# Impact of Nanoparticles on Nematic Ordering in Square Wells

Authors: M. Slavinec; E. Klemenčič; M. Ambrožič; M. Krašna
Journal: Advances in Condensed Matter Physics
 (2015)

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## Abstract

Nematic liquid crystalline structures within square wells are studied numerically using both Lebwohl-Lasher lattice semimicroscopic and the Landau-de Gennes mesoscopic approach. At lateral boundary wall strong planar anchoring is enforced. The cell thicknessh along the z Cartesian coordinate is assumed to be smaller than the characteristic square well size R. Using semimicroscopic modelling we restrict to effectively two-dimensional systems which we study in terms of the tensor nematic order parameter. We consider impact of appropriate nanoparticles (NPs) on nematic configuration for cases where R becomes comparable to the biaxial order parameter correlation length. In this case a star-like order reconstruction biaxial profile could be formed in absence of NPs. We demonstrate existence of a rich variety of different nematic structures, including topological defects, which are enabled by presence of appropriate NPs.


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## Body

## 1. Introduction

Confined thermotropic liquid crystals (LCs) are of constant interest for years [1–3]. Due to their softness, liquid character, optical anisotropy, and rich variety of different phases and structures, they are interesting both from applicational and fundamental perspective [1, 4, 5]. Complexity of their behavior could be greatly enhanced by doping them with various nanoparticles [6–8]. The resulting configurations are referred to as soft-nanocomposites and are promising to play key role in future nano-based technology.Uniaxial bulk nematic phase represents the simplest LC configuration exhibiting only long range orientational ordering. It is typically reached on lowering temperature from isotropic (ordinary liquid) phase via 1st order continuous symmetry breaking phase transition [1, 9, 10]. In case of weak distortions nematic local ordering could be well described by uniaxial director field n→ and uniaxial order parameter S. The unit vector n→ determines local uniaxial direction where the states ±n→ are equivalent. The order parameter S describes extent of fluctuations about n→. It equals zero in the isotropic phase and exhibits maximal value S=1 for a rigid local alignment. In bulk nematic n→ is homogeneously aligned along a symmetry breaking direction and S is spatially homogeneous. In general, due to continuous symmetry breaking nature [11–13] of the isotropic-nematic phase nematic patterns could contain topological defects (TDs) in the center of which n→ is not uniquely defined [14–18]. Furthermore, strong local distortions within cores of defect could give rise to local biaxial ordering [15, 19, 20] which necessitates description in terms of the tensor order parameter Q. Note that TDs are in general energetically costly. Consequently in nonchiral LCs they could be stabilized only in confined geometries or in appropriate mixtures [21, 22].In most LC-based electrooptic applications nematic configurations are confined to regular geometries [20, 23–28]. Confinement walls break the nematic translational symmetry. In addition specific anchoring and wetting conditions at walls impose different nematic patterns, including patterns possessing TDs, with potential to display rich variety of different optic textures. Recently, several studies have focused on bistable LC devices consisting of an array of micrometer-sized wells [29, 30]. Diversity of LC states is commonly enabled by boundary condition imposed frustrations. If these frustrations are imposed on a characteristic linear scale R that is large with respect to a relevant order parameter correlation length ξ then essentially uniaxial nematic patterns are formed [31]. Consequently, local orientational order was well described in terms of n→. However, in cases where R and ξ are comparable essentially biaxial states also appear [24, 25]. For example, for sufficiently strong antagonistic uniaxial anchorings order reconstruction (OR) structures [23–25, 30] appear to reconcile surface-imposed frustrations. Description of OR mechanism requires picturing of nematic ordering in terms of Q. Order reconstruction refers to cases where spatially conflicting ordering tendencies are resolved via exchange of Q eigenvalues without involving a rotation of the tensor eigen frame. Typical OR structure mediating two competing regions exhibiting contradicting positive uniaxial ordering consists of a sheet possessing negative uniaxiality lying in between. This sheet is further enclosed by two sheets possessing maximal biaxiality. This sequence of nematic configurations is required topologically [4, 14, 19, 32].In this paper we study impact of NPs on nematic structures within a square well where boundary conditions impose planar anchoring. We show that complex OR structures could appear as positions of appropriate NPs are varied within a square well. The plan of the paper is as follows. The geometry of the problem and theoretical models is introduced in Section2. Both lattice Lebwohl-Lasher semimicroscopic-type and mesoscopic Landau-de Gennes approaches are used. In Section 3 we present simulations which result that effectively two dimensional (2D) ordering appears in systems of our interest. We also present results obtained using mesoscopic modelling where we focus on impact of NPs on LC structures. In the last section we summarize our results.


## 2. Modelling

Of our interest is impact of specific nanoparticles on nematic LC configurations confined to a rectangular cavity. In our study we use both lattice-type semimicroscopic and Landau-de Gennes mesoscopic modelling. In the following we first describe geometry of our interest and afterwards both theoretical approaches.


### 2.1. Geometry of the Problem

We consider a nematic LC sample confined within a rectangular planar (see Figure1) cell in the Cartesian coordinate system (x,y,z), where the corresponding unit vectors pointing along coordinate axes are labeled as (e→x,e→y,e→z). The confining plates are placed at x=0, x=R, y=0, y=R, z=0, z=h. The lateral boundaries enforce strong homogeneous tangential anchoring along corresponding Cartesian coordinates. Therefore, the plates at x=0, x=R enforce orientation along ±e→y, and the plates y=0, y=R enforce orientation along ±e→x. On the other hand we set that the top (z=0) and bottom (z=h) surface enforce degenerate tangential anchoring. Consequently, no orientation within a (x,y) plane is singled out.Figure 1
Nematic LC sample confined within a rectangular planar cell in the Cartesian coordinate system(x,y,z). The confining plates are placed at x=0, x=R, y=0, y=R, z=0, z=h.


