On equilibrium configurations of nematic liquid crystals droplet with anisotropic elastic energy
 Dong An^{1},
 Wei Wang^{2} and
 Pingwen Zhang^{1}Email author
DOI: 10.1186/s4068701600945
© The Author(s) 2017
Received: 30 June 2016
Accepted: 30 December 2016
Published: 10 March 2017
Abstract
We investigate the effect of anisotropic elastic energy on defect patterns of liquid crystals confined in a threedimensional spherical domain within the framework of Landau–de Gennes model. Two typical strong anchoring boundary conditions, namely homeotropic and mirrorhomeotropic anchoring conditions, are considered. For the homeotropic anchoring, we find three different configurations: uniaxial hedgehog, ring and splitcore, in both cases with or without the anisotropic energy. For the mirrorhomeotropic anchoring, there are also three analogue solutions: the uniaxial hyperbolic hedgehog, ring and splitcore for the isotropic energy case. However, when the anisotropic energy is taken into account, the numerical results and rigorous analysis reveal that the uniaxial hyperbolic hedgehog is no longer a solution. Indeed, we find ring solution only for negative \(L_2\) (the elastic coefficient of the anisotropic energy), while both splitcore and ring solutions can be stable minimizers for positive \(L_2\). More precisely, the uniaxial hyperbolic hedgehog for \(L_2=0\) bifurcates to a splitcore solution when \(L_2\) increases and to a ring solution when \(L_2\) decreases. This example shows that the anisotropic energy may significantly affect the symmetry of point defects with degree \(1\) whenever it is introduced.
1 Background
Configurations and defect patterns of nematic liquid crystals, subject to some topological constraints, remain to be one of the most attractive topics among research on liquid crystals, as predicting defect patterns is important both in practical and in theoretical points of view [4]. There are lots of studies on configurations and structures of defects in liquid crystals by various different mathematical models, such as Oseen–Frank model [15], Ericksen’s model [5] and Landau–de Gennes (LdG) model [7, 10, 18]. The first two models postulate that one director preferred by molecules at each point, while the Landau–de Gennes theory allows the molecular orientation to have two preferred directions at each point. Since the Landau–de Gennes theory can capture biaxial behavior of liquid crystals near defect points, there are many studies on the defect patterns under this framework, see [11, 16–21] and the references therein.
From mathematical viewpoint, the presence of anisotropic energy will bring analytical difficulty due to its asymmetric structure; thus, some powerful tools, such as maximum principle, cannot be used to study the minimizers or equilibrium solutions. A wellknown example is that the minimizers of isotropic Oseen–Frank energy, which are harmonic maps, have only finite singular points in threedimensional domain, while it is quite difficult to prove a similar result for minimizers of anisotropic Oseen–Frank energy [9]. The same difficulty occurs in studying related problems within the Landau–de Gennes model. Therefore, in many existing studies, the elastic energy is assumed to be isotropic, which is referred as oneconstant approximation. However, there are few concrete liquid crystal materials that have isotropic elastic coefficients. Hence, it becomes important to understand whether the anisotropic energy could affect the static or dynamic behaviors of liquid crystals. A typical example arises from the isotropic–nematic interface problem, in which it is found that whether the elastic energy is isotropic or anisotropic corresponds to different boundary conditions on the interface [6].
In this paper, we study how anisotropic energy affects the configuration with certain given boundary conditions within the framework of the Landau–de Gennes model, by combining numerical simulations and theoretical analysis. In particular, we focus on static equilibrium configurations of liquid crystals confined in a threedimensional ball with strong anchoring conditions at the boundary.
The paper is organized as follows: In Sect. 2, we introduce the scaling and boundary conditions. In Sect. 3, we state the numerical methods we implement, followed by our main numerical results in Sect. 4. Section 5 presents a preliminary theoretical analysis on the behavior of uniaxial solutions under the mirrorhomeotropic boundary condition. Finally, we summarize and discuss our results in Sect. 6, along with some open problems.
2 Models and boundary conditions
2.1 Models and scaling

effective temperature: \(t=\frac{27AC}{B^2}\),

characteristic length: \(\xi _0 = \frac{\sqrt{27CL_1}}{B}\),

“normalized” elastic constant: \(\varepsilon =\frac{\xi _0}{R}= \frac{\sqrt{27CL_1}}{BR}\),

anisotropic rate: \(L_{21} = \frac{L_2}{L_1}\),
2.2 Boundary conditions
3 Method for numerical simulation
To obtain numerical results both efficiently and accurately enough, we start with small N, L, M and gradually increase some or all of them until the numerical results converge, i.e., no significant change in the value of free energy.
4 Numerical results
4.1 Homeotropic anchoring condition

Radial hedgehog solution: A uniaxial state with Q satisfying the profile (7). The center of the ball is the isolated isotropic point. Hedgehog solution is stable only for large t and \(\varepsilon \). For small t and \(\varepsilon \), the isotropic point broadens into a biaxial ring.

