A Numerical Investigation on Configurational Distortions in Nematic Liquid Crystals

Transcription

1 DOI /s A Numerical Investigation on Configurational Distortions in Nematic Liquid Crystals Anna Pandolfi Gaetano Napoli Received: 16 September 2010 / Accepted: 28 April 2011 Springer Science+Business Media, LLC 2011 Abstract When subjected to magnetic or electric fields, nematic liquid crystals confined between two parallel glass plates and initially uniformly oriented may undergo homogeneous one-dimensional spatial distortions (Fréedericksz and Zolina, Trans. Faraday Soc. 29:919, 1933) or periodic distortions (Lonberg and Meyer, Phys. Rev. Lett. 55(7): , 1985; and Srajer et al., Phys. Rev. Lett. 67(9): , 1991). According to the experimental observations, periodic phases are stable configurations at intermediate intensity of the acting field, while homogeneous phases are stable at higher strengths. We present a fully nonlinear finite element approach able to describe homogeneous and periodic configurational phases in a cell of confined nematic liquid crystal with strong planar anchoring boundary conditions. Stationary configurations are obtained by setting to zero the first variation of the discretized total energy of the system. Unstable configurations are identified by evaluating the behavior of the solution under small numerical perturbations. Numerical calculations are able to describe the evolution of the configurational distortions as a function of the applied field and are able to capture the critical points between homogeneous and periodic phases. The proposed Communicated by P. Newton. A. Pandolfi ( ) Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, Italy pandolfi@stru.polimi.it G. Napoli Dipartimento di Ingegneria dell Innovazione, Università del Salento, Via per Monteroni, Edificio Corpo O, Lecce, Italy gaetano.napoli@unisalento.it

2 approach has been proved to be an excellent tool to predict the existence of unstable or metastable distortions, characterized by higher energy levels. Keywords Nematic liquid crystal Stable configuration Metastable configuration Phase transition Finite elements Mathematics Subject Classification (2000) 76A15 82B21 65N30 1 Introduction Nematic liquid crystals (NLC) are aggregates of rodlike molecules whose mass center does not exhibit any positional order (de Gennes and Prost 1993). NLC can flow as liquid, but maintain some optical features typical of solid crystals. Their uniaxial optical properties are strictly related to the average orientation of the molecules, which spontaneously tend to lie along a common direction. The particular shape of NLC renders the molecules very sensitive to the presence of physical boundaries and to the action of magnetic or electric fields. Depending on the magnetic properties of the nematic, molecules reorient along, or normally to, the direction of the field. In particular, the sudden application of a field switches on off the transmission of polarized light; thus NLC may find wide applications in displays for computers, flat-panel televisions, cell phones, calculators and watches. In the typical configuration of the most wide-spread applications, thin layers ( 100 µm) of a nematic are confined between two flat glass substrates. The glass surfaces are treated suitably so as to assign the orientation of the molecules on the boundaries. The condition where the average molecular orientation on the boundary is orthogonal to the delimiting surfaces is called homeotropic anchoring. The anchoring is called planar when the molecular axes are constrained to lie parallel to the boundary. Alternative anchoring conditions are possible with the expenditure of an additional energy. For example, the situations where the molecules are close to the boundary naturally aligned along an energetically favorable axis direction and reorient when a sufficient energy is available are called weak anchoring conditions (Rapini and Papoular 1969). Consider a confined nematic, initially disposed to planar homogeneous alignment. As first observed by Fréedericksz and Zolina (1933), at a critical value H c1 of the applied magnetic field, liquid crystal molecules reorient along the field direction, assuming an alternative stable distortion. This phenomenon, well described, e.g., in Virga (1994), is called a homogeneous Fréedericksz transition (HD), and it is observed under both strong and weak anchoring boundary conditions. Indeed, the distortion is homogeneous in the direction orthogonal both to the field and to the initial alignment of the molecules. The post-critical response of nematic cells under the action of high strength fields has been investigated deeply in theoretical studies (Deuling 1972; Gruler and Meier 1972; Self et al. 2002; Napoli 2006). More recently, a new type of field induced configurational transition has been observed experimentally by Lonberg and Meyer (1985) on a nematic made of a mixture of levo and dextro versions of synthetic polypeptide polybenzylglutamate (PBG), with a length/width ratio