### 2.2. Semimicroscopic Modelling

The 3D bulk LC spin model is represented by a rectangular cell, a lattice ofM×N×L sites. Each site is occupied by a unit vector S→(i,j,k)≡S→ijk (called briefly thenematic spin) representing the local nematic director, where 1≤i≤M, 1≤j≤N and 1≤k≤L. In order to study the dependence of the LC structure on the cell thickness, it is convenient to take lattice dimensions M=N>L, and to vary L. In addition, we artificially widen the lattice to the size (M+2) × (N+2) × (L+2) in the simulations to include the desired boundary conditions. The boundary planes, i = 0 and M+1, or j=0 and N+1, or k=0 and L+1, do not represent LC nematic director and are thus not counted in the calculation of various macroscopic parameters but are used only to fix certain spin directions.The strength of the interaction between the nearest nematic spins is given by positive coupling constantJ. The free energy functional is modeled by the sum of the terms of all spin sites [33]:
(1)F=∑ijkfijk,
where the energy term fijk consists of the sum
(2)fijk=-12J∑n.n.S→ijk·S→n.n.2.
This sum represents the interactions of the nematic spin with its six nearest neighbors: the denotation Sn.n. for the nearest neighbor thus includes S→(i+1,j,k), S→(i-1,j,k), S→(i,j+1,k), S→(i,j-1,k), S→(i,j,k+1), and S→(i,j,k-1). We must take an additional treatment of nematic spins at the simulation cell boundary. Take, for example, the spins with the first index i = 1. These spins lack the left neighbor spin to interact with it; they interact with the left wall corresponding to index i = 0, instead. Thus, in formula (2) we take theanchoring strength W instead of coupling constant J for the corresponding interaction. We suppose different boundary anchoring conditions for the four “side” walls (i = 0 or i = M+1 or j = 0 or j = N+1) as compared to “base,” that is, bottom and top walls (k=0 or k=L+1). For side walls the anchoring direction is planar horizontal; that is, S→ is fixed to (1, 0, 0) or (0, 1, 0), for the two pairs of opposite walls, correspondingly; the corresponding anchoring strength is denoted by W4s. However, for the pair of base walls the anchoring direction is degenerate planar; that is, any direction in that plane is equivalent. This condition is considered by S→ being fixed to (0, 0, 1) for the two walls at k=0 and L+1; the corresponding anchoring strength is W2b. But the corresponding interaction must be now taken with positive sign, not negative as in (2). This ensures that in order to reduce the free energy the nematic spins tend to align perpendicularly to the direction (0, 0, 1), at the base walls, that is, planar direction without any preferred easy axis.The free energy and the three interaction magnitudes (J, W4s, and W2b) are next renormalized with respect to the coupling constant J by setting J = 1. The equilibrium spin configuration is obtained by minimizing the total free energy with respect to all the nematic spins. In order to satisfy the normalization of the spin vectors, S→ijk2=1, the “operational” total free energy must be rewritten as
(3)F*=∑ijkfijk*,
where
(4)fijk*=λijkS→ijk2-1+fijk,
with additional multiplication “constants” λijk, which also have to be found in order to solve the system.When thermal fluctuations of the spins have to be considered the final equilibrium state of the system is calculated through the “real-time” relaxation process including thermal disturbance. The change of spin components in the time stepΔt is according to [34]
(5)Sijk,nnew=Sijk,nold+DΔtkBT·∂F∂Sijk,nold+ΔSTijk,n,
where Sijk,n denotes nth component of the spin, D is the appropriate degenerated rotation diffusion tensor of the system, and kB is the Boltzmann constant. The first term on the right side of (5) corresponds to mechanical torque which tends to rotate the spin towards equilibrium, while the second term ΔS(T)ijk,n represents random thermal fluctuation. If we introduce dimensionless time step and temperature, Δt*=Dt, T*=kBT/J, and use (5) with dimensionless residuum vector we obtain the corresponding numerical equation:
(6)S→ijknew=S→ijkold+2Δt*T*·R→ijkold+ΔS→T,
where we have derived the “residuum” vector quantity:
(7)R→ijk=∑n.n.g→(S→ijk,S→n.n.)
and the vector g→ is a function of two vectors which is defined as
(8)g→v→1,v→2=v→1·v→2v→2-v→1·v→2v→1.The appropriate dimensionless time intervalΔt* is chosen as the input parameter according to literature: Δt*≈0.01. A convenient criterion of stopping the numeric iterations is that the values of some macroscopic quantities such as free energy F or the scalar parameter S below do not change (apart from very small fluctuations) any more.After the numeric iteration is stopped and the spins are calculated, various macroscopic quantities can be obtained. One of them is the equilibrium total energy which is conveniently normalized to one spin site:(9)f=FMNL
and is the average energy term per spin, Sijk from (2).The order of the nematic spin system can be characterized by the traceless symmetric order parameter tensor:(10)Qmn=32Sijk,mSijk,n-12I,
where Sijk,m is the mth component of the spin S→ijk. The brackets ⋯ denote the average of the quantity through the simulation cell, and I is the identity matrix. The eigenvalues of the Q→mn tensor satisfy the equation: λ1+λ2+λ3=0 and are limited to the interval (−1/2, 1). The scalar order parameter S is defined as the largest eigenvalue. The degree of biaxiality is measured with thebiaxiality parameter[35]
(11)β2=1-6Tr⁡Q32Tr⁡Q23=1-6λ13+λ23+λ332λ12+λ22+λ323,
where 0 ≤ β2 ≤ 1. The uniaxial states are characterized by β2 = 0 and the states exhibiting maximal biaxiality by β2=1.


### 2.3. Mesoscopic Modelling

We describe liquid crystalline orientational ordering using the nematic tensor order parameter [30]
(12)Q=∑i=13sie→i⊗e→i,
where e→i and si are eigenvectors and eigenvalues of Q, respectively. In case of uniaxial order Q is conventionally expressed as [1]
(13)Q=Sn→⊗n→-13I.
The unit vector n→ lies along the local uniaxial ordering direction and is referred to as the nematic director field. The scalar order parameter S expresses the magnitude of fluctuations about n→. Note that nematic LC ordering exhibits at mesoscopic level the so called head-to-tail invariance; that is, the orientations ±n→ are equivalent.The degree of biaxiality could be measured by the scalar parameter [35]
(14)β2=1-6tr⁡Q32tr⁡Q23∈0,1.
Here β2=0 signals uniaxial states, while an ordering with the maximum degree of biaxiality corresponds to β2=1. Equality tr⁡Q3=3det⁡Q reveals that maximum value of β2 is reached when at least one eigenvalue si vanishes.We consider confined nematic LC and express the free energy of the system as(15)F=∫(fc+fe+ff)d3r→+∫fsd2r→,
where d3r→ and d2r→ stand for the volume and area measures, respectively. The condensation (fc), elastic (fe), external field (ff) and surface (fs) free energy densities are expressed as [30]
(16)fc=A0(T-T*)2tr⁡Q2-B3tr⁡Q3+C4tr⁡Q22,fe=L2∇Q2,ff=ε0Δε2E→·QE→.
Here A0, B, and C are material constants, T stands for temperature and T* is the supercooling temperature of the isotropic phase. The condensation term enforces S(T<TIN)≡Seq(T)=B/4C1+1-24A0(T-T*)C/B21/2 and S(T>TIN)=0. The bulk isotropic-nematic phase transition temperature is given by TIN=T*+B2/27A0C. The superheating temperature of the nematic phase is realized at T**=T*+B2/24A0C. The elastic term fe penalizing departures from a spatially homogeneous orientational ordering is weighted by a positive elastic constant L. The so called single constant approximation is used. The external field term introduces the impact of an electric field E→=Ee→, where ε0 stands for the electrical permittivity constant, Δε is the field anisotropy constant, and the unit vector e→ determines the orientation of the field. At the lateral surfaces we strongly impose tangential ordering. Therefore, we assume that the corresponding surface free energy density imposes strong anchoring conditions.The important length scales [30] entering our analysis are the biaxial order parameter correlation length ξb~LC/B and external field coherence length ξf~LS/(ε0ΔεE2). These lengths are in general temperature dependent [30].


#### 2.3.1. Parametrization

For a geometry given equilibrium configurations are not expected to exhibit variations along thez-axis and rotation of Q tensor eigen frame is restricted to (x,y) planes. A corresponding suitable order parameter tensor Q parametrization is given by [30, 36]
(17)Qx,y=(q3+q1)e→x⊗e→x+(q3-q1)e→y⊗e→y+q2(e→x⊗e→y+e→y⊗e→x)-2q3e→z⊗e→z,
where q1, q2, and q3 are functions only of x and y. Hence, e→3=e→z is always an eigenvector of Q. The remaining two eigenvectors are allowed to rotate with respect to the reference (e→x,e→y) by an angle φ∈[0,π). It holds
(18)e→1=cos⁡φe→x+sinφe→y,e→2=-sinφe→x+cos⁡φe→y,e→3=e→z.
The parameter q3 reveals departures of the order parameter eigen frame (e→1,e→2,e→3) from the Cartesian frame (e→x,e→y,e→z). The two frames coincide when q2=0. The order parameters (q1,q2,q3) and eigenvalues s1,s2,s3 are related via s1=q3+q12+q22, s2=q3-q12+q22,s3=-2q3. Note that the condition q12+q22=0 corresponds to the exchange of eigenvalues s1↔s2 in the nematic phase.In the case of positive uniaxial ordering alonge→1 (i.e., n→=e→1 in (13)) it holds
(19)q1=S2cos⁡2φ,q2=S2sin2φ,q3=S6.


