shows the z-dependence of the order parameters q1 and q2 across thecell for d = 5 ξb. In this case, the behaviour typical of thick cells (Sect. 3.1) emerges; accordingly, severalvalues of U 0 are considered. Figure 3 shows the corresponding order tensor eigenvalues λ1, λ2 and λ3 for thesame U 0 and d/ξb as in Fig. 2. For completeness, the non trivial profile of the third order parameter q3, andthe corresponding rotation angle ϕ are displayed in Fig. 4, with an appropriate set of values of the boundarypotential. As a further illustration of the equilibrium textures traversed by the cell upon increasing U 0, in Fig. 5we draw the graphs of both the reduced scalar order parameter s defined by (44) and the degree of biaxiality

β2 defined by (3), for d = 5 ξb. For U 0 > U(cr)

0,2 = 7.97, the value of s decreases considerably at the planesz = ±z0, where β2 vanishes, as λ2 = λ3 and the local nematic state is then uniaxial. The degree of biaxialityreaches its maximum where one eigenvalue of Q vanishes; this occurs at two pairs of planes, each delimitinga biaxial wall centred around z = ±z0. These walls are a distinctive sign of order reconstruction [3,12,13].They move towards the plates upon increasing U0, for fixed d/ξb. For completeness, we draw for d = 3.5 ξbin Figs. 6–9 the graphs drawn in Figs. 2–5 for d = 5 ξb: they show the same qualitative features of the order


−1 −0.5 0 0.5 1

0.7

0.75

0.8

z/d

q1 /s

*

U0

−1 −0.5 0 0.5 1−2

−1

0

1

2

z/d

q2 /s

*

U0

Fig. 2 The profile of the order parameters q1 = q1/s∗ (left) and q2 = q2/s∗ (right) across the cell as a function of z = z/d ,with d/ξb = 5, εa = 0.307, θ = −8 and several values of the scaled potential U 0 = U0/U∗. In detail, thin lines correspond toU 0 = 0, 1, 2, 3, 4, 5, 5.5, 5.75, 6, 6.5, 7, 7.5, 9, 10, with green shades for low values of U 0 (lower graphs) changing into red for

larger U 0 (upper graphs). Thick lines correspond to the critical potentials U(cr)

0,1 = 5.66 (brown lower graph) and U(cr)

0,2 = 7.97(red upper graph). An arrow shows the direction of increase of U0 and allows identifying the different plots

−1 −0.5 0 0.5 1

−1.65

−1.6

−1.55

−1.5

−1.45

−1.4

−1.35

z/d

λ1/s

*

U0

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

1.5

2

2.5

3

z/d

λ2/s

*, λ

3/s

*

U0

Fig. 3 The profile of the eigenvalues λ1 (left), and λ2 (solid lines, right) and λ3 (dashed lines, right) of the order tensor Q = Q/s∗across the cell as a function of z = z/d , with d/ξb = 5, εa = 0.307, θ = −8 and several values of the scaled potential U 0 = U0/U∗.In detail, thin lines correspond to U 0 = 0, 1, 2, 3, 4, 5, 5.5, 5.75, 6, 6.5, 7, 7.5, 9, 10, with green shades for low values of U 0

(upper graphs on the left, bumpier ones on the right), changing into red for larger U 0 (lower graphs on the left, bumpier graphs

on the right). Thick lines correspond to the critical potentials U(cr)

0,1 = 5.66 (brown graph) and U(cr)

0,2 = 7.97 (red graph). Anarrow shows the direction of increase of U0, and allows identifying the different plots

texture; only the critical values of the electric potential change. In particular, by comparing Figs. 5 and 9, oneeasily sees that at the critical voltage the biaxial walls are closer to the bounding plates for the smaller valueof d/ξb.

Figure 10 displays the non-trivial dependence of the first two order parameters, q1 and q2, for d = ξb,well within the thin-cell regime (Sect. 3.2). Several values of U 0 are considered. In this case, q3 vanishesthroughout the cell for any value of the bounding potential U 0. For a given small U 0, for which the H texturestill prevails at equilibrium, the distortion of the eigenvalues of Q is larger in thinner cells than in thicker ones.Upon increasing U 0, both q1 and q2 are exalted.