3 70:1. When subjected to intermediate strength fields, polymeric liquid crystals characterized by very long molecules may undergo spatially periodic instabilities. Within thin cells, and under strong planar anchoring boundary conditions, periodic domains form in the plane orthogonal to the initially undistorted and uniform molecule alignment. Such an effect is called periodic Fréedericksz transition (PD) as opposed to the homogeneous Fréedericksz transitions described above. The periodic distortion, characterized by a wavelength of the order of magnitude of the cell thickness, occurs at a critical threshold H c smaller than H c1. An analysis of the PD critical threshold performed via linearized equilibrium equations is reported in the pioneering article (Lonberg and Meyer 1985). Subsequent works (Zimmermann and Kramer 1986; Miraldietal.1986) generalized the original analysis to weak anchoring boundary conditions. With respect to the undeformed state, both HD and PD are second-order phase transitions. In fact, the transitions from the uniform state occur continuously, as a consequence of the variation of the intensity of the external field. By examining the response of the system to sudden changes of the external field, switching from very high to intermediate-high, Srajer and collaborators (1991) identified the existence of a further transition from HD to PD at a critical field strength H t. In particular, it was observed that at intermediate high field strengths both states can exist, the PD state being characterized by a lower energy than the HD state. The behavior of the samples suggested the presence of a finite energy barrier between the two distorted states, therefore the experimentalists classified the HD to PD transition as a first-order phase transition. Other types of transitions, involving different aspect of nematic energies or different distortion patterns, have been observed or predicted in liquid crystals, see, e.g., Gooden et al. (1985), Allender et al. (1987), Virga (1994). The current literature is relatively poor of numerical approaches for the analysis of NLC distortions. A discrete approach to the study of NLC distortions was proposed in Gartland (1995). Frank s theory was adopted to study numerically the development of stripe or ripple phases in nematic thin films under sufficiently strong magnetic fields, considering the bend-fréedericksz geometry at temperatures slightly above that of the nematic to the smectic-a phase transition. The approach was used subsequently to analyze the stability of HD in highly anisotropic thin nematic films (Golovaty et al. 2001). An analytical treatment of the same problem was presented later in Barbero and Evangelista (2006); an interesting analogy with the present investigation is that the wavelength of PD was computed through energy minimization. Features of PD have been analyzed numerically in a relatively recent study (Krzyzanski and Derfel 2000). Periodic structures were described in terms of orientation of the director at regularly distributed points of a unit periodic cell. To reduce the computational effort, in all the calculations eight-fold symmetry of the periodic structures was enforced. At each value of the field strength, the total energy of the system was evaluated for different distributions of the director orientation. Thus, the configuration leading to the minimum value of the free-energy was assumed to be the equilibrium configuration; models of the numerical solution in terms of the director field have been reported before (Krzyzanski and Derfel 2000). Finite element approaches to analyze the behavior of NLC have been discussed or used in Davis and Gartland (1998), Di Pasquale et al. (1996, 1999), Fernández et al. (2002), James et al. (2006), Amoddeo et al. (2011). Davis and Gartland (1998) de-