#### 2.3.2. Scaling and Euler-Lagrange Equations

We use the scaling introduced in [35, 37], where dimensionless temperature is defined as t=(T-T*)/(T**-T*). Therefore, t(T=T*)=0, t(T=T1N)=8/9, t(T=T**)=1. For latter convenience it is customary to introduce
(20)τ=1+1-t.
In this scaling the equilibrium uniaxial order parameter can be expressed as
(21)Seq=S**τ,
where S**=B/4C=Seq(T**).The key material length of systems of our interest is the biaxial correlation length. Its value is well estimated by(22)ξb=ξb(0)τ,
where ξb(0)=2LC/B is the bare biaxial correlation length. For conventional nematics, ξb~20nm. Furthermore, we introduce the external field coherence
(23)ξf=LS**ε0ΔεE2
which is for latter convenience expressed at temperature T=T**.We further introduce the following scaled and dimensionless quantities:Q~=Q/S**, x~=x/R, y~=y/R, z~=z/R, ∇~=R∇, F~=F/F0, where F0=LS**2R3/(2ξb(0)2). We henceforth skip the tildes bearing in mind that all quantities are scaled. The resulting dimensionless free energy F is expressed as in (15), where dimensionless free energies are given by
(24)fc=τ6tr⁡Q2-23tr⁡Q3+18tr⁡Q22=τ3q12+q22+3q32-4q3q12+q22-q32+12q12+q22+3q322,fe=ξb(0)R2∇Q2=2ξb(0)R23∇q32+∇q12+∇q22,ff=-ΔεΔεξb(0)ξf2e→·Qe→=-ΔεΔεξb(0)ξf2·q3+q1e→x·e→2+q3-q1e→y·e→2ssssssq3+q1e→x·e→2+q3-q1e→y·e→2+2q2e→x·e→e→y·e→.
Minimization of the free energy yields the following bulk Euler-Lagrange equations for variational parameters q0, q1, and qm:
(25)Δ⊥q1ξb(0)R2-τ6q1+2q2q1-q123q22+q12+q32+14ξb(0)ξf2e→x·e→2-e→y·e→2=0,Δ⊥q2ξb(0)R2-τ6q2+13q12+q32-3q22-q223q22+q12+q32+112ξb(0)ξf2=0,Δ⊥q3ξb(0)h2-τ6q3+2q2q3-q323q22+q12+q32+12ξb(0)ξf2e→x·e→e→y·e→=0,
where Δ⊥=∂2/∂x2+∂2/∂y2.


## 2.1. Geometry of the Problem

We consider a nematic LC sample confined within a rectangular planar (see Figure1) cell in the Cartesian coordinate system (x,y,z), where the corresponding unit vectors pointing along coordinate axes are labeled as (e→x,e→y,e→z). The confining plates are placed at x=0, x=R, y=0, y=R, z=0, z=h. The lateral boundaries enforce strong homogeneous tangential anchoring along corresponding Cartesian coordinates. Therefore, the plates at x=0, x=R enforce orientation along ±e→y, and the plates y=0, y=R enforce orientation along ±e→x. On the other hand we set that the top (z=0) and bottom (z=h) surface enforce degenerate tangential anchoring. Consequently, no orientation within a (x,y) plane is singled out.Figure 1
Nematic LC sample confined within a rectangular planar cell in the Cartesian coordinate system(x,y,z). The confining plates are placed at x=0, x=R, y=0, y=R, z=0, z=h.


## 2.2. Semimicroscopic Modelling

The 3D bulk LC spin model is represented by a rectangular cell, a lattice ofM×N×L sites. Each site is occupied by a unit vector S→(i,j,k)≡S→ijk (called briefly thenematic spin) representing the local nematic director, where 1≤i≤M, 1≤j≤N and 1≤k≤L. In order to study the dependence of the LC structure on the cell thickness, it is convenient to take lattice dimensions M=N>L, and to vary L. In addition, we artificially widen the lattice to the size (M+2) × (N+2) × (L+2) in the simulations to include the desired boundary conditions. The boundary planes, i = 0 and M+1, or j=0 and N+1, or k=0 and L+1, do not represent LC nematic director and are thus not counted in the calculation of various macroscopic parameters but are used only to fix certain spin directions.The strength of the interaction between the nearest nematic spins is given by positive coupling constantJ. The free energy functional is modeled by the sum of the terms of all spin sites [33]:
(1)F=∑ijkfijk,
where the energy term fijk consists of the sum
(2)fijk=-12J∑n.n.S→ijk·S→n.n.2.
This sum represents the interactions of the nematic spin with its six nearest neighbors: the denotation Sn.n. for the nearest neighbor thus includes S→(i+1,j,k), S→(i-1,j,k), S→(i,j+1,k), S→(i,j-1,k), S→(i,j,k+1), and S→(i,j,k-1). We must take an additional treatment of nematic spins at the simulation cell boundary. Take, for example, the spins with the first index i = 1. These spins lack the left neighbor spin to interact with it; they interact with the left wall corresponding to index i = 0, instead. Thus, in formula (2) we take theanchoring strength W instead of coupling constant J for the corresponding interaction. We suppose different boundary anchoring conditions for the four “side” walls (i = 0 or i = M+1 or j = 0 or j = N+1) as compared to “base,” that is, bottom and top walls (k=0 or k=L+1). For side walls the anchoring direction is planar horizontal; that is, S→ is fixed to (1, 0, 0) or (0, 1, 0), for the two pairs of opposite walls, correspondingly; the corresponding anchoring strength is denoted by W4s. However, for the pair of base walls the anchoring direction is degenerate planar; that is, any direction in that plane is equivalent. This condition is considered by S→ being fixed to (0, 0, 1) for the two walls at k=0 and L+1; the corresponding anchoring strength is W2b. But the corresponding interaction must be now taken with positive sign, not negative as in (2). This ensures that in order to reduce the free energy the nematic spins tend to align perpendicularly to the direction (0, 0, 1), at the base walls, that is, planar direction without any preferred easy axis.The free energy and the three interaction magnitudes (J, W4s, and W2b) are next renormalized with respect to the coupling constant J by setting J = 1. The equilibrium spin configuration is obtained by minimizing the total free energy with respect to all the nematic spins. In order to satisfy the normalization of the spin vectors, S→ijk2=1, the “operational” total free energy must be rewritten as
(3)F*=∑ijkfijk*,
where
(4)fijk*=λijkS→ijk2-1+fijk,
with additional multiplication “constants” λijk, which also have to be found in order to solve the system.When thermal fluctuations of the spins have to be considered the final equilibrium state of the system is calculated through the “real-time” relaxation process including thermal disturbance. The change of spin components in the time stepΔt is according to [34]
(5)Sijk,nnew=Sijk,nold+DΔtkBT·∂F∂Sijk,nold+ΔSTijk,n,
where Sijk,n denotes nth component of the spin, D is the appropriate degenerated rotation diffusion tensor of the system, and kB is the Boltzmann constant. The first term on the right side of (5) corresponds to mechanical torque which tends to rotate the spin towards equilibrium, while the second term ΔS(T)ijk,n represents random thermal fluctuation. If we introduce dimensionless time step and temperature, Δt*=Dt, T*=kBT/J, and use (5) with dimensionless residuum vector we obtain the corresponding numerical equation:
(6)S→ijknew=S→ijkold+2Δt*T*·R→ijkold+ΔS→T,
where we have derived the “residuum” vector quantity:
(7)R→ijk=∑n.n.g→(S→ijk,S→n.n.)
and the vector g→ is a function of two vectors which is defined as
(8)g→v→1,v→2=v→1·v→2v→2-v→1·v→2v→1.The appropriate dimensionless time intervalΔt* is chosen as the input parameter according to literature: Δt*≈0.01. A convenient criterion of stopping the numeric iterations is that the values of some macroscopic quantities such as free energy F or the scalar parameter S below do not change (apart from very small fluctuations) any more.After the numeric iteration is stopped and the spins are calculated, various macroscopic quantities can be obtained. One of them is the equilibrium total energy which is conveniently normalized to one spin site:(9)f=FMNL
and is the average energy term per spin, Sijk from (2).The order of the nematic spin system can be characterized by the traceless symmetric order parameter tensor:(10)Qmn=32Sijk,mSijk,n-12I,
where Sijk,m is the mth component of the spin S→ijk. The brackets ⋯ denote the average of the quantity through the simulation cell, and I is the identity matrix. The eigenvalues of the Q→mn tensor satisfy the equation: λ1+λ2+λ3=0 and are limited to the interval (−1/2, 1). The scalar order parameter S is defined as the largest eigenvalue. The degree of biaxiality is measured with thebiaxiality parameter[35]
(11)β2=1-6Tr⁡Q32Tr⁡Q23=1-6λ13+λ23+λ332λ12+λ22+λ323,
where 0 ≤ β2 ≤ 1. The uniaxial states are characterized by β2 = 0 and the states exhibiting maximal biaxiality by β2=1.