Figure 3 shows the onset of the OR texture in thick cells, marked by the tendency for λ2 and λ3 to coincideat two planes in the cell, placed symmetrically with respect to the centre of the cell. Upon increasing U 0, theeigenvalues λ2 and λ3, which are well separate for U 0 sufficiently small, with λ2 > λ3 everywhere in the cell,eventually coalesce at z = ±z0 := z0/d , and come across one another when U 0 reaches the critical value


−1 −0.5 0 0.5 10

0.5

1

1.5

z/d

q3 /s

*

5.75

6

6.5 7

7.5

7.95

7.97

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

z/d

φ/(π/2)

U0

Fig. 4 The profile of the order parameter q3 = q3/s∗ (left) across the cell as a function of z = z/d , flanked by the correspondingprofile for the rotation angle ϕ with d/ξb = 5, εa = 0.307, θ = −8 and U 0 = U0/U∗ = 5.75, 6, 6.5, 7, 7.5, 7.95 (thin lines)

and U 0 = U(cr)

0,2 = 7.97 (thick line). Lines have green (light) shades for low values of U 0 (lower graphs on the right) gettingblue (dark) for larger U 0 (upper graphs on the right). For the graphs on the left, labels single out different plots; for the graphson the right, an arrow shows the direction of increase of U0, and allows identifying the different plots. All the plots refer to theDF texture; there, the rotation angle tends to reach the limiting value π/2 in the middle of the cell. However, upon increasing U 0,the graphs of q3 first rise and then they fall back as U 0 reaches the upper limit of existence for the equilibrium DF texture

−1 −0.5 0 0.5 1

2.5

3

3.5

4

4.5

z/d

s/s* 7.97

6

7

109

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

z/d

β2

6

7

7.97910

Fig. 5 The scaled reduced order parameter s = s/s∗, defined by Eq. (44), and the degree of biaxiality β2, defined by Eq. (3),

shown across the cell as functions of z = z/d , with d/ξb = 5, εa = 0.307, θ = −8 and U 0 = U0/U∗ = 6, 7, U(cr)

0,2 , 9, 10. Lineshave green (light) shades for low values of U 0 (flatter graphs) getting blue (dark) for larger U 0 (bumpier graphs). Labels allow

different plots to be singled out. The graph for the critical potential U(cr)

0,2 = 7.97 is thicker than the others. The behaviour of β2

clearly shows the creation of a biaxial wall in the neighbourhood of the planes z = ±z0, where order reconstruction takes place

−1 −0.5 0 0.5 1

0.7

0.75

0.8

0.85

0.9

z/d

q1 /s

*

U0

−1 −0.5 0 0.5 1−2

−1

0

1

2

z/d

q2 /s

*

U0

Fig. 6 The profile of the order parameters q1 = q1/s∗ (left) and q2 = q2/s∗ (right) across the cell as a function of z = z/d ,with d/ξb = 3.5, εa = 0.307, θ = −8 and several values of the scaled potential U 0 = U0/U∗. In detail, thin lines correspondto U 0 = 0, 1, 2, 3, 4, 5, 5.5, 6, 6.25, 6.5, 6.75, 7.25, 8, 9, 10, with green shades for low values of U 0 (lower graphs) changing

into red for larger U 0 (upper graphs). Thick lines correspond to the critical potentials U(cr)

0,1 = 5.75 (brown lower graph) and

U(cr)

0,2 = 7.05 (red upper graph). An arrow shows the direction of increase of U0 and allows identifying the different plots


−1 −0.5 0 0.5 1−1.85

−1.8

−1.75

−1.7

−1.65

−1.6

−1.55

−1.5

−1.45

−1.4

−1.35

z/d

λ1/s

*

U0

−1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

z/d

λ2/s

*, λ

3/s

*

U0

Fig. 7 The profile of the eigenvalues λ1 (left), and λ2 (solid lines, right) and λ3 (dashed lines, right) of the order tensorQ = Q/s∗ across the cell as a function of z = z/d , with d/ξb = 3.5, εa = 0.307, θ = −8 and several values of the scaledpotential U 0 = U0/U∗. In detail, thin lines correspond to U 0 = 0, 1, 2, 3, 4, 5, 5.5, 6, 6.25, 6.5, 6.75, 7.25, 8, 9, 10, with greenshades for low values of U 0 (upper graphs on the left, flatter ones on the right), changing into red for larger U 0 (lower graphs on the

left, bumpier graphs on the right). Thick lines correspond to the critical potentials U(cr)

0,1 = 5.75 (brown graph) and U(cr)

0,2 = 7.05(red graph). An arrow shows the direction of increase of U0, and allows identifying the different plots

−1 −0.5 0 0.5 10

0.5

1

1.5

z/d

q3 /s

*

5.8

6

6.25 6.5

6.75

7

7.05

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

z/d

φ/(π/2)