4 scribed the minimization of the Landau de Gennes free-energy functional for computing equilibrium configurations of the tensor order parameter in liquid crystals. Their study addressed analytical and numerical issues and the feasibility of applying finite elements, but do not provide any numerical applications. Two- and threedimensional combined finite element and finite difference methods for the numerical analyses of NLC devices have been presented by Day and collaborators. Twodimensional steady state analyses of LC structures under nonuniform electric fields are reported in Di Pasquale et al. (1996), where linear finite elements are adopted. In Di Pasquale et al. (1999) NLC test cells with interdigital electrodes are analyzed. In order to study the coupling between electric field and director, they employed a simplified form of the Frank s energy, where one single constant accounted for the elasticity properties of the nematic, and used finite element to obtain the solution for the electric field, while finite differences and central difference time integration provide the solution for the director. Three-dimensional finite element analysis of the optical transmittance of NLC devices are documented in Fernández et al. (2002), where the authors used a compact expression of the free-energy density of the system to simplify the numerical implementation. James et al. (2006) used two-dimensional finite elements and a modified expression of the Ericksen Leslie formulation, including the variation of the order parameter, to describe disclinations. Lin and Liu (2006), Lin et al. (2008) used finite elements to simulate the two-dimensional hydrodynamics of liquid crystals, adopting a simplified version of the Ericksen Leslie theory, which retains the most important mathematical and physical features of the original but it is more appealing for numerical applications. Adaptive finite elements have been used in combination with the Landau de Gennes free-energy functional in Amoddeo et al. (2011). That model is able to describe the nematic order dynamics within a one-dimensional cell under the action of rapid electric pulses of different amplitudes. The aim of our work is to set up a numerical instrument able to explore numerically the existence of alternative equilibrium configurations and capture the transitions between different phases via finite elements analysis and Frank s theory. Our application deals with a fully nonlinear finite element analysis of the onset and evolution of distortions in nematic liquid crystals using Frank s theory. With the remarkable exception of Di Pasquale et al. (1996), where the study was devoted to different problems, the FE approaches documented in the literature are based on simplified forms of the energy or, alternatively, on linearized versions of the equilibrium problem; therefore, such approaches are only able to capture one of the possible distortions and cannot detect bifurcations. In the literature no explicit expressions of the matrices and arrays in their general form have been reported; only the special expressions for linear triangles or tetrahedra (Fernández et al. 2002). Finally, alternative numerical studies based on discrete formulations to analyze nematic distortions did not make use of FE (Krzyzanski and Derfel 2000; Barbero and Evangelista 2006), or used a different form of energy (Amoddeo et al. 2011). To describe the behavior of a nematic, we adopt the classic Frank s continuum theory and discretize the energy of the system by quadratic finite elements. By imposing stationarity of the energy functional, we identify numerically the existence of several equilibrium configurations of the system. Equilibrium configurations are

5 characterized by the local orientation of the director all over the integration domain and, in the case of striped pattern periodicity, by their wavelength. Thus, by exploring a wide range of forcing fields, we are able to identify different classes of configurational distortions and to capture the transitions or critical points of the material. The peculiarity of this study is that the numerical configurations, not constrained to satisfy a predetermined form of periodicity neither to respect symmetries, show a strong similarity with the experimental findings reported in Srajer et al. (1991). A second remarkable result is that the procedure is able to detect secondary, metastable configurations, representing equilibrium bifurcation branches at higher energy levels that cannot be observed in experiments. The organization of the paper is as follows. The classic Frank s continuum theory (Frank 1958) is summarized in Sect. 2. Then we identify a representative cell where the problem can be posed in dimensionless form; see Sect The finite element discretization of the problem and the solution strategy of the nonlinear system is described in Sect. 3, and in Sect. 4 we present our numerical results. We conclude with a brief discussion of the method, its abilities and its limits, in Sect Frank s Theory of NLC In the formalism proposed by Frank (1958) a nematic is treated as a continuum; molecular details are ignored completely. Rather, this theory considers perturbations to a presumed oriented sample. A rigorous mathematical description of Frank s theory can be found in Virga (1994). In Frank s theory, the assumed local free-energy function is dependent on the unit vector field n, pointing in the average molecular orientation, and on its gradient n. The director n adjusts throughout the sample in order to minimize the deformation energy of the system, according to the boundary conditions. Since n is physically equivalent to n, the local stored energy density ψ N is assumed to be ψ N (n, n) = K 1 2 (div n)2 + K 2 2 (n curl n)2 + K 3 2 n curl n 2. (1) In this expression, the response of the material is decomposed into three additive terms characterized by the elastic constants K 1, K 2 and K 3 corresponding to three types of distortions that can occur in an oriented sample: splay, twist and bend, respectively. In order to guarantee the attainment of a stable undistorted configuration in absence of external fields or confinements, we assume K i (i = 1, 2, 3) non-negative, according to Ericksen s inequalities (Ericksen 1966). The saddle-splay term has been disregarded a priori, since it does not induce effects under strong anchoring boundary conditions. In the presence of a magnetic field, the energy includes an additional term which describes the field director interaction ψ H (Deuling 1972; de Gennes and Prost 1993): ψ(n, n) = ψ N (n, n) + ψ H (n). (2) In the following, we assume that the external field is magnetic. The treatment in the presence of an electric field is analogous. Additionally, we make the simplifying