## 2.3. Mesoscopic Modelling

We describe liquid crystalline orientational ordering using the nematic tensor order parameter [30]
(12)Q=∑i=13sie→i⊗e→i,
where e→i and si are eigenvectors and eigenvalues of Q, respectively. In case of uniaxial order Q is conventionally expressed as [1]
(13)Q=Sn→⊗n→-13I.
The unit vector n→ lies along the local uniaxial ordering direction and is referred to as the nematic director field. The scalar order parameter S expresses the magnitude of fluctuations about n→. Note that nematic LC ordering exhibits at mesoscopic level the so called head-to-tail invariance; that is, the orientations ±n→ are equivalent.The degree of biaxiality could be measured by the scalar parameter [35]
(14)β2=1-6tr⁡Q32tr⁡Q23∈0,1.
Here β2=0 signals uniaxial states, while an ordering with the maximum degree of biaxiality corresponds to β2=1. Equality tr⁡Q3=3det⁡Q reveals that maximum value of β2 is reached when at least one eigenvalue si vanishes.We consider confined nematic LC and express the free energy of the system as(15)F=∫(fc+fe+ff)d3r→+∫fsd2r→,
where d3r→ and d2r→ stand for the volume and area measures, respectively. The condensation (fc), elastic (fe), external field (ff) and surface (fs) free energy densities are expressed as [30]
(16)fc=A0(T-T*)2tr⁡Q2-B3tr⁡Q3+C4tr⁡Q22,fe=L2∇Q2,ff=ε0Δε2E→·QE→.
Here A0, B, and C are material constants, T stands for temperature and T* is the supercooling temperature of the isotropic phase. The condensation term enforces S(T<TIN)≡Seq(T)=B/4C1+1-24A0(T-T*)C/B21/2 and S(T>TIN)=0. The bulk isotropic-nematic phase transition temperature is given by TIN=T*+B2/27A0C. The superheating temperature of the nematic phase is realized at T**=T*+B2/24A0C. The elastic term fe penalizing departures from a spatially homogeneous orientational ordering is weighted by a positive elastic constant L. The so called single constant approximation is used. The external field term introduces the impact of an electric field E→=Ee→, where ε0 stands for the electrical permittivity constant, Δε is the field anisotropy constant, and the unit vector e→ determines the orientation of the field. At the lateral surfaces we strongly impose tangential ordering. Therefore, we assume that the corresponding surface free energy density imposes strong anchoring conditions.The important length scales [30] entering our analysis are the biaxial order parameter correlation length ξb~LC/B and external field coherence length ξf~LS/(ε0ΔεE2). These lengths are in general temperature dependent [30].


### 2.3.1. Parametrization

For a geometry given equilibrium configurations are not expected to exhibit variations along thez-axis and rotation of Q tensor eigen frame is restricted to (x,y) planes. A corresponding suitable order parameter tensor Q parametrization is given by [30, 36]
(17)Qx,y=(q3+q1)e→x⊗e→x+(q3-q1)e→y⊗e→y+q2(e→x⊗e→y+e→y⊗e→x)-2q3e→z⊗e→z,
where q1, q2, and q3 are functions only of x and y. Hence, e→3=e→z is always an eigenvector of Q. The remaining two eigenvectors are allowed to rotate with respect to the reference (e→x,e→y) by an angle φ∈[0,π). It holds
(18)e→1=cos⁡φe→x+sinφe→y,e→2=-sinφe→x+cos⁡φe→y,e→3=e→z.
The parameter q3 reveals departures of the order parameter eigen frame (e→1,e→2,e→3) from the Cartesian frame (e→x,e→y,e→z). The two frames coincide when q2=0. The order parameters (q1,q2,q3) and eigenvalues s1,s2,s3 are related via s1=q3+q12+q22, s2=q3-q12+q22,s3=-2q3. Note that the condition q12+q22=0 corresponds to the exchange of eigenvalues s1↔s2 in the nematic phase.In the case of positive uniaxial ordering alonge→1 (i.e., n→=e→1 in (13)) it holds
(19)q1=S2cos⁡2φ,q2=S2sin2φ,q3=S6.


### 2.3.2. Scaling and Euler-Lagrange Equations

We use the scaling introduced in [35, 37], where dimensionless temperature is defined as t=(T-T*)/(T**-T*). Therefore, t(T=T*)=0, t(T=T1N)=8/9, t(T=T**)=1. For latter convenience it is customary to introduce
(20)τ=1+1-t.
In this scaling the equilibrium uniaxial order parameter can be expressed as
(21)Seq=S**τ,
where S**=B/4C=Seq(T**).The key material length of systems of our interest is the biaxial correlation length. Its value is well estimated by(22)ξb=ξb(0)τ,
where ξb(0)=2LC/B is the bare biaxial correlation length. For conventional nematics, ξb~20nm. Furthermore, we introduce the external field coherence
(23)ξf=LS**ε0ΔεE2
which is for latter convenience expressed at temperature T=T**.We further introduce the following scaled and dimensionless quantities:Q~=Q/S**, x~=x/R, y~=y/R, z~=z/R, ∇~=R∇, F~=F/F0, where F0=LS**2R3/(2ξb(0)2). We henceforth skip the tildes bearing in mind that all quantities are scaled. The resulting dimensionless free energy F is expressed as in (15), where dimensionless free energies are given by
(24)fc=τ6tr⁡Q2-23tr⁡Q3+18tr⁡Q22=τ3q12+q22+3q32-4q3q12+q22-q32+12q12+q22+3q322,fe=ξb(0)R2∇Q2=2ξb(0)R23∇q32+∇q12+∇q22,ff=-ΔεΔεξb(0)ξf2e→·Qe→=-ΔεΔεξb(0)ξf2·q3+q1e→x·e→2+q3-q1e→y·e→2ssssssq3+q1e→x·e→2+q3-q1e→y·e→2+2q2e→x·e→e→y·e→.
Minimization of the free energy yields the following bulk Euler-Lagrange equations for variational parameters q0, q1, and qm:
(25)Δ⊥q1ξb(0)R2-τ6q1+2q2q1-q123q22+q12+q32+14ξb(0)ξf2e→x·e→2-e→y·e→2=0,Δ⊥q2ξb(0)R2-τ6q2+13q12+q32-3q22-q223q22+q12+q32+112ξb(0)ξf2=0,Δ⊥q3ξb(0)h2-τ6q3+2q2q3-q323q22+q12+q32+12ξb(0)ξf2e→x·e→e→y·e→=0,
where Δ⊥=∂2/∂x2+∂2/∂y2.