U0

Fig. 8 The profile of the order parameter q3 = q3/s∗ (left) across the cell as a function of z = z/d , flanked by the correspondingprofile for the rotation angle ϕ with d/ξb = 3.5, εa = 0.307, θ = −8 and U 0 = U0/U∗ = 5.8, 6, 6.25, 6.5, 6.75, 7 (thin lines)

and U 0 = U(cr)

0,2 = 7.05 (thick line). Lines have green (light) shades for low values of U 0 (lower graphs on the right) gettingblue (dark) for larger U 0 (upper graphs on the right). For the graph on the left labels single out different plots; for the graph onthe right, an arrow shows the direction of increase of U0, and allows identifying the different plots. All the plots refer to the DFtexture; there, the rotation angle tends to reach the limiting value π/2 in the middle of the cell. However, upon increasing U 0,the graphs of q3 first rise and then they fall back as U 0 reaches the upper limit of existence for the equilibrium DF texture

U(cr)

0,2 = 7.97. At U 0 = U(cr)

0,2 , a dramatic change occurs in the eigenvalue profiles, which makes both λ2 andλ3 experience larger excursions and forces the corresponding eigenvectors e2 and e3 to alternate in the role ofthe effective nematic director n where λ2 = λ3. Figure 11 shows a similar behaviour for d/ξb = 1, but here

the H-to-OR transition is direct, with no intervening DF texture, and it takes place at U 0 = U(cr)

0,3 = 5.84.Moreover, the two eigenvalue λ2 and λ3 coincide at z = 0 on the onset of the order reconstruction. In thiscase, only two biaxial walls form, one in any half of the cell, instead of two, as is the case of thick cells; at

the critical voltage U(cr)

0,3 , upon increasing d/ξb, the biaxial walls move towards the plates and away from thecentre of the cell. It should be noted that maximum biaxiality corresponds to one of the eigenvalues vanishing.From the graphs in Figs. 3–11, we notice that, upon increasing U0, in any case λ3 vanishes for values slightly


−1 −0.5 0 0.5 12.5

3

3.5

4

4.5

z/d

s/s*

7.05

6

6.5

98

10

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

z/d

β2

6.5

6

7.058

10

9


shown across the cell as functions of z = z/d , with d/ξb = 3.5, εa = 0.307, θ = −8 and U 0 = U0/U∗ = 6, 6.5, U(cr)

0,2 , 8, 9, 10.Lines have green (light) shades for low values of U 0 (flatter graphs) getting blue (dark) for larger U 0 (bumpier graphs). Labels

allow different plots to be singled out. The graph for the critical potential U(cr)

0,2 = 7.05 is thicker than the others. The behaviourof β2 clearly shows the creation of a biaxial wall in the neighbourhood of the planes z = ±z0, where order reconstruction takesplace

−1 −0.5 0 0.5 1

0.8

1

1.2

1.4

z/d

q1 /s

*

U0

−1 −0.5 0 0.5 1−2

−1

0

1

2

z/d

q2 /s

*

U0

Fig. 10 The profile of the order parameters q1 = q1/s∗ (left) and q2 = q2/s∗ (right) across the cell as a function of z = z/d ,with d/ξb = 1, εa = 0.307, θ = −8 and several values of the scaled potential U 0 = U0/U∗. In detail, thin lines correspond toU 0 = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, with green shades for low values of U 0 (lower graphs) changing into red for larger U 0 (uppergraphs). An arrow shows the direction of increase of U0, and allows identifying the different plots. Thick brown lines correspond

to the scaled critical potentials U(cr)

0,3 = 5.84; as described in the text, in correspondence of this value, q2 crosses zero at z = 0

smaller than U(cr)

0,2 or U(cr)

0,3 ; therefore, a state with maximum biaxiality somehow announces the onset of orderreconstruction. As we have done for thick cells, we draw in Fig. 12 the graphs of s and β2, for d/ξb = 1,

and U 0 > U(cr)

0,3 = 5.84. The value of s decreases considerably at the planes z = ±z0, where β2 vanishes,as λ2 = λ3 and the local nematic state is then uniaxial. The degree of biaxiality reaches its maximum where

one eigenvalue of Q vanishes; for U ≥ U(cr)

0,3 , this occurs at two pairs of planes, each delimiting a biaxialwall centred around z = ±z0. Similarly, in Fig. 13, are reported the graphs of s and β2 for d/ξb = 1 when

U 0 ≤ U(cr)

0,3 . As U 0 decreases, both graphs tend to become uniform within the cell. First the two biaxialwalls move toward one another, until they merge in a single wall in the middle of the cell, which is eventuallydepressed until it disappears.