6 Fig. 1 Orthonormal reference system adopted for the analysis of a nematic confined between two glass plates. The plates are parallel to the plane xz,andthe surface treatment constrains the LC molecules adjacent to the plates to orient in the z direction. The magnetic field acts in the direction of the y axis. The plane xy defines the domain where the periodic distortions of the LC appear assumption that, while the magnetic field may induce distortions in the nematic, the magnetic field itself is not altered by the presence of the distortion. Under such a hypothesis, called the magnetic approximation (Self et al. 2002), the field assumes a prescribed value, and H will not appear as unknown in the subsequent discussion. Denoting by χ a the diamagnetic anisotropy and with m and H the orientation and the intensity of the applied field, respectively, the interaction energy can be expressed as ψ H (n) = 1 2 χ a(h n) 2 = 1 2 χ a H 2 (m n) 2. (3) One can easily verify that, whenever χ a is positive, the molecules tend to align their axes along the magnetic field; on the contrary, if χ a is negative, they tend to align their axis normally to the magnetic field. Considering a nematic sample occupying the volume B, equilibrium configurations correspond to stationary points of the total energy of the system. In order to account for the constraint of the unit length director, the total energy Ψ can be written in the extended form: [ Ψ(n, n,l)= ψn (n, n) + ψ H (n) + l ( n 2 1 )] dv (4) B where l denotes a Lagrangian multiplier scalar field. Thus, the equilibrium configurations are sought as the solutions of the stationarity problem: δψ (n, n, l) = 0, (5) where δ denotes the first variation of the total energy, according to the boundary conditions. We are concerned here with a very common and practical situation, i.e., a nematic is confined between two parallel plates at a small distance d. We adopt an orthonormal reference system x,y,z (with basis vectors e 1, e 2 and e 3 ) where the x and z axes define the directions with infinite extension and the y axis the direction orthogonal to the plates, see Fig. 1. The plane xy defines the domain where the periodic distortions become manifest. The applied magnetic field acts in the y direction: H = H e 2, ψ H (n) = 1 2 χ ah 2 n y 2. (6)