## 2.3.1. Parametrization

For a geometry given equilibrium configurations are not expected to exhibit variations along thez-axis and rotation of Q tensor eigen frame is restricted to (x,y) planes. A corresponding suitable order parameter tensor Q parametrization is given by [30, 36]
(17)Qx,y=(q3+q1)e→x⊗e→x+(q3-q1)e→y⊗e→y+q2(e→x⊗e→y+e→y⊗e→x)-2q3e→z⊗e→z,
where q1, q2, and q3 are functions only of x and y. Hence, e→3=e→z is always an eigenvector of Q. The remaining two eigenvectors are allowed to rotate with respect to the reference (e→x,e→y) by an angle φ∈[0,π). It holds
(18)e→1=cos⁡φe→x+sinφe→y,e→2=-sinφe→x+cos⁡φe→y,e→3=e→z.
The parameter q3 reveals departures of the order parameter eigen frame (e→1,e→2,e→3) from the Cartesian frame (e→x,e→y,e→z). The two frames coincide when q2=0. The order parameters (q1,q2,q3) and eigenvalues s1,s2,s3 are related via s1=q3+q12+q22, s2=q3-q12+q22,s3=-2q3. Note that the condition q12+q22=0 corresponds to the exchange of eigenvalues s1↔s2 in the nematic phase.In the case of positive uniaxial ordering alonge→1 (i.e., n→=e→1 in (13)) it holds
(19)q1=S2cos⁡2φ,q2=S2sin2φ,q3=S6.


## 2.3.2. Scaling and Euler-Lagrange Equations

We use the scaling introduced in [35, 37], where dimensionless temperature is defined as t=(T-T*)/(T**-T*). Therefore, t(T=T*)=0, t(T=T1N)=8/9, t(T=T**)=1. For latter convenience it is customary to introduce
(20)τ=1+1-t.
In this scaling the equilibrium uniaxial order parameter can be expressed as
(21)Seq=S**τ,
where S**=B/4C=Seq(T**).The key material length of systems of our interest is the biaxial correlation length. Its value is well estimated by(22)ξb=ξb(0)τ,
where ξb(0)=2LC/B is the bare biaxial correlation length. For conventional nematics, ξb~20nm. Furthermore, we introduce the external field coherence
(23)ξf=LS**ε0ΔεE2
which is for latter convenience expressed at temperature T=T**.We further introduce the following scaled and dimensionless quantities:Q~=Q/S**, x~=x/R, y~=y/R, z~=z/R, ∇~=R∇, F~=F/F0, where F0=LS**2R3/(2ξb(0)2). We henceforth skip the tildes bearing in mind that all quantities are scaled. The resulting dimensionless free energy F is expressed as in (15), where dimensionless free energies are given by
(24)fc=τ6tr⁡Q2-23tr⁡Q3+18tr⁡Q22=τ3q12+q22+3q32-4q3q12+q22-q32+12q12+q22+3q322,fe=ξb(0)R2∇Q2=2ξb(0)R23∇q32+∇q12+∇q22,ff=-ΔεΔεξb(0)ξf2e→·Qe→=-ΔεΔεξb(0)ξf2·q3+q1e→x·e→2+q3-q1e→y·e→2ssssssq3+q1e→x·e→2+q3-q1e→y·e→2+2q2e→x·e→e→y·e→.
Minimization of the free energy yields the following bulk Euler-Lagrange equations for variational parameters q0, q1, and qm:
(25)Δ⊥q1ξb(0)R2-τ6q1+2q2q1-q123q22+q12+q32+14ξb(0)ξf2e→x·e→2-e→y·e→2=0,Δ⊥q2ξb(0)R2-τ6q2+13q12+q32-3q22-q223q22+q12+q32+112ξb(0)ξf2=0,Δ⊥q3ξb(0)h2-τ6q3+2q2q3-q323q22+q12+q32+12ξb(0)ξf2e→x·e→e→y·e→=0,
where Δ⊥=∂2/∂x2+∂2/∂y2.


## 3. Results

### 3.1. Semimicroscopic Results

In this section we analyze simulation results obtained using the lattice-type semimicroscopic approach within a cubic cell of volumeR3. In Figure 2 we plot spatial averaged squared z-component of the nematic spins Sz2xy for two values of L=R/a0. Here ⋯ stands for the spatial average and a0 stands for the lattice constant (i.e., equals separation between neighboring spins). One sees that on decreasing temperature the structure is becoming increasingly planar, that is, effectively two-dimensional. Sudden drop of Sz2xy value at critical temperature corresponding to kBTc/J~1.24 reveals isotropic-nematic phase transition.Figure 2
Temperature variation ofSz2xy for two different system sizes L=R/a0. The isotropic-nematic phase transition takes place at kBTc/J~1.24. Deep in the nematic phase the system is effectively two-dimensional. Lines are used to guide the eye.In Figure3(a) we plot β2xz as a function of k=z/a0 in the cells characterized by L = 40 and L = 20. Here ⋯xz marks spatial averaging within the cell. Note that at the edges of surface we enforce uniaxial ordering. One sees that β2xz relatively weakly depends on the z-coordinate spatially. Note that for large enough cells the ordering is uniaxial, corresponding to β2xz=0. However, in our cells we see relative large value of β2xz which increases with decreasing L. This reveals that the structures become increasingly biaxial on average for such severe confinements despite the fact that the phase transition still exists.(a) Plot ofβ2xy as a function of k=z/a0 for L=20 and L=40 deep in the nematic phase. (b) Plot of Sz2xy as a function of k=z/a0 for L=20 and L=40 deep in the nematic phase. One sees that the nematic structure is effectively two-dimensional in the whole cell. Lines are used to guide the eye.
(a)
(b)Finally, in Figure3(b) we plot Sz2xy as a function of k=z/a0 deep in the nematic phase. One witnesses relatively small values Sz2xy in the whole simulational cell. Therefore, the ordering can be considered as effectively two-dimensional for low enough temperatures.


### 3.2. Mesoscopic Results

In this section we analyze effectively 2D nematic structures within square wells where the lateral confinement sizeR is below μm and therefore comparable to the biaxial nematic order parameter correlation length ξb. We focus on the impact of square-shaped NPs enforcing strong planar boundary conditions on nematic configurations in absence and presence of an external electric field.