Ring disclination: A biaxial state containing a ring which is a combination of point defects with degree \(+1/2\). Around the ring is shelled by a strong biaxial region. This solution is rotationally symmetric. For small t and \(\varepsilon \), ring solution is more energetically favored than radial hedgehog solution.

Splitcore solution: A biaxial state containing a short +1 disclination line connecting two isotropic points. This solution is also rotationally symmetric. It seems to be metastable for all considered parameters, i.e., the ring solution or radial hedgehog solution always has lower free energy under the same parameters.

There still exists a radial hedgehog solution which satisfies the profile (7), but the scalar functions h for \(L_{21} \ne 0\) are different from those in \(L_{21} = 0\) (see Fig. 2). The radial hedgehog solution has been studied in [8], in which the stability/instability for different \(L_{21}\) are discussed when t and \(\varepsilon \) are small.

One can also obtain a ring solution and a splitcore solution for \(L_{21} \ne 0\) similar to the case of \(L_{21}=0\). Our numeral results indicate that they are still rotationally symmetric. To verify it, we define the error functionIf Q is axially symmetric, \(\text {err}\) will be 0 everywhere. Due to the existence of numerical error, a small maximum of error function will support the axial symmetry. Numerical verifications on the axial symmetry are listed in Table 1.$$\begin{aligned} \text {err}(r,\theta ,\varphi ) = Q(r,\theta ,\varphi )  P_{\varphi }Q(r,\theta ,0)P_{\varphi }^\mathrm{T}. \end{aligned}$$(23)
Numerical verifications of rotational symmetries of the ring and the splitcore solutions with nonzero \(L_{21}\) for homeotropic anchoring
\({t} ={0.3},{{\varepsilon }} = {0.1}, {L}_{{21}} = {3}\)  \({t} ={3}, {{\varepsilon }} = {0.2}, {L}_{21} = {0.2}\)  

N  L  M  \(\Vert {\varvec{err}}\Vert _{\mathbf{L}}^{\varvec{\infty }}\)  N  L  M  \(\Vert {\varvec{err}}\Vert _{\mathbf {L}}^{\varvec{\infty }}\) 
Ring  
32  16  4  6.06e−7  32  16  4  1.70e−6 
32  16  8  4.85e−7  32  16  8  3.70e−6 
32  32  16  4.72e−7  32  32  16  2.22e−6 
32  32  32  4.71e−7  32  32  32  2.22e−6 
\({t} = { 7}, {{\varepsilon }} = {0.2}, {L}_{21} = {0.3}\)  \({t} = {7}, {{\varepsilon }} = {0.2}, {L}_{21} = {0.1}\)  

N  L  M  \(\Vert {\varvec{err}}\Vert _{\mathbf{L}}^{\varvec{\infty }}\)  N  L  M  \(\Vert {\varvec{err}}\Vert _{\mathbf{L}}^{\varvec{\infty }}\) 
Splitcore  
32  16  4  3.03e−6  32  16  4  8.37e−7 
32  16  8  1.64e−6  32  16  8  1.30e−6 
32  32  16  9.71e−6  32  32  16  1.07e−6 
32  32  32  2.13e−6  32  32  32  1.06e−6 
4.2 Mirrorhomeotropic anchoring condition
4.2.1 The case of \(L_{21} = 0\)

Hyperbolic hedgehog solution: This solution is a “mirror” version of the radial hedgehog. Thus, it is uniaxial everywhere (except an isolated isotropic point in the center), and the order parameter is a radial symmetric function. Precisely, it has the following formHowever, the orientation directors in this solution are no longer aligned as “hedgehog,” so we call it hyperbolic hedgehog. This solution is illustrated in Fig. 4a.$$\begin{aligned} Q(\mathbf {x}) = s(r)\left( \mathbf {m}\otimes \mathbf {m} \frac{1}{3}I\right) ,\quad \mathbf {m}=(x,y,z)/r. \end{aligned}$$(25)

The ring solution and the splitcore solution of mirrorhomeotropic anchoring case are quite similar to the corresponding solutions of homeotropic anchoring case, with the same distributions on \(\beta \) but different eigenvectors. These solutions are illustrated in Fig. 4.
4.2.2 The case of \(L_{21} > 0\)
Numerical verification of ring’s and splitcore’s axial symmetry with mirrorhomeotropic anchoring condition
\(\mathbf{t}= \mathbf{2}, {\varvec{\varepsilon }}= \mathbf{0.05}, L_{21}=\mathbf{2}\)  \(\mathbf{t}=\mathbf{0.5}, {\varvec{\varepsilon }}= \mathbf{0.1}, L_{21}=\mathbf{2}\)  