7 Strong boundary conditions and elastic stiffness of the nematic tend to maintain the molecules in the uniform planar alignment along z, while the applied field tries to realign the director parallel to its direction. At high intensities of the external field, above a critical value, HD or PD distortions occur. When the three elastic constants of the nematic are of the same order of magnitude, HD is the sole admissible distorted state compatible with the external field. In such a state, the molecules reorient in a splay configuration, maintaining their axes in the plane spanned by the unit vectors e 2 and e 3. The misalignment with respect to the e 3 axis reaches a maximum at the center of the cell and decreases monotonically and symmetrically towards both the plate boundaries (Self et al. 2002). At the glass boundary, the director is perfectly aligned with the e 3 axis, according to the strong anchoring conditions. Conversely, periodic distortions may appear only at intermediate field strengths in nematics characterized by long molecules, for which the elastic coefficient K 1 associated to the splay configuration is bigger than K 2, the one associated to the twist configuration. In such materials, the energy requested for a splay distortion is higher than the energy requested to accommodate a twist dominated distortion. Therefore, twist configurations prevail, and a plane distortion characterized by periodic structures along the e 1 direction occurs (Srajer et al. 1991). In order to capture the periodic distortions in the plane (x, y), we discard the dependence on the z direction, which is assumed to be infinite, and limit our attention to a planar domain, although the tackled problem is fully three-dimensional. Under such restrictions, the energy of the nematic is defined per unit length in direction z and is a function of the two coordinates x and y only. 2.1 Scaled Energy A further simplification is suggested by the special shape of the periodic distortions. The periodicity of such patterns is defined by a wavelength related intrinsically to the specimen thickness d and to the field strength, although the value of the wavelength is not known a priori. In particular, in Fig. 1 of their paper, Srajer et al. (1991) describe periodic patterns formed by two adjacent twist dominated domains. The first domain is characterized by clockwise twist, while the second domain shows counterclockwise twist. Evidently, for symmetry reasons, the two domains possess the same energy. We focus on one domain only, which is characterized by half wavelength of the whole solution. The wavelength λ in Srajer et al. (1991) is expressed here as λ = 2 d/β, where the parameter β is an outcome of the analysis. The size of the domain where the configurational distortions are sought is restricted to the undetermined rectangle Ω ={(x, y) 0 x d/β,0 y d}. To build a more tractable expression of the energy, where the unknown width of the domain does not appear in the boundaries, we restate the problem in a dimensionless form by introducing the scaled coordinates: and the dimensionless elastic constants: ξ = β d x, η = 1 d y (7) κ 2 = K 2 K 1, κ 3 = K 3 K 1. (8)

8 The domain of interest becomes the square Σ ={(ξ, η) 0 ξ 1, 0 η 1} and we deal with a dimensionless energy per unit length Ψ : Ψ = β { ( Ψ = ψ(n, n) + l n 2 1 )} dξ dη (9) K 1 Σ where l denotes the dimensionless Lagrangian multiplier. Note that the same treatment was used in Gartland (1995) to analyze numerically bend-fréedericksz transitions. By introducing the forcing field R, χa R = Hd, (10) K 1 the dimensionless free-energy density reads 2ψ(n, n) = (div n) 2 + κ 2 (n curl n) 2 + κ 3 n curl n 2 R 2 (n e 2 ) 2, (11) where the gradient operator is now =β ξ e 1 + η e 2. (12) Thus, according to the scaling, the unknown wavelength β appears explicitly in the bulk energy as a coefficient of the derivatives with respect to ξ. The Euler Lagrange equations of the scaled functional (9) are ( ) ψ ψ n div + 2 l n = 0, n 2 = 1, (13) n to be solved under the proper boundary conditions. As alternative approach, we could make use of the Rayleigh Ritz method, which leads to the matrix equations directly from the variational form of the free-energy. We prefer to use the Euler Lagrange approach, since it allows for the inclusion of the boundary conditions in a more efficient way. Boundary conditions are defined with reference to the first domain in Fig. 1 in (Srajer et al. 1991). On each cell, the director n must satisfy the strong planar anchoring conditions imposed by the pair of glass plates, i.e. n must parallel to e 3 : n(ξ, 0) = n(ξ, 1) = e 3. (14) On the vertical sides of each domain, the component in e 1 of the director is null, while the component in e 2 on the right is opposite to the one on the left: n ξ (0,η)= n ξ (1,η)= 0, n η (0,η)= n η (1,η), n ζ (0,η)= n ζ (1,η). (15) Note that the boundary conditions (15) 2 and (15) 3 are not standard Dirichlet or Neumann conditions, since neither the value of n nor the value of n is assigned. We conclude by observing that, up to this point, we did not make any assumptions on the