#### 3.2.1. Absence of NPs

First, we consider cases without NPs in absence and presence of a statical ordering external fieldE→=Ee→z. Note that the lateral walls impose strong tangential anchoring. Consequent conflicting boundary conditions at corners of a well enforce topological defects. Typical nematic structures are shown in Figures 4 and 5. In Figures 4(a) and 5 we show the so calleddiagonal structure(DS) which is stable for R≫ξb and E = 0. In most parts of the cell the nematic structure is essentially uniaxial and the nematic director field is preferentially aligned along the diagonal of the square; see Figure 5. Substantial departures from uniaxial order are surrounding defects placed at the corners of the well. Their presence is clearly seen by plotting the β2(x,y) spatial profile (see Figure 4(a)). For visualization purpose the structure is shown in a regime where R is comparable to ξb. Defects locally enforce biaxial ordering and topological characteristics of the resulting defect-core structure are known. Main defect-core features are as follows. The center of each defect exhibits negative uniaxial order along e→z and is surrounded by a rim possessing maximal degree of biaxiality β2=1. Far from centers of isolated defects essentially positive uniaxial order is established in bulk. The characteristic linear size of defect-core structure is roughly determined by the biaxial length ξb, which reflects LC material properties at a given temperature.The degree of biaxialityβ2(x,y) within a square well, τ=4, strong anchoring condition. (a) DS, R/ξb=7, R/ξf=0; (b) DS, R/ξb=7, R/ξf=10; (c) DS, R/ξb=7, R/ξf=100; (d) WORS, R/ξb=3, R/ξf=0.
(a)
(b)
(c)
(d)Figure 5
Orientational distribution of maximal eigenvectors ofQ in DS. In the uniaxial limit these eigenvectors equal the nematic director field n→. The corresponding β2(x,y) dependence is plotted in Figure 4(a).   R/ξb=7, τ=4, strong anchoring condition.On decreasing the ratioη=R/ξb the so calledwell order reconstruction structure(WORS) is formed which is shown in Figure 4(d). Deep in the nematic phase this structure is stable for η<ηc=3.28±0.01. In it the local eigen frames of Q coincide with the e→x,e→y,e→z laboratory frame orientation in the whole (x,y) plane. The β2=1 rim exhibiting maximal biaxiality adopts an open cross-like profile. It separates regions exhibiting essentially positive uniaxial and essentially negative uniaxial ordering outside and inside the rim, respectively.The impact of an external field for LCs with positive field anisotropy is demonstrated in Figures4(b) and 4(c), where the field is aligned along the y-axis. The field tends to align e→1 (the eigenvalue corresponding to the largest eigenvalue of Q) along its direction. Consequently, on increasing the field strength the elastic distortions become increasingly accumulated at the top-bottom plates, because these plates enforce contradicting orientations with respect to E→. A typical pattern for a relatively weak field strength is shown in Figure 4(b). The top-bottom/left-right symmetry realized in absence of the field is broken. If the external field is strong enough it triggers the surface order reconstruction type of structural transition at the top-bottom plates, which is shown in Figure 4(c). We study also an alternative case where LC exhibits negative field anisotropy for the same orientation of the external field (i.e., e→=e→y). In this case the field tends to orient e→1 perpendicular to its direction. Consequently, the elastic distortions now accumulate at left-right plates. In both Δε>0 and Δε<0 cases the external field enforces similar biaxial textures which are rotated with respect to each other for an angle π/2 in the (x,y) plane.


#### 3.2.2. Presence of NPs

Next, we study impact of a nanoparticle on ordering within the cell exhibiting effectively 2D behavior. Of our interest is to demonstrate existence of a rich variety of qualitatively different patterns just on varying the relative position of the nanoparticle within the cell and using different external ordering electric fieldE→=Ee→y strengths. For this purpose we choose LC exhibiting positive field anisotropy. We choose square-shaped NP of typical length a=R/10. Furthermore, we set that it enforces strong tangential anchoring to surrounding LC. We place it either (1) in the center of the cell, (2) close to the left bottom well edge, or (3) close to the left boundary, and (4) close to the bottom boundary of the well. These positions of NP are schematically sketched in Figure 6. We henceforth refer to these positions as the (1)central, (2)edge, (3)left boundary,and (4)bottom boundaryposition.Figure 6
Schematic presentation of different positions of square-shaped NP of lengtha=R/10 within the square well. (1)centralposition, (2)edgeposition, (3)left boundaryposition, and (4)bottom boundaryposition.In Figure7 we analyze structures for η=7 on varying E. In the first row we show cases for E=0. The edges of NP are sources of elastic distortions, giving rise to locally induced biaxiality. We see that for all four positions of NPs the global LC structure remains qualitatively the same, although the structures are slightly different. Note that in this case patterns emerging from left boundary and bottom boundary position are the same if the patterns are rotated for π/2 about the z-axis. Next we switch on the external field of strength R/ξf=50. Presence of the field breaks the symmetry of the problem. Its effect is increasingly pronounced for thebottomposition of NP. In this case the mutual impact of NP and external field enables order reconstruction (OR) type transition at the bottom plate. Note that the external field alone is too weak to stabilize this type of structure. For still stronger field, that is, R/ξf=100, we get four qualitatively different patterns. Among them thebottomandedgeposition support OR-type patterns.Nematic patterns for different positions of NP forη=7. (a) R/ξf=0. (b) R/ξf=50. (c) R/ξf=100.
(a)
(b)
(c)In Figure8 we set η=5 which is above the threshold value ηc=3.28 below which the WORS pattern is stable. However, we see that for the central position WORS-type pattern is nevertheless stabilized for E=0 due to the presence of NP. If the external field is switched on the WORS-type pattern vanishes and is replaced by “common” OR-type structures. In this case R/ξf=50 and R/ξf=100 yield qualitatively similar patterns.Nematic patterns for different positions of NP forη=5. (a) R/ξf=0. (b) R/ξf=50. (c) R/ξf=100.
(a)
(b)
(c)Finally, in Figure9 we set η=3<ηc. For such condition the WORS pattern is stable in absence of NP. However, presence of NP does not support this pattern for theedgeposition for E=0. For R/ξf=50 and R/ξf=100 we obtain again qualitatively similar patterns. None of them exhibits WORS-type structure.Nematic patterns for different positions of NP forη=3. (a) R/ξf=0. (b) R/ξf=50. (c) R/ξf=100.
(a)
(b)
(c)


## 3.1. Semimicroscopic Results

In this section we analyze simulation results obtained using the lattice-type semimicroscopic approach within a cubic cell of volumeR3. In Figure 2 we plot spatial averaged squared z-component of the nematic spins Sz2xy for two values of L=R/a0. Here ⋯ stands for the spatial average and a0 stands for the lattice constant (i.e., equals separation between neighboring spins). One sees that on decreasing temperature the structure is becoming increasingly planar, that is, effectively two-dimensional. Sudden drop of Sz2xy value at critical temperature corresponding to kBTc/J~1.24 reveals isotropic-nematic phase transition.Figure 2
Temperature variation ofSz2xy for two different system sizes L=R/a0. The isotropic-nematic phase transition takes place at kBTc/J~1.24. Deep in the nematic phase the system is effectively two-dimensional. Lines are used to guide the eye.In Figure3(a) we plot β2xz as a function of k=z/a0 in the cells characterized by L = 40 and L = 20. Here ⋯xz marks spatial averaging within the cell. Note that at the edges of surface we enforce uniaxial ordering. One sees that β2xz relatively weakly depends on the z-coordinate spatially. Note that for large enough cells the ordering is uniaxial, corresponding to β2xz=0. However, in our cells we see relative large value of β2xz which increases with decreasing L. This reveals that the structures become increasingly biaxial on average for such severe confinements despite the fact that the phase transition still exists.(a) Plot ofβ2xy as a function of k=z/a0 for L=20 and L=40 deep in the nematic phase. (b) Plot of Sz2xy as a function of k=z/a0 for L=20 and L=40 deep in the nematic phase. One sees that the nematic structure is effectively two-dimensional in the whole cell. Lines are used to guide the eye.
(a)
(b)Finally, in Figure3(b) we plot Sz2xy as a function of k=z/a0 deep in the nematic phase. One witnesses relatively small values Sz2xy in the whole simulational cell. Therefore, the ordering can be considered as effectively two-dimensional for low enough temperatures.


## 3.2. Mesoscopic Results

In this section we analyze effectively 2D nematic structures within square wells where the lateral confinement sizeR is below μm and therefore comparable to the biaxial nematic order parameter correlation length ξb. We focus on the impact of square-shaped NPs enforcing strong planar boundary conditions on nematic configurations in absence and presence of an external electric field.