N  L  M  \(\Vert {\varvec{err}}\Vert _{L}^{\varvec{\infty }}\)  N  L  M  \(\Vert {\varvec{err}}\Vert _{\mathbf{L}}^{\varvec{\infty }}\) 
Splitcore  
32  16  4  1.79e−6  32  16  4  6.62e−7 
32  16  8  1.92e−6  32  16  8  6.47e−7 
32  32  16  2.04e−6  32  32  16  4.38e−7 
32  32  32  2.10e−6  32  32  32  4.38e−7 
\(\mathbf{t} = \mathbf{2}, {\varvec{\varepsilon }} = \mathbf{0.1}, L_{21}=\mathbf{0.2}\)  \(\mathbf{t} = \mathbf{8}, {\varvec{\varepsilon }} = \mathbf{0.1}, L_{21}=\mathbf{0.5}\)  

N  L  M  \(\Vert {\varvec{err}}\Vert _\mathbf{L}^{\varvec{\infty }}\)  N  L  M  \(\Vert {\varvec{err}}\Vert _\mathbf{L}^{\varvec{\infty }}\) 
Ring  
32  16  4  1.67e−6  32  16  4  2.61e−5 
32  16  8  1.67e−6  32  16  8  2.61e−5 
32  32  16  1.47e−6  32  32  16  2.60e−5 
32  32  32  1.52e−6  32  32  32  2.57e−5 
As for stability, phase diagrams are shown in Fig. 5. Here for large t and \(\varepsilon \), splitcore solution is stable, and ring solution is energetically favored if taking small t and \(\varepsilon \). We remark that when \(L_{21}>0\) we cannot find the hyperbolic hedgehog solution. Actually, when \(L_{21}\ne 0\), it can be proved that such uniaxial solution cannot be an equilibrium solution of the Landau–de Gennes energy functional. We will carry out a detailed analysis in the next section.
4.2.3 The case of \(L_{21} < 0\)
For \(L_{21}<0\) case, our numerical results suggest that there exists only one stable equilibrium solution, ring solution, whatever coefficients t and \(\varepsilon \) vary. Not only hyperbolic hedgehog solution but also splitcore solution disappears.
This interesting phenomenon leads to a natural conjecture that there exists at least one splitcore solution, which is stable when \(L_{21} > 0\); however, all splitcore solutions are unstable when \(L_{21} < 0\) and are at most metastable for \(L_{21} = 0\).
4.2.4 More detailed transition behavior of hyperbolic hedgehog solution near \(L_{21}=0\)
According to our aforementioned numerical results, we find out that there is no hyperbolic hedgehog solution when \(L_{21}\ne 0\). Now, we study how the hyperbolic hedgehog solution evolves when \(L_{21}\) varies near zero. For this, we focus on the values of all components of Qtensor at the center of ball, namely Q(0). From the definitions of three basic solutions, we know that: \(Q(0)=0\) for hyperbolic hedgehog solution, Q(0) is positive uniaxial for ring solution and Q(0) is negative uniaxial for splitcore solution.
Numerical value of Q(0) with parameters \(t = 0.2, \varepsilon = 0.2\) and \(t = 2, \varepsilon = 0.05\)
\(\mathbf {t} = \mathbf {0.2}, {\varvec{\varepsilon }} = \mathbf {0.2}\) (\(\mathbf {N}=\mathbf {64},\mathbf {L}=\mathbf {32},\mathbf {M}=\mathbf {32}\))  

\(\mathbf {L}_\mathbf {21}\)  \(\mathbf {Q}_\mathbf {11}\)  \(\mathbf {Q}_\mathbf {12}\)  \(\mathbf {Q}_\mathbf {13}\)  \(\mathbf {Q}_\mathbf {22}\)  \(\mathbf {Q}_\mathbf {23}\)  \(\parallel {\varvec{\nabla }}{\mathbf {F}}\parallel \) 
−0.10  −3.88e−2  5.33e−12  −6.9e−14  −3.88e−2  −3e−15  9.85e−5 
−0.05  −1.68e−2  3.19e−13  2e−15  −1.68e−2  −2e−15  9.34e−5 
−0.01  −3.04e−3  1.11e−13  −6e−15  −3.04e−3  1e−15  1.00e−5 
0  −8.57e−6  −3.27e−12  −6.0e−14  −8.57e−6  −3.1e−14  1.26e−4 
0.01  2.93e−3  1.57e−13  −1.1e−14  2.93e−3  0  3.71e−4 
0.05  1.37e−2  2.78e−13  1.76e−13  1.37e−2  −5.51e−13  1.04e−4 
0.10  2.52e−2  2.31e−12  2.1e−14  2.52e−2  −5e−15  1.96e−4 
\(\mathbf {t}= \mathbf {2}, {\varvec{\varepsilon }}= \mathbf {0.05}\) (\(\mathbf {N}=\mathbf {64},\mathbf {L}=\mathbf {32},\mathbf {M}=\mathbf {32}\))  