Figure 14 illustrates the scaled potential U within the cell for different values of the boundary potentialU 0, and two different values of the thickness of the cell (d/ξb = 1, 5). These graphs confirm the conclusionof our analysis in Appendix B, according to which U is a monotonic function of z.

In all cases studied, the deviation of the electric potential from the linear z-dependence in the cell is rathersmall and appreciable only in the vicinity of the plates. Therefore an insignificant change in the results for thenematic texture is expected if the electric field is set to be homogeneous in the cell and solving the differentialequation (54d) is spared.


−1 −0.5 0 0.5 1

−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

z/d

λ1/s

*

U0

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

z/d

λ2/s

*, λ

3/s

*

U0

Fig. 11 The profile of the eigenvalues λ1 (left), and λ2 (solid lines, right) and λ3 (dashed lines, right) of the order tensorQ = Q/s∗ across the cell as a function of z = z/d , with d/ξb = 1, εa = 0.307, θ = −8 and several values of the scaled potentialU 0 = U0/U∗. In detail, thin lines correspond to U 0 = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, with green shades for low values of U 0

(flatter graphs) changing into red for larger U 0 (bumpier graphs). An arrow shows the direction of increase of U0, and allows

identifying the different plots. Thick brown lines correspond to the critical potentials U(cr)

0,3 = 5.84; as described in the text, incorrespondence of this value, λ2 = λ3 at z = 0, where a uniaxial state with director n = ex is attained. This marks the onset ofthe order reconstruction

−1 −0.5 0 0.5 1

3.5

4

4.5

5

5.5

6

6.5

z/d

s/s*

U0

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

z/d

β2

U0


shown across the cell as functions of z = z/d , with d/ξb = 1, εa = 0.307, θ = −8 and U 0 = U0/U∗ = U(cr)

0,3 , 6, 7, 8, 9, 10.Lines have green (light) shades for low values of U 0 (flatter graphs) getting blue (dark) for larger U 0 (bumpier graphs). An arrow

shows the direction of increase of U0, and allows identifying the different plots. The graph for the critical potential U(cr)

0,3 = 5.84is thicker than the others. The biaxial walls move towards the plates upon increasing the voltage applied

Figure 15 summarises the outcomes of all our computations; it shows the phase diagram of the threeequilibrium textures encountered here. For each point in the plane (d/ξb, U 0) we read off the correspondingtexture in the cell. The plane is parted into three distinct regions by the curves representing the potentials

U(cr)

0,1 , U(cr)

0,2 , and U(cr)

0,3 ; these curves meet at a triple point, whose coordinate, for εa = 0.307, are d/ξb = 2.31,

U 0 = 6.23. The curves for U(cr)

0,2 and U(cr)

0,3 have been obtained with matcont with a continuation of eitherbranching or limiting points; in fact, each critical value corresponds to a branching point in a bifurcationanalysis with d/ξb fixed and U 0 as continuation parameter; these also turn out to be limiting points. Withinthis scenario, the equilibrium solution associated with the DF texture appears and becomes the stable solutionfor U 0 ranging between the values corresponding to the two branching points.


−1 −0.5 0 0.5 1

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4s/s

*

U0

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1β2

2

3

5.84

45

10

z/d z/d


shown across the cell as functions of z = z/d , with d/ξb = 1, εa = 0.307, θ = −8 and U 0 = U0/U∗ = 0, 1, 2, 3, 4, 5, U(cr)

0,3 .Lines have green (light) shades for low values of U 0 (flatter graphs) getting blue (dark) for larger U 0 (bumpier graphs). An arrow

shows the direction of increase of U0 and allows identifying the different plots. The graph for the critical potential U(cr)

0,3 = 5.84is thicker than the others

−1 −0.5 0 0.5 1−10

−5

0

5

10

z/d

U/U*

d/ξb = 1 U

0

−1 −0.5 0 0.5 1−10

−5

0

5

10

z/d

U/U*

d/ξb = 5 U

0

Fig. 14 The profile of electric potential U = U/U∗ across the cell as a function of z = z/d , with d/ξb = 1 (left) or d/ξb = 5(right), εa = 0.307, θ = −8. The values of U 0 chosen and the corresponding colour shades of the graphs are the same as inFigs. 2 and 10, respectively. An arrow shows the direction of increase of U0, and allows identifying the different plots

We have also determined the triple point for other values of the reduced dielectric anisotropy, rangingfrom 0.1 to 1. Whilst the value of the potential U 0 changes (it decreases with increasing εa , as expected), the

coordinate d/ξb of the triple point remains virtually the same. The asymptotic value of U(cr)