### 3.2.1. Absence of NPs

First, we consider cases without NPs in absence and presence of a statical ordering external fieldE→=Ee→z. Note that the lateral walls impose strong tangential anchoring. Consequent conflicting boundary conditions at corners of a well enforce topological defects. Typical nematic structures are shown in Figures 4 and 5. In Figures 4(a) and 5 we show the so calleddiagonal structure(DS) which is stable for R≫ξb and E = 0. In most parts of the cell the nematic structure is essentially uniaxial and the nematic director field is preferentially aligned along the diagonal of the square; see Figure 5. Substantial departures from uniaxial order are surrounding defects placed at the corners of the well. Their presence is clearly seen by plotting the β2(x,y) spatial profile (see Figure 4(a)). For visualization purpose the structure is shown in a regime where R is comparable to ξb. Defects locally enforce biaxial ordering and topological characteristics of the resulting defect-core structure are known. Main defect-core features are as follows. The center of each defect exhibits negative uniaxial order along e→z and is surrounded by a rim possessing maximal degree of biaxiality β2=1. Far from centers of isolated defects essentially positive uniaxial order is established in bulk. The characteristic linear size of defect-core structure is roughly determined by the biaxial length ξb, which reflects LC material properties at a given temperature.The degree of biaxialityβ2(x,y) within a square well, τ=4, strong anchoring condition. (a) DS, R/ξb=7, R/ξf=0; (b) DS, R/ξb=7, R/ξf=10; (c) DS, R/ξb=7, R/ξf=100; (d) WORS, R/ξb=3, R/ξf=0.
(a)
(b)
(c)
(d)Figure 5
Orientational distribution of maximal eigenvectors ofQ in DS. In the uniaxial limit these eigenvectors equal the nematic director field n→. The corresponding β2(x,y) dependence is plotted in Figure 4(a).   R/ξb=7, τ=4, strong anchoring condition.On decreasing the ratioη=R/ξb the so calledwell order reconstruction structure(WORS) is formed which is shown in Figure 4(d). Deep in the nematic phase this structure is stable for η<ηc=3.28±0.01. In it the local eigen frames of Q coincide with the e→x,e→y,e→z laboratory frame orientation in the whole (x,y) plane. The β2=1 rim exhibiting maximal biaxiality adopts an open cross-like profile. It separates regions exhibiting essentially positive uniaxial and essentially negative uniaxial ordering outside and inside the rim, respectively.The impact of an external field for LCs with positive field anisotropy is demonstrated in Figures4(b) and 4(c), where the field is aligned along the y-axis. The field tends to align e→1 (the eigenvalue corresponding to the largest eigenvalue of Q) along its direction. Consequently, on increasing the field strength the elastic distortions become increasingly accumulated at the top-bottom plates, because these plates enforce contradicting orientations with respect to E→. A typical pattern for a relatively weak field strength is shown in Figure 4(b). The top-bottom/left-right symmetry realized in absence of the field is broken. If the external field is strong enough it triggers the surface order reconstruction type of structural transition at the top-bottom plates, which is shown in Figure 4(c). We study also an alternative case where LC exhibits negative field anisotropy for the same orientation of the external field (i.e., e→=e→y). In this case the field tends to orient e→1 perpendicular to its direction. Consequently, the elastic distortions now accumulate at left-right plates. In both Δε>0 and Δε<0 cases the external field enforces similar biaxial textures which are rotated with respect to each other for an angle π/2 in the (x,y) plane.


### 3.2.2. Presence of NPs

Next, we study impact of a nanoparticle on ordering within the cell exhibiting effectively 2D behavior. Of our interest is to demonstrate existence of a rich variety of qualitatively different patterns just on varying the relative position of the nanoparticle within the cell and using different external ordering electric fieldE→=Ee→y strengths. For this purpose we choose LC exhibiting positive field anisotropy. We choose square-shaped NP of typical length a=R/10. Furthermore, we set that it enforces strong tangential anchoring to surrounding LC. We place it either (1) in the center of the cell, (2) close to the left bottom well edge, or (3) close to the left boundary, and (4) close to the bottom boundary of the well. These positions of NP are schematically sketched in Figure 6. We henceforth refer to these positions as the (1)central, (2)edge, (3)left boundary,and (4)bottom boundaryposition.Figure 6
Schematic presentation of different positions of square-shaped NP of lengtha=R/10 within the square well. (1)centralposition, (2)edgeposition, (3)left boundaryposition, and (4)bottom boundaryposition.In Figure7 we analyze structures for η=7 on varying E. In the first row we show cases for E=0. The edges of NP are sources of elastic distortions, giving rise to locally induced biaxiality. We see that for all four positions of NPs the global LC structure remains qualitatively the same, although the structures are slightly different. Note that in this case patterns emerging from left boundary and bottom boundary position are the same if the patterns are rotated for π/2 about the z-axis. Next we switch on the external field of strength R/ξf=50. Presence of the field breaks the symmetry of the problem. Its effect is increasingly pronounced for thebottomposition of NP. In this case the mutual impact of NP and external field enables order reconstruction (OR) type transition at the bottom plate. Note that the external field alone is too weak to stabilize this type of structure. For still stronger field, that is, R/ξf=100, we get four qualitatively different patterns. Among them thebottomandedgeposition support OR-type patterns.Nematic patterns for different positions of NP forη=7. (a) R/ξf=0. (b) R/ξf=50. (c) R/ξf=100.
(a)
(b)
(c)In Figure8 we set η=5 which is above the threshold value ηc=3.28 below which the WORS pattern is stable. However, we see that for the central position WORS-type pattern is nevertheless stabilized for E=0 due to the presence of NP. If the external field is switched on the WORS-type pattern vanishes and is replaced by “common” OR-type structures. In this case R/ξf=50 and R/ξf=100 yield qualitatively similar patterns.Nematic patterns for different positions of NP forη=5. (a) R/ξf=0. (b) R/ξf=50. (c) R/ξf=100.
(a)
(b)
(c)Finally, in Figure9 we set η=3<ηc. For such condition the WORS pattern is stable in absence of NP. However, presence of NP does not support this pattern for theedgeposition for E=0. For R/ξf=50 and R/ξf=100 we obtain again qualitatively similar patterns. None of them exhibits WORS-type structure.Nematic patterns for different positions of NP forη=3. (a) R/ξf=0. (b) R/ξf=50. (c) R/ξf=100.
(a)
(b)
(c)