9 wavelength β, treated as a parameter. Given the explicit presence of β in the energy expression, its optimal value can be determined by minimizing the dimensionless energy (9) written in the form ψ(β)= 1 2 a(n, n)β2 + b(n, n)β + c(n, n), (16) which gives b(n, n) β opt (n, n) = a(n, n) ; (17) see a similar treatment in Gartland (1995). The integral expressions of the non-trivial functions a(n, n) and b(n, n) are reported in Appendix A. When periodic distortions appear, β>0 is of the order of the unity. The positiveness of β is consequence of the condition a(n, n) >0, following from the definition of (A.3) and the restriction on the value of the elastic constants, and the additional b(n, n)<0. If b(n, n) 0, the optimal value of β is lower-bounded by zero, condition that characterizes both undistorted and homogeneous distortions. Details of the algorithm used in the numerical applications are reported in the next section. 3 Finite Element Formulation We resort to the finite element method and obtain the weak form of (13) by introducing the admissible fields n and l: { [ ( ) ] ψ ψ n n div + 2 l n + l ( n 2 1 )} dξ dη = 0. (18) n Σ By using integration by parts and the divergence theorem, (18) becomes { [ ] ( ) ψ ψ n Σ n + 2 l n + n + l ( n 2 1 )} dξ dη n 1 [( β n ψ ) ( n ψ ) n n 0 + (1,η) (0,η) ] e 1 dη, (19) where we account for the Dirichlet boundaries at η = 0 and η = 1. The boundary integrals in (19) account for the contribution of the unknown components of n on the vertical boundaries. They will disappear only when β = 0, i.e. no periodicity is observed, or when the integrals over the two vertical boundaries are opposite in the weak sense. Next, we discretize the domain with finite elements and, in each element, introduce the interpolation functions for the director field in the form: n(s, t) = a N a (s, t) u a. (20)

10 The shape functions N a are assigned polynomial (linear or parabolic) functions with a non-zero value in the sole finite element domain; s, t are the local coordinates of the finite element, and u ai, (i = ξ,η,ζ) are the unknown components of the director at the nodal points. The same interpolation applies to the Lagrangian multipliers: l(s, t) = a N a (s, t) λ a, (21) where λ a are the scalar nodal values of the Lagrangian multiplier fields. In order to compute the integrals in (19) it is necessary to specify the discretization of the director along the boundary of the integration volume: n(s) = a N Γ a (s) u a, (22) where the surface shape functions Na Γ must be consistent with the one adopted in the volume interpolation. The sum of the two discretized line integrals will differ from zero only when periodic boundary conditions are not satisfied. Upon discretization, the stationarity conditions furnish a system of nonlinear algebraic equations. In particular, listing the four unknowns of each nodal point a in a vector U a = [u a,λ a ], the corresponding nodal equations read [ F a = Σ f a dσ + Γ t ] a dγ Σ g = 0. (23) a dσ The integrals in (23) extend over volume and boundaries of all the elements connected to the node a. In the code, the integrals in (23) are computed on each triangular element through 3-point Gauss quadrature. The expressions of t a, f a and g a are reported in Appendix B. Finally, the assembling leads to the global system of nonlinear equations: F (U) = 0, (24) where the vector U collects the nodal unknowns U a over the np nodal points of the discretization. In view of the solution of the system (24) through consistent linearization, we derive the Hessian matrix of the system as F (U) K(U) = U. (25) The components of K are reported in Appendix C. The Hessian matrix is used inside an iterative Newton Raphson algorithm to evaluate the solution up to the desired precision. In the present application we reach a relative error below Atthek iteration, we compute the expression: K k[ U k+1 U k] = F k, F k = F ( U k), K k = K ( U k) (26) and perform the inversion of the Hessian matrix, using a general purpose library for the direct solution of the sparse system of linear equations, based on a LU decomposition with partial pivoting (Demmel et al. 1999).