0,3 as d/ξb → 0,which in Fig. 15 has been computed along a discrete set of points approaching the origin, can also be estimatedanalytically, as explained in Appendix E:

limd→0

U(cr)

0,3 =√

3

εa

√1 + 2εaq0

2∫

1

dq

[ 1q0

+ 2εa(2q − 3)](2 − q), (58)

with q0 := 16 (

√1 − θ + 1) and 1 − 2εaq0 > 0. When this estimate is applied to the case illustrated in Fig. 15

(εa = 0.307, θ = −8), it delivers the value U(cr)

0,3 = 5.81, which agrees well with the limit obtained numeri-cally. In the light of the phase diagram shown in Fig. 15, formula (58) is also a good estimate for the criticalvoltage at the triple point.

4 Conclusions

The Landau-de Gennes theory of liquid crystals describes the local molecular organisation through the ordertensor Q. Within this theory, we studied the equilibrium order textures in a nematic cell confined between two


0 5 10 15 200

5

10

15

20

d/ξb

U0 /U

*

OR

H

DF

Fig. 15 Phase diagram in the plane (d/ξb, U 0) for the occurrence of nematic equilibrium textures H, DF, OR, for εa = 0.307and θ = −8. The lines dividing the plane into three region, meet at the triple point, at which the three textures would coexist atequilibrium; the coordinates of the triple point are (2.31, 6.23)

parallel plates enforcing one and the same uniaxial state, under the action of an antagonistic electric field. Thissetting is known to give rise to a continuous transition from the uniform molecular alignment compatible withthe surface anchoring to a bent molecular arrangement, for a critical threshold field. This is the Freedericksztransition; for sufficiently thick cells, it can be described in terms of purely elastic director distortions. Theobject of this paper has been to explore cells with thickness comparable to the biaxial coherence length ξb,which nanoscience has made experimentally accessible. At this length scale, the purely uniaxial descriptionof the molecular nematic ordering becomes insufficient and Landau-de Gennes’s theory is instead the electedtool.

We have found two different regimes, distinguished by the half-thickness d of the cell. For d large enough,typically d > 2.5ξb, upon increasing the voltage 2U0 applied across the cell, the homogeneous alignedtexture H is first changed into a distorted Freedericksz texture (DF), which mainly exhibits a director rotation,though the eigenvalues of Q are also affected by the electric field. Therefore, the transition from H to DFcorresponds to the well-known classical Freedericksz transition. Then, upon further increasing U0, a secondthreshold is met, where the electric field saturates in the bulk of the cell and creates two biaxial walls atthe bounding surfaces, through which the uniaxial order induced by the field in the bulk is destroyed andthe antagonistic uniaxial order of the plates is instead established. We called OR the resulting texture, as itexhibits order reconstruction. In the complementary regime of small thickness (d � 2.5ξb), the OR textureis established directly from the H texture upon increasing the potential U0; no director rotation takes place inbetween. Here the electric field need not saturate in the bulk to prompt the order reconstruction on the surfaceof the cell. In this regime, a given surface potential U0 is much more effective in altering the eigenvaluesof Q than in the other regime. The main conclusions of our study are summarised in the phase diagram ofFig. 15. This has been computed for a specific scaled dielectric anisotropy, εa = 0.307; however, the thick-ness of the cell corresponding to the triple point (d � 2.5ξb), which marks the transition between the tworegimes outlined above, appeared sufficiently independent of that material parameter. We have shown thatboth director rotation and order reconstruction are mechanisms necessary to explain molecular reorganisationin nematic liquid crystal cells exhibiting the Freedericksz transition. In particular, the latter prevails when thethickness of the cell falls in the nanoscopic range. Recently developed techniques, involving electric currentand optical measurements, have allowed observing time-resolved order reconstruction, both in the homoge-neous case, i.e in the absence of defects, and in the inhomogeneous case, which is defect mediated [16]. Inthese experiments, the order reconstruction is observed even in the presence of a small surface pretilt, andstrong anchoring is not broken; these techniques might be adapted to reveal the thresholds predicted in ourmodel.

Acknowledgments We are grateful to G. E. Durand for discussions on the subject of this paper.


Appendix A: Variational analysis

Here and in the following appendices we collect further details of our analysis not included in the main bodyof the paper to ease its readability.

