Zhou Xuan, Chen Si-Bo, Zhang Zhi-Dong. Fine structures of defect cores induced by elastic anisotropy and biaxiality in hybrid alignment nematics. Chinese Physics B, 2018, 27(5): 056103
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Fine structures of defect cores induced by elastic anisotropy and biaxiality in hybrid alignment nematics
Zhou Xuan1, Chen Si-Bo2, 3, Zhang Zhi-Dong1, †
Department of physics, Hebei University of Technology, Tianjin 300401, China
School of Electronic and Information Engineering, Hebei University of Technology, Tianjin 300401, China
Tianjin Key Laboratory of Electronic Materials and Devices, Tianjin 300401, China
Based on Landau–de Gennes theory and two-dimensional finite-difference iterative method, the spontaneous distortion in hybrid alignment nematic cells with M = ±1/2 disclination lines is investigated by establishing two models. The fine structures of defect cores are described in the order space S2/Z2. The joint action of elastic anisotropy (L2/L1) and biaxiality of defects induces the spontaneous twist distortion, accompanied by the movement of the defect center to the upper or lower plate. For each model, four mixed defect structures appear with the same energy, which are defined as energetically degenerated quadruple states.
Topological defects, found in systems with broken continuous symmetry, are ubiquitous in nature, from microscopic condensed matter systems governed by quantum mechanics to a universe in which gravity plays a decisive role.[1–3] Defects in liquid crystals (LCs) have been the subject of much interest, still offering unsolved problems. A topological charge M of integer or half-integer can be assigned to each defect. One value of topological charge M = ±1/2 characterizes the entire class of stable defects in nematics.[1] All other half-integer-valued lines, whether positive or negative, can be continuously deformed into a line with charge ± 1/2 and integer-valued lines can “escape in the third dimension”.[4]
Topological defects can also be classified by homotopy groups.[5–7] In a three-dimensional (3D) nematic liquid crystal (NLC) system, due to the equivalence between the head and the tail of the director (or that n and −n), the order space is S2/Z2 (where S2 is a homotopy group of the sphere, and Z2 is the Abelian group with two elements). Each defect can be represented by a contour in S2/Z2. The director n rotates by an angle of along the contour encircling the defect. Contours of defects with integer charge are actually closed (e.g., circles) and can shrink into a point, indicating that the defects are topologically unstable.[8] While the contours of defects with half-integer charge terminate at two diametrically opposite points; under any continuous deformations, the ends of these contours remain fixed at the diametrically opposite points, indicating that the defects are topologically stable.[5]
A defect line parallel to the rotation axis of n is referred to as wedge disclination; while a defect line perpendicular to the rotation axis of n is referred to as twist disclination.[5] Note that the intermediate case where the defect line neither parallel nor perpendicular to the rotation axis of n is possible, which we refer to as mixed disclination.[9–12] Fukuda et al.[6] have realized the transformation of a −1/2 wedge disclination line to a +1/2 one in a numerical simulation, and mixed disclinations appeared during the transformation.
The region where the presence of a defect causes the apparent deviation from bulk ordering is referred to as defect core, which is difficult to explore experimentally, as their spatial extension is very small, usually a few molecular lengths, which makes it practically impossible to use standard optical or electron microscopy in studying the properties of the core.[13,14] Because of very limited experimental data, modern theories of the topological defects treat the core problem under a number of strong simplified assumptions,[15–17] such as the one-elastic-constant approximation (, the director n is confined in a two-dimensional (2D) plane. Zhou et al.[18] explored the cores of disclinations in the chromonic nematics that extend over macroscopic length scales accessible for optical characterization, but they did not give the precise director profile around the core. So far the fine structure of the core region of defects is also one of the most difficult problems, especially on condition that the director is not confined in a 2D plane, which has gained increasing research attention.
In this study, we investigate the structural transition in a hybrid alignment nematic (HAN) layer with M = ±1/2 topological defect theoretically. Our study is based on the Landau–de Gennes theory describing the orientational order of an LC in terms of a second-rank tensor Q, which encompasses both uniaxial and biaxial states.[19] Mixed defect structures induced by elastic anisotropy are found and the fine structures of defect core are given. The rest of this paper is organized as follows. In Section 2, we introduce the phenomenological model to be employed here in this paper and describe the geometry of the problem and our parameterization. The results are presented in Section 3, and the conclusions are drawn from the present studies in Section 4.