## 3.2.1. Absence of NPs

First, we consider cases without NPs in absence and presence of a statical ordering external fieldE→=Ee→z. Note that the lateral walls impose strong tangential anchoring. Consequent conflicting boundary conditions at corners of a well enforce topological defects. Typical nematic structures are shown in Figures 4 and 5. In Figures 4(a) and 5 we show the so calleddiagonal structure(DS) which is stable for R≫ξb and E = 0. In most parts of the cell the nematic structure is essentially uniaxial and the nematic director field is preferentially aligned along the diagonal of the square; see Figure 5. Substantial departures from uniaxial order are surrounding defects placed at the corners of the well. Their presence is clearly seen by plotting the β2(x,y) spatial profile (see Figure 4(a)). For visualization purpose the structure is shown in a regime where R is comparable to ξb. Defects locally enforce biaxial ordering and topological characteristics of the resulting defect-core structure are known. Main defect-core features are as follows. The center of each defect exhibits negative uniaxial order along e→z and is surrounded by a rim possessing maximal degree of biaxiality β2=1. Far from centers of isolated defects essentially positive uniaxial order is established in bulk. The characteristic linear size of defect-core structure is roughly determined by the biaxial length ξb, which reflects LC material properties at a given temperature.The degree of biaxialityβ2(x,y) within a square well, τ=4, strong anchoring condition. (a) DS, R/ξb=7, R/ξf=0; (b) DS, R/ξb=7, R/ξf=10; (c) DS, R/ξb=7, R/ξf=100; (d) WORS, R/ξb=3, R/ξf=0.
(a)
(b)
(c)
(d)Figure 5
Orientational distribution of maximal eigenvectors ofQ in DS. In the uniaxial limit these eigenvectors equal the nematic director field n→. The corresponding β2(x,y) dependence is plotted in Figure 4(a).   R/ξb=7, τ=4, strong anchoring condition.On decreasing the ratioη=R/ξb the so calledwell order reconstruction structure(WORS) is formed which is shown in Figure 4(d). Deep in the nematic phase this structure is stable for η<ηc=3.28±0.01. In it the local eigen frames of Q coincide with the e→x,e→y,e→z laboratory frame orientation in the whole (x,y) plane. The β2=1 rim exhibiting maximal biaxiality adopts an open cross-like profile. It separates regions exhibiting essentially positive uniaxial and essentially negative uniaxial ordering outside and inside the rim, respectively.The impact of an external field for LCs with positive field anisotropy is demonstrated in Figures4(b) and 4(c), where the field is aligned along the y-axis. The field tends to align e→1 (the eigenvalue corresponding to the largest eigenvalue of Q) along its direction. Consequently, on increasing the field strength the elastic distortions become increasingly accumulated at the top-bottom plates, because these plates enforce contradicting orientations with respect to E→. A typical pattern for a relatively weak field strength is shown in Figure 4(b). The top-bottom/left-right symmetry realized in absence of the field is broken. If the external field is strong enough it triggers the surface order reconstruction type of structural transition at the top-bottom plates, which is shown in Figure 4(c). We study also an alternative case where LC exhibits negative field anisotropy for the same orientation of the external field (i.e., e→=e→y). In this case the field tends to orient e→1 perpendicular to its direction. Consequently, the elastic distortions now accumulate at left-right plates. In both Δε>0 and Δε<0 cases the external field enforces similar biaxial textures which are rotated with respect to each other for an angle π/2 in the (x,y) plane.


## 3.2.2. Presence of NPs

Next, we study impact of a nanoparticle on ordering within the cell exhibiting effectively 2D behavior. Of our interest is to demonstrate existence of a rich variety of qualitatively different patterns just on varying the relative position of the nanoparticle within the cell and using different external ordering electric fieldE→=Ee→y strengths. For this purpose we choose LC exhibiting positive field anisotropy. We choose square-shaped NP of typical length a=R/10. Furthermore, we set that it enforces strong tangential anchoring to surrounding LC. We place it either (1) in the center of the cell, (2) close to the left bottom well edge, or (3) close to the left boundary, and (4) close to the bottom boundary of the well. These positions of NP are schematically sketched in Figure 6. We henceforth refer to these positions as the (1)central, (2)edge, (3)left boundary,and (4)bottom boundaryposition.Figure 6
Schematic presentation of different positions of square-shaped NP of lengtha=R/10 within the square well. (1)centralposition, (2)edgeposition, (3)left boundaryposition, and (4)bottom boundaryposition.In Figure7 we analyze structures for η=7 on varying E. In the first row we show cases for E=0. The edges of NP are sources of elastic distortions, giving rise to locally induced biaxiality. We see that for all four positions of NPs the global LC structure remains qualitatively the same, although the structures are slightly different. Note that in this case patterns emerging from left boundary and bottom boundary position are the same if the patterns are rotated for π/2 about the z-axis. Next we switch on the external field of strength R/ξf=50. Presence of the field breaks the symmetry of the problem. Its effect is increasingly pronounced for thebottomposition of NP. In this case the mutual impact of NP and external field enables order reconstruction (OR) type transition at the bottom plate. Note that the external field alone is too weak to stabilize this type of structure. For still stronger field, that is, R/ξf=100, we get four qualitatively different patterns. Among them thebottomandedgeposition support OR-type patterns.Nematic patterns for different positions of NP forη=7. (a) R/ξf=0. (b) R/ξf=50. (c) R/ξf=100.
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(c)In Figure8 we set η=5 which is above the threshold value ηc=3.28 below which the WORS pattern is stable. However, we see that for the central position WORS-type pattern is nevertheless stabilized for E=0 due to the presence of NP. If the external field is switched on the WORS-type pattern vanishes and is replaced by “common” OR-type structures. In this case R/ξf=50 and R/ξf=100 yield qualitatively similar patterns.Nematic patterns for different positions of NP forη=5. (a) R/ξf=0. (b) R/ξf=50. (c) R/ξf=100.
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(c)Finally, in Figure9 we set η=3<ηc. For such condition the WORS pattern is stable in absence of NP. However, presence of NP does not support this pattern for theedgeposition for E=0. For R/ξf=50 and R/ξf=100 we obtain again qualitatively similar patterns. None of them exhibits WORS-type structure.Nematic patterns for different positions of NP forη=3. (a) R/ξf=0. (b) R/ξf=50. (c) R/ξf=100.
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## 4. Conclusions

We studied numerically combined impact of NPs and external aligning electric field on nematic structures within a square well where walls enforce planar uniaxial boundary conditions [29, 30]. Most simulations were performed at mesoscopic scale where we described nematic structures in terms of the 3D tensor order parameter Q [1]. We focus on regime where order reconstruction type solutions might exist. These patterns could be stabilized if a characteristic lateral well size R is comparable with the biaxial order parameter correlation length ξb [16].We first analyzed behavior in a 3D cubic well in absence of external field and NPs cell using a lattice-type semimicroscopic model [34]. Our simulations reveal that for given boundary conditions deep in the nematic phase LC ordering exhibits essentially 2D behavior [22]. Furthermore, results reveal that on decreasing the characteristic linear scale R of the system degree of biaxiality is apparently increasing for studied values of R. It is to be mentioned that in all cases the system size was large enough to exhibit the isotropic-nematic phase transition [34].Based on gained knowledge in 3D semimicroscopic simulations we studied in more detail impact of NPs on structural behavior in square wells in which the heighth≪R, where we neglected spatial variations along thez-coordinate [29, 30]. We analyzed structures deep in the nematic phase for ratios η=R/ξb=7, η=5, and η=3. We consider square-shaped NPs where their walls enforce strong tangential anchoring to surrounding LC. Note that in absence of external field and NPs the OR-type structure exists below the critical value ηc~3.28 [30].  Our simulations show that, in particular, close to the critical value of η one could get variety of different structures just by placing the nanoparticle at different points within the well.Note that essential features of the problem could be reproduced using 2D modelling within the(x,y) Cartesian plane using 2D tensor order parameter Q2D=q1e→x⊗e→x-q1e→y⊗e→y+q2e→x⊗e→y+e→y⊗e→x. This parametrization is commonly used to study defect textures in nematic shells [22]. The relationship between commonly used 2D and our 3D parametrization can be expressed as Q=Q2D+Qu, where Qu=q3e→x⊗e→x+q3e→y⊗e→y-2q3e→z⊗e→z is the uniaxial tensor. Therefore, both descriptions are equivalently deep in the nematic phase for a spatially constant value of q3. In our simulations q3 exhibits apparent spatial variations at sites exhibiting order reconstruction changes. Therefore, the 2D and 3D approaches are expected to be different at least from quantitative point of view. Furthermore, in 2D modelling the cubic invariant is absent (i.e., tr⁡Q2D3=0) [22]. Consequently, using 2D modelling theI-Ntransition would be of 2nd order character. Therefore, results obtained using the 2D or 3D approach could substantially quantitatively differ close to theI-Nphase transition.



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