11 As observed before, the unknown wavelength β enters in the discretized expression of the energy as a scaling factor for the derivatives with respect to ξ. Therefore, for the evaluation of the correct value of β, it is necessary to introduce an additional iterative procedure, nested outside the loop on U. In particular, for an assigned and fixed value of the magnetic field, we initialize the value of β opt with 1, or with the optimal value obtained for the previous value of the magnetic field. Then, once the convergence on U has been reached, the optimal β opt associated to the stationarity condition is computed as β opt = b k a k, a k = a ( U k ), b k = b ( U k ), (27) where k denotes the last iteration index in (26) (see again Appendix A for the operative expression of the coefficients a and b). With this new guess for β opt, a new cycle on U is started. Obviously, the problem being highly nonlinear, the sequence of calculations (26) (27) has to be repeated a few times, until convergence in the value of β opt is observed. In general, three to ten iterations suffice to reduce the relative error on β below Additionally, when the computed value becomes β 10 3, it is forced to zero. As an alternative to the approach by Lagrangian multipliers, the enforcement of the unit constraint may be done through a penalty approach, i.e. the constraints are added to the energy through an assigned large constant p: { Ψ = ψ(n, n) + p( n 2 1) 2} dξ dη. (28) Σ The penalty approach reduces the number of unknowns up to two thirds, but it provides less accurate solutions. We resort to this method when very fine meshes are adopted, but only when the solution is far from critical points. 4 Numerical Results The method previously illustrated is used to seek the equilibrium configurations of a nematic cell at different values of the applied field. Thus, the critical field H c1 for the appearance of HD in short molecule nematics is identified with R c1 = π; the critical field for the transition from undistorted to PD is denoted with R c, while the critical field for the transition from PD to HD is denoted R t. Besides the field strength and the field orientation, the only input data for the simulation of the experiments reported in Lonberg and Meyer (1985) are the material property ratios κ 2 = and κ 3 = 2.0. The iterative procedure (26) allows one to obtain the equilibrium (stationary) points in the configuration space, provided that the initial configuration is a close enough guess. Therefore, we explore sequentially the equilibrium configurations by assuming the solution of the previous step as starting guess for the next one, according to a technique also used in Di Pasquale et al. (1996). If no convergence is reached, a small numerical alteration of the current configuration is enough to switch towards an alternative, lower energy, stationary configuration. Obviously, since we

12 Table 1 Mesh size comparison: data and results Model h-size Nodes Elements CPU [s] R t β t Err % M1 1/ M2 1/ M3 1/ M4 1/ Fig. 2 Plot of the dimensionless total energy as a function of the dimensionless field strength R, for the particular NLC considered in Lonberg and Meyer (1985). Comparison between the results obtained with three different meshes. Note that coarser meshes do not have enough degrees of freedom to capture the transition between periodic and homogeneous distortions and provide an incorrect solution are dealing with a non-convex energy, stationary points may not correspond to stable configurations of the system. Stable configurations of the system are then obtained as the envelope of the lowest energy states obtained in the analysis. To rule out spurious mesh dependency effects, it is mandatory to identify an adequate size of the discretization. In the proximity of the transition point R t, the spatial approximation introduced by finite elements may shade or eventually miss the behavior of the nematic as allowed by Frank s theory. Only sufficiently fine meshes would be able to capture all the configurations. Solutions with coarse meshes show an artifact in the energy plot, suggesting a jump around the transition point that cannot be true. Under such conditions, the system would rather prefer undergoing a dynamic instability than increase its total energy; but the quasi-static calculations here performed cannot follow a dynamic evolution and cannot capture this effect. To select the correct size of grid, we performed the same simulation with four different uniform meshes discretized with quadratic elements. The characteristics of the meshes and the average computational cost in terms of CPU time per analysis are reported in Table 1. In the same table are collected the computed critical field R t, the wavelength β, and the relative error on R t with respect to the solution provided by the finest mesh, i.e. simulation M4. We explored the range of field strengths between 1.8 R 4.5, at regular steps R = Figure 2 shows the plot of the total energy of the system as a function of the field strength. The energies of the four meshes agree for low and high field strengths. Low strengths, up to R c = 2 for the four meshes, provide a flat