2. Theoretical basis
2.1. Free energy
Our theoretical argument is based on Landau–de Gennes theory,[19] in which the orientational order of LC is described by a second-rank symmetric and traceless tensor[20]Here, is the orthogonal unit vector representing the eigenvector of Q, and is its eigenvalue. The range of eigenvalues must be to explain Q as a traceless second-moment tensor of the molecular distribution function. In the isotropic phase Q vanishes, whereas this parameter has two degenerate eigenvalues in the uniaxial ordering and can be represented bywhere n is the nematic director pointing along the local uniaxial ordering direction, and S is the uniaxial scalar parameter expressing the magnitude of fluctuations about the nematic director. In Eq. (2), S can be either positive or negative. The ensemble of molecules represented by Q tends to align along n when S is positive and tends to lie in the plane orthogonal to n when S is negative. Finally, the LC is in a biaxial state when all the eigenvalues of Q are different from each other. The degree of biaxiality is expressed by the parameter defined as[21]This equation is of a convenient parameter for illustrating spatial inhomogeneities of Q, and the value of is in a range of [0, 1]. All uniaxial states with two degenerate eigenvalues correspond to , whereas states with maximal biaxiality correspond to . As tr(Q3) = 3detQ, states with are those with detQ = 0, which implies that at least one eigenvalue of Q vanishes. As is well known, the regions of defect cores are biaxially textured.[15–17]
The Landau–de Gennes free energy density of LC is given by , in which
is the bulk energy that describes a homogeneous phase. In , B and C are positive constants, and A is assumed to vary with temperature T in the form ), where a is a positive constant and is the nematic supercooling temperature. Equation (4) gives the bulk equilibrium value of the uniaxial scalar order parameter in Eq. (2),which also depends on temperature.
The free energy, fe, which penalizes gradients in the tensor order parameter field, is given bywhere L1 and L2 are elastic constants. According to the Frank theory, we obtain the relationship between Li and Kii asHere, we assume just for simplicity.
2.2. Models and geometry of the problem
We choose a HAN cell in strong anchoring boundary conditions, and the nematic director is parallel to one wall and perpendicular to the other (see Fig. 1). The plates are placed at of a Cartesian coordinate system. The lengths, dx and dy, of the cell along the x- and the y-axes are much larger than d (.
Fig. 1. Geometries of (a) Model I and (b) Model II.
We study HAN structures with M = −1/2 wedge disclination (Model I), and M = +1/2 wedge disclination (Model II). The disclination line is parallel to the y-axis of the simulation cells, and we seek solutions independent of y.
To describe the configuration, the nematic director is given bywhere θ is the polar angle between the z-axis and the director n, and φ is the twist angle between the x-axis and the x–y projection of the director. In the coordinate system, Q can be written as
At , we enforce the uniaxial strong anchoring represented by , and , , that is
On the lateral walls at , we denote the free boundary conditions with initial uniaxial ordering. For Model I, the total rotation at is −π/2, while at it is π/2. We obtain Model II by exchanging the lateral walls (see Fig. 1).
2.3. Scaling and dimensionless evolution equations
We introduce the following dimensionless quantities:where is the superheating order parameter at the nematic superheating temperature T**, andis the characteristic length for order parameter changes. Equations (4) and (5) can be reduced respectively towhere the reduced parameter is defined as the temperature scale. The isotropic-nematic transition occurs at . Thus, the reduced uniaxial ordering has the formwhere is the reduced uniaxial scalar parameter in equilibrium.
We compute the evolution of LC with a dynamic theory for tensor order-parameter field Q(r, t). The local values of the scalar order parameter S and the director n can be calculated from Q by using the highest eigenvalue and the associated eigenvector, respectively. According to [22], the evolution equation describing the dynamics of Q can be written aswhere , with D* being the rotational diffusion for the nematic; is assumed to be symmetrical.