To obtain the differential equations describing our model, we need perform variations on Q and U in Eq. (21)that comply with Eqs. (20):

Qε := Q + εK with K|∂B = 0Uε := U + εu with u|∂B = 0 so that ∇su|∂B = 0,

(59)

where K is an arbitrary traceless tensor, u is an arbitrary scalar function, and ∇s denotes the surface gradient.With these postulations, the first variation of F ∗ in (21) is

d

dεF ∗[Qε, Uε]|ε=0 =

∫

B

{L∇Q · ∇K + ∂ fb

∂Q· K − ε0(εi ∇u · ∇U + εa∇u · Q(∇U )) − ε0

2εa ∇U ⊗ ∇U · K

}dV

−∫

B

{∇λ · [ε0(εi ∇U + εaQ(∇U )) + εaK(∇U )

]} dV −∫

∂B

λε0(εi ∇U + εaQ(∇U )) · ν dA,

(60)

where V and A are the volume and area measures, ν is the unit outer normal to ∂B, and A denotes thesymmetric traceless part of the tensor A,

A := 1

2

(A + AT

)− 1

3(tr A) I .

Recalling that (u ⊗ v)T = v ⊗ u, tr (u ⊗ v) = u · v, and ∇u|∂B = ∂u∂ν

ν, as ∇su ≡ 0, integrating by parts andusing Eqs. (20) we obtain from Eq. (60) that

d

dεF ∗[Qε, Uε]|ε=0 =

∫

B

{[−L∇2Q + ∂ fb

∂Q− ε0

2εa ∇U ⊗ ∇U − ε0εa ∇λ ⊗ ∇U

]· K

−ε0[εi (∇U + ∇λ) + εaQ(∇U + ∇λ)] · ∇u} dV −∫

∂B

λ∂u

∂νε0(εiν + εaQν) · ν dA

=∫

B

{[−L∇2Q + ∂ fb

∂Q− ε0

2εa ∇U ⊗ ∇U − ε0εa ∇λ ⊗ ∇U

]· K

+ div (ε0ε∇(U + λ)) u

}dV

− ε0

∫

∂B

λ∂u

∂νν · εν dA = 0

(61)

Since K and u are arbitrary in B and ∂u∂ν

is arbitrary on ∂B, whilst ε is a positive definite tensor, from Eq. (61)we obtain Eqs. (22a), (23), and also with the aid of Eq. (12),

div [(εi I + εaQ)∇(U + λ)] = 0 . (62)

By combining this equation with Eqs. (16) and (15), we arrive at Eqs. (22b) and (22c).


Appendix B: Lagrange multiplier

We prove here that in our case study the Lagrange multiplierλvanishes identically at equilibrium (see Eq. (54e)).Equations (53d) and (53e) can easily be integrated once:

[εi + εa(q1 + q2)] U ′(z) = KU , (63a)

[εi + εa(q1 + q2)] λ′(z) = Kλ, (63b)

where KU and Kλ are integration constants. For U ′ to be determined by (63a) as a continuous function in theinterval [−d, d], neither KU nor the function in brackets on the left-hand side of Eq. (63a) can vanish. Thisand the boundary conditions (1) for U0 �= 0 imply that U is an increasing function in [−d, d]. Equations (63)can also be given the following form in terms of differentials:

dU = KU dz

[εi + εa(q1 + q2)], (64a)

dλ = Kλ dz

[εi + εa(q1 + q2)]. (64b)

Integrating both sides of Eqs. (64a) and (64b) with boundary conditions (1), we obtain

U (d)∫

U (−d)

dU = KU

d∫

−d

dz

[εi + εa(q1 + q2)]= 2U0, (65a)

λ(d)∫

λ(−d)

dλ = Kλ

d∫

−d

dz

[εi + εa(q1 + q2)]= 0 . (65b)

Now, the quantity

I :=d∫

−d

dz

[εi + εa(q1 + q2)],

can be written from Eq. (65a) as

I = 2U0

KU.

This means that I �= 0, unless U0 vanishes, which is not the physically relevant case. Thus for Eq. (65b) tobe satisfied, Kλ must vanish, that is,

λ(z) = 0 ∀ z ∈ [−d, d] . (66)

Appendix C: Alternative parametrisation

The class of nematic states considered in our model can be represented equivalently by another set of param-eters, (q1, r, ϕ), instead of (q1, q2, q3). The angle ϕ has already played a role in our development; here weestablish a closer connection between these representations. For given r and ϕ,

q2 = −r cos 2ϕ (67a)

q3 = r sin 2ϕ . (67b)


Conversely,

r =√

q22 + q2

3 (68a)

ϕ =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

−sgn(q3)π4 if q2 = 0 and q3 �= 0,

− 12 arctan q3

q2if q2 > 0,

π2 if q2 < 0 and q3 = 0,

− 12 arctan q3

q2− sgn(q3)

π2 if q2 < 0 and q3 �= 0,

(68b)

while ϕ remains undefined for q2 = q3 = 0. Here sgn(x) is the sign function, defined for x ∈ R − {0}, as

sgn(x) ={

+1 if x > 0,

−1 if x < 0 .