\(\mathbf {L}_\mathbf {21}\)  \(\mathbf {Q}_\mathbf {11}\)  \(\mathbf {Q}_\mathbf {12}\)  \(\mathbf {Q}_\mathbf {13}\)  \(\mathbf {Q}_\mathbf {22}\)  \(\mathbf {Q}_\mathbf {23}\)  \(\parallel {{\varvec{\nabla }}}{\mathbf {F}}\parallel \) 
−0.10  −4.75e−2  −7.06e−11  2.07e−13  −4.75e−2  −1.42e−12  1.53e−4 
−0.05  −2.10e−2  −1.31e−11  −2.86e−13  −2.10e−2  −1.16e−13  2.06e−4 
−0.01  −3.93e−3  −2.59e−12  1.04e−13  −3.93e−3  7.3e−14  1.92e−4 
0  6.99e−6  −1.26e−11  4.93e−13  −6.99e−6  −5.53e−13  2.93e−4 
0.01  3.72e−3  −4.16e−12  2.06e−13  3.72e−3  9.0e−14  1.37e−4 
0.05  1.81e−2  6.62e−13  4.56e−13  1.81e−2  2.65e−13  3.85e−4 
0.10  3.41e−2  5.23e−13  6.8e−14  3.41e−2  7.5e−14  5.63e−4 
We also investigate the qualitative relationship between the size of biaxial region and \(L_{21}\). We call the distance between two isotropic points “split length.” The relationship between split length and \(L_{21}\) is shown in Figs. 6 and 7. It can be observed that split length is proportional to \(L_{21}\) near \(L_{21}=0\). Thus, the hyperbolic hedgehog solution for \(L_{21}\) can be viewed as a special splitcore solution with zero split length.
5 Nonexistence of uniaxial hyperbolic hedgehog solutions for mirrorhomeotropic anchoring condition with \(L_{21}\ne 0\)
6 Discussion and conclusion
In this paper, we investigate the effect of anisotropic energy to the local structure of point defects in a threedimensional ball. Both homeotropic and mirrorhomeotropic anchoring conditions, which correspond to degree +1 and 1 point defects, respectively, are considered.
Our numerical results reveal that the anisotropic energy will affect the phase behavior significantly for the mirrorhomeotropic anchoring condition even if the absolute value of anisotropic elastic coefficient \(L_{21}\) is small. When \(L_{21}\ne 0\), there is no uniaxial solution with radial symmetric order parameter which may be stable minimizer for certain parameters. More precisely, the uniaxial solution will deform into a splitcore solution for \(L_{21}>0\) and into a ring solution for \(L_{21}<0\) in hightemperature region. This is very different from the homeotropic anchoring case, in which the three basic configurations still exist and their stabilities are not essentially affected by \(L_{21}\). In particular, the radially symmetric solution is preserved when \(L_{21}\ne 0\) in homeotropic anchoring case.
We further perform analysis on the hyperbolic hedgehog solution for the mirrorhomeotropic anchoring. We prove that the hyperbolic hedgehog solution cannot be a solution to the Euler–Lagrange equation when \(L_{21}\ne 0\). Based on this result, it is quite reasonable to make the following conjecture.
Conjecture 1
Uniaxial solution with degree \(1\) cannot be a stable minimizer once anisotropic elastic energy is considered.
We have also found out that there exist stable splitcore solutions for high temperature when \(L_{21}>0\). This is different from the isotropic energy case, in which splitcore solutions are shown to be only metastable. More interestingly, the stable splitcore solution will reduce to hyperbolic hedgehog when \(L_{21}\) goes to zero and will deform to the ring solution if \(L_{21}\) decreases to be negative. This inspires us to make the following conjecture.
Conjecture 2
For the mirrorhomeotropic anchoring condition, there exists stable splitcore solution for high temperature when \(L_{21}\) is positive, but splitcore solution is unstable everywhere once \(L_{21}\) is negative.
As a final point, we remark that all the solutions obtained in this paper are axisymmetric although such a symmetric constrain is not imposed in our simulation. Whether there is any other solution without axisymmetry is worth being investigated. We leave it to future works.
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments. WW was supported by NSF of China under Grant 11501502 and “the Fundamental Research Funds for the Central Universities” 2016QNA3004. PZ was supported by NSF of China under Grants 11421101 and 11421110001.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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