Numerical calculations are performed using the reduced variables. When the functional derivatives in Eq. (10) are evaluated, the following evolution equations for can be obtained:with . The value of L2/L1 represents the magnitude of the elastic anisotropy. A larger value of L2/L1 corresponds to a smaller twist elastic constant K22.
We adopt the two-dimensional finite-difference method developed in our previous studies to obtain the numerical simulation results.[23–25] Here, the system is relaxed from an initial boundary condition given in Subsection 2.2, and the initial conditions of the left half and right half in the bulk are consistent with the boundary conditions at the left wall and right wall given above. The initial perturbation of twist angle is given as follows: from to , the twist angle changes as or . That means that the initial twist angle is zero on the lower and upper plates, and in the middle of the nematic cell. For each model, there are four forms of the initial perturbations distinguished by the signs of the initial twist angle on the left and right half. For example, if the initial twist angles are both positive on the left and right half in the bulk, we express it as +(left) +(right) (Form A); similarly, we can write other three forms as −(left)−(right) (Form B), +(left)−(right) (Form C), and −(left)+(right) (Form D).
We consider only the equilibrium configurations that correspond to the global minimum F. In our numerical calculations, a discretization in time steps of 10−8 s is sufficient to guarantee the stability of the numerical procedure. In addition, our equilibration runs take 2×106 steps in total which is adequate for the system to reach the equilibrium.
3. Results
According to parameters given in [26], , , , can be easily obtained. In our simulations, the scaled temperature is set to be , which corresponds to . The rotational diffusion D* is set to be 0.35, which is the value used in [26]. The cell gap is set to be . We then obtain the exact value and the cell gap . The elastic anisotropy is set to be L2/, corresponding to K22/. We focus on the dependence of structure transition on elastic anisotropy.
In order to present the results in the order space S2/Z2, we introduce Θ and Φ to redefine the director n at equilibrium state, where is the angle between the director n and the x–z plane, to show the deviation of the director from the x–z plane induced by elastic anisotropy L2/L1, and is the angle between the x-axis and the x–z projection of the director (see Fig. 2). Here, all directors are redefined into an upper hemisphere representing the order space S2/Z2, according to the y-component of the director, ny. If , the director n maintains to be in the upper hemisphere; while if , n changes into −n to ensure it in the upper hemisphere.
Fig. 2. (color online) Order space S2/Z2 for the nematic phase.
3.1. Structure transition of Model I
3.1.1. Structures of Forms A and B
Figure 3 shows the profiles of Θ and Φ of the director at equilibrium state for the case of Form A. It is seen clearly that the defect center moves to the upper plate. In addition, the elastic anisotropy induces the spontaneous distortion, and the maximum distortion exists near the defect center (see Fig. 3(a)). For further explanation of the spontaneous distortion, we give the y-component of the director, ny, in Fig. 4, and the figure also shows that the maximum distortion exists around the defect center. Along the contour that we plotted, the new structure can be explained in the order space S2/Z2 by Fig. 7(a). The point o on the contour is the starting point, and is also the terminal point o′ of the contour. This point corresponds to two diametrically opposite points on the bottom circle of the hemisphere in Fig. 7(a). The value of Θ changes from 0 to a maximum value between 0 and π/2[, and then turns back to 0; meanwhile, Φ changes from to 0 and then to . Here we should notice that the value of depends on the contour that we plotted. It is seen clearly in Fig. 7(a) that the rotation axis of n is neither parallel nor perpendicular to the defect line (y-axis), thus the new defect structure is a mixed disclination.
Fig. 7. (color online) Descriptions of mixed defects in the order space S2/Z2 for (a) Form A and (b) Form B.
Figure 5 and 6 shows the profiles of Θ, Φ, and ny component of the director for the case of Form B. Like Form A, the defect center also moves to the upper plate and it generates spontaneous distortion (see Figs. 3(a) and 5(a)). Moreover, the ny component is also the same as that of Form A (see Figs. 4 and 6). The difference is that along the contour that we ploted, Φ changes from to −π and then to π/2. The defect structure can be shown in the order space S2/Z2 by Fig. 7(b), also representing a mixed disclination. Comparing Form A with Form B, the difference in initial perturbation leads to the difference in ny. For our presentation in the order space S2/Z2, all of ny components are redefined to be positive, thus the changes of the quadrants about the director n happen.