With this choice, ϕ ranges in [−π2 , π

2 ], and it is easily identified with the angle that the unit vector e2 makeswith ey in Fig. 1.

Now, for completeness, we also write the free energy functional in the parametrisation (q1, r, ϕ). To thisend, Eqs. (67) need to be differentiated with respect to z to give:

q ′2 = −r ′ cos 2ϕ + 2ϕ′r sin 2ϕ, (69a)

q ′3 = r ′ sin 2ϕ + 2ϕ′r cos 2ϕ, (69b)

which easily yield

q ′22 + q ′2

3 = r ′2 + 4ϕ′2r2 . (70)

A further differentiation allows us to write

q ′′2 = −r ′′ cos 2ϕ + 4ϕ′r ′ sin 2ϕ + 2ϕ′′r sin 2ϕ + 4ϕ′2r cos 2ϕ, (71a)

q ′′3 = r ′′ sin 2ϕ + 4ϕ′r ′ cos 2ϕ + 2ϕ′′r cos 2ϕ − 4ϕ′2r sin 2ϕ . (71b)

By inserting Eqs. (67) and (70) into Eq. (37), we obtain that

F =d∫

−d

{L

[3q ′2

1 + r ′2 + 4ϕ′2r2] + A(3q21 + r2) + 2Bq1(q

21 − r2) + C(3q2

1 + r2)2

−ε0

2(εi + εa(q1 − r cos 2ϕ)) U ′2} dz . (72)

New equilibrium equations for F , equivalent to Eqs. (53), are now easily obtained from Eq. (72):

Lq ′′1 = Aq1 + B

3(3q2

1 − r2) + 2C(3q21 + r2)q1 − ε0εa

12U ′2, (73a)

Lr ′′ = Ar + 4Lϕ′2r − 2Bq1r + 2C(3q21 + r2)r + ε0εa

4cos 2ϕU ′2, (73b)

L(ϕ′r2)′ = L(ϕ′′r2 + 2ϕ′r ′) = −ε0εa

8r sin 2ϕU ′2, (73c)

{[εi + εa(q1 − r cos 2ϕ)] U ′}′ = 0, (73d)

where U has been treated as an unconstrained field.


While clearly Eq. (73d) is the same as Eq. (53d), and Eq. (53a) reduces to Eq. (73a) upon using Eq. (68a),showing the correspondence between the pairs of equations (73b), (73c) and (53b), (53c) requires some furthermanipulation. In particular, the following relations are necessary:

r ′ = q2q ′2 + q3q ′

3

r, (74a)

ϕ′ = −1

2

q ′3q2 + q ′

2q ′3

q22 + q2

3

= q ′2q3 − q ′

3q2

2r2 , (74b)

r ′′ = q2q ′′2 + q3q ′′

3

r+ 4(ϕ′)2r, (74c)

(ϕ′r2)′ = q ′′2 q3 − q ′′

3 q2

2. (74d)

With the aid of these formulae, Eq. (73b) is obtained by multiplying both sides of Eq. (53b) by q2 and bothsides of Eq. (53c) by q3, and then adding the two resulting equations. Similarly, Eq. (73c) is obtained bymultiplying both sides of Eq. (53b) by q3 and both sides of Eq. (53c) by q2, and then subtracting the tworesulting equations.

In the (q1, r, ϕ) parametrisation the boundary conditions (56) would read as

q1(−d) = q1(+d) = sb

6, (75a)

r(−d) = r(+d) = sb

2, (75b)

ϕ(−d) = ϕ(+d) = 0, (75c)

as one can easily verify by Eqs. (68). In the next appendix, Eqs. (73) and (75) will lead us to estimate thecritical voltage of the classical Freedericksz transition.

Appendix D: Classical Freedericksz transition

We derive now an analytic estimate for the critical value U(cr)

0,1 of U0 at which the classical Freedericksz tran-sition takes place. If the eigenvalues of Q are constrained throughout the cell to their bulk equilibrium values,it readily follows from Eq. (75a) and (75b) that

q1 ≡ sb

6and r ≡ sb

2. (76)

Inserting these equations in Eqs. (73c) and (73d), in the limit of small deviations of ϕ from nought, also byEq. (1), we arrive at

U ′ ≡ U0

dand (77a)

ϕ′′ = − ε0εa

2Lsb

U 20

d2 ϕ. (77b)

Rescaling z, sb, U0, and εa as in Eqs. (43), (49), and (50), we write Eq. (77b) in the form

ϕ′′ = −εaU20

sbϕ, (78)

subject to the boundary conditions

ϕ(−1) = ϕ(1) = 0 . (79)


The least value of U0 for which this problem has a non-trivial solution is

U(cr)

0 = π

2

√sb

εa. (80)

For sb = 4 and εa = 0.307, we obtain U(cr)

0 = 5.67, which negligibly differs from the value computed

numerically for d = 10ξb, that is, U(cr)

0,1 = 5.66.

Appendix E: Asymptotic estimate for U(cr)0,3

We estimate in this appendix the asymptotic value of the critical potential U(cr)

0,3 in the limit as d/ξb → 0. Insuch a limit, the equilibrium equations (54) become

q ′′1 = −1

6εaU

′2, (81a)

q ′′2 = −1

2εaU

′2, (81b)

q ′′3 = 0, (81c)

{[1 + εa(q1 + q2)]

U′}′ = 0. (81d)

Equations (81) are subject to the boundary conditions

q1(−1) = q1(1) = q0 := 1

6(√

1 − θ + 1) > 0, (82a)

q2(−1) = q2(1) = −3q0, (82b)

q3(−1) = q3(1) = 0, (82c)

−U (−1) = U (1) = U 0. (82d)

It follows immediately from Eqs. (81c) and (82c) that q3 ≡ 0, whilst Eqs. (81a) and (81b) imply that

(q2 − 3q1)′′ = 0 . (83)

By Eqs. (82a) and (82b), we obtain from Eq. (83) that

q2 = 3q1 − 6q0 . (84)

Equation (81d) can be integrated at once:

U′ = c

1 + 2εa(2q1 − 3q0), (85)

where c is an integration constant, and use has also been made of Eq. (84). Inserting Eq. (85) into Eq. (81a)gives

q ′′1 = −1

6εa

c2

[1 + 2εa(2q1 − 3q0)]2 . (86)

By multiplying both sides of this Eq. (86) by q ′1, we reduce it to quadratures:

q ′21 = c2

12

1

1 + 2εa(2q1 − 3q0)+ b, (87)

where b is another integration constant. Since εa > 0, by Eq. (81a), q1 is a concave function; furthermore,since Eq. (87) is autonomous and the boundary conditions in Eq. (82a) are symmetric, q1 is even in [−1, 1],


and attains its maximum at z = 0. If the term [1 + 2εa(2q1 − 3q0)] vanishes somewhere in [−1, 1], there U′

diverges. For U′

to remain bounded, and with it the electric field, the function [1 + 2εa(2q1(z) − 3q0)] mustnot change its sign in [−1, 1]. Then, since U 0 > 0 in Eq. (82d), U must be an increasing odd function of z,and so U

′(−1) = U

′(1) ≥ 0. Therefore, if c > 0 then

1 + 2εa(2q1(±1) − 3q0) = 1 − 2εaq0 > 0, (88)

whereas, if c < 0 then

1 + 2εa(2q1(0) − 3q0) < 0 . (89)

As explained in Section 3, since q3 ≡ 0, U(cr)

0,3 is the least value of U 0 for which q2 vanishes in the interval

[−1, 1]. By Eq. (84), U(cr)

0,3 is attained when q2(0) = 0, that is, when

q1(0) = 2q0 . (90)

For εa, q0 > 0, Eq. (90) violates the inequality (89), and so we assume that Eq. (88) is valid instead, and wetake c > 0. Thus, by Eqs. (82a), (87), and (90), after determining b, we find that q1 is given in the subintervals[−1, 0] and [0, 1] by

q ′1 = ±c

√εa

3

1√1 + 2εaq0

√2q0 − q1

1 + 2εa(2q1 − 3q0), (91)

respectively. Integrating both sides of Eq. (85) in the interval [−1, 0], we change the variable z into q1 in theright-hand side, thus obtaining

U(cr)

0,3 = c

2q0∫

q0

dq1

q ′1[1 + 2εa(2q1 − 3q0)]

, (92)

since U (0) = 0. By writing here q ′1 as in the increasing branch of Eq. (91), we readily obtain

U(cr)

0,3 =√

εa

3

√1 + 2εaq0

2q0∫

q0

dq1√[1 + 2εa(2q1 − 3q0)](2q0 − q1), (93)

whence, by setting q := q1/q0, we arrive at Eq. (58) in the text.

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Director reorientation and order reconstruction: competing mechanisms in a nematic cell

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