Fig. 6. (color online) Prifile of ny component of the director at equilibrium state for Form B of Model I.
3.1.2. Structures of Forms C and D
In the cases of Forms C and D, the equilibrium states show that the center of the defect moves to the lower plate. Figure 8 and 9 show the profiles of Θ, Φ, and ny component of the director at equilibrium state for the case of Form C. It is also shown that the elastic anisotropy induces the spontaneous distortion, and the maximum distortion exists near the defect center (see Figs. 8(a) and 9). Along the contour that we plotted, Θ changes from 0 to a maximum value between 0 and π/2, and then turns back to 0; meanwhile, Φ changes from π to π/2 and then to 0. The structure can be explained in the order space S2/Z2 by Fig. 10(a). Form D generates a similar defect structure to Form C with Φ changing from 0 to and then to −π, which can be shown in the order space S2/Z2 by Fig. 10(b). Figure 1010 shows other two mixed disclinations. We need to note here that the starting point o and terminal point o′ for Forms C and D are different from those of Forms A and B. This is because o and o′ are actually the same point, represented by two diametrically opposite points on the bottom circle of the hemisphere, thus we need to choose the points with Θ = 0 as the starting and terminal point.
Fig. 10. (color online) Descriptions of mixed defects in the order space S2/Z2 for (a) Form C and (b) Form D.
We make an energy comparison among the four mixed defect structures, where the energy is calculated from our numerical results. The results show that the four new defect structures are quadruple energetically degenerated states.
3.2. Structure transition of Model II
For Model II, four forms of initial twist angles give four similar mixed defect structures to those for Model I, respectively. The defect centers of Forms A and B move upward, while those of Forms C and D move downward. The only difference is the rotation direction of the director n. Figure 11 describes the four structures in the order space S2/Z2. Compared with Figs. 7 and 10, the opposite rotation directions of the director for Model I and Model II are clearly seen, resulting from the opposite rotation directions of initial n.
Fig. 11. (color online) Descriptions of mixed defects for (a) Form A, (b) Form B, (c) Form C, and (d) Form D in the order space S2/Z2 for Model II.
The energy calculation shows that the four mixed structures are also quadruple energetically degenerated states. Further, the energies of Model I and Model II are identical, indicating the consistency of defects, and they will appear with the same probability.
For comparison, a normal HAN cell without defect is also simulated with the same parameters. For this cell, the elastic anisotropy will not induce the spontaneous distortion, and the structure remains unchanged for different values of L2/L1. That is, and in the normal HAN system. Considering the initial defect structures in Models I and II, we can easily obtain that the joint action of elastic anisotropy and defect biaxiality induces the twist distortion as well as the mixed defect structures. The inexistence of biaxiality in normal HAN leads to no twist distortion.
4. Conclusions
Within Landau-de Gennes theory, we have investigated the spontaneous distortion in HAN cells with M = ±1/2 defects, using the 2D finite-difference iterative method. Two models are studied, with an M = −1/2 wedge defect and M = +1/2 wedge defect, respectively. A normal HAN structure with no defect is utilized for conducting comparative studies.
The joint action of elastic anisotropy and defect biaxiality induces spontaneous distortion phenomenon, accompanied by the movement of the defect center to upper or lower plate. Four mixed defect structures appear with the same energy, which we called quadruple energetically degenerated states. The new defect structures do not confine the director n in a 2D plane. It can be deduced that the movement of the defect center will increase with increasing . A pure HAN system has no twist distortion under the individual effect of elastic anisotropy.
According to the parameters given, we can easily obtain the characteristic length , which gives the cell gap . Maybe the transition structure will change under nano-scale cells. It is our future task to acquire the detailed results. We expect that the main findings of the present work would appear in other theoretical models, and will provide theoretical guidance for the experimental investigation of the core structures.