13 energy diagram, and correspond to undistorted configurations. High strengths, R> 4.2, correspond to homogeneous (pure splay) configurations and are characterized by a negative energy. Both undeformed and HD states are evaluated equally well by all the meshes. The four meshes provide the same value also for the critical field R c = 2, and similar values for the corresponding β c Contrariwise, intermediate-high strength fields in proximity of the critical point H t show a strong mesh dependency. In particular, the three coarser meshes cannot capture correctly the location of the transition zone between PD and HD. Indeed, the coarse meshes suddenly switch from a lower energy branch corresponding to the PD to a higher energy level corresponding to HD. Only the finest mesh seems to possess enough resolution to capture a correct critical point of the transition between PD and HD, at about R t = 4.2. Note that the difference on R t between the two finest meshes is quite small, about 1.2%, see Table 1. It is evident, however, that the cost of simulation M4 in terms of CPU time is very high. Nevertheless, it is important to observe that, in proximity of the critical point R t, the distribution of the director in the four meshes is very similar, and the periodicity is characterized by β<1. Note that previous numerical studies on nematic configurational problems (Gartland 1995; Krzyzanski and Derfel 2000) used coarse grids, with sizes four times larger than the one used in the present work. The refinement here adopted is necessary in order to detect correctly intermediate equilibrium configurations, not constrained to satisfy symmetries or simplified forms of periodicity. In the subsequent discussion we show the results obtained with both the simulation M3 and M4. In particular, all the plots refer to the results of simulation M4, while the maps of director orientations and energy always refer to simulation M3, since the finest mesh does not clearly show the director textures. 4.1 Stable Configurations We start by considering the solutions corresponding to least energy configurations, i.e. the stable solutions of the problem, identified by the solid energy line in Fig. 2. Figure 3(a) reports for the simulation M4 the contribution of splay, twist, field and total energy of the system as a function of the field strength, while Fig. 3(b) reports the wavelength β as a function of the field strength. Obviously, non-zero values of the parameter β testify the presence of PD. In agreement with experimental observations (Srajer et al. 1991), for the particular value of κ 2 here considered, we capture the two critical fields R c and R t that identify the transition between different states; and a third field R m corresponding to a change of the behavior for the PD. The magnetic field, interacting with the nematic, applies a torque to the director and tries to align it parallel to the field. Above R t, the energy associated to the periodic distortion exceeds the energy associated with the splay distortion, and HD become the preferred configuration. Figure 4 shows the distribution of the director in the pure splay configuration for R = 4.25 and the corresponding total energy density distribution. High positive values of nematic energy (due to splay) are observed at the glass boundaries; in the central part of the sample, negative values of the energy are due to strong field interaction. Below R c, the restoring elastic torque is effective in maintaining the original orientation of the nematic in the direction of the z

14 Fig. 3 Plot of (a) dimensionless energies and (b) inverseofthe wavelength, β, as a function of the dimensionless field strength R, for the particular NLC considered in Lonberg and Meyer (1985). Results of simulation M4 axis. At R c, we observe the first numerical appearance of PD, characterized, for intermediate strength fields, by a low energy with respect to the one associated to the undistorted configuration. For R>R c, PDs are characterized by values of β greater than 1, originating with vortices elongated in the η direction; see Fig. 5. Except for more or less pronounced inclination of the molecules towards the η direction, at different strengths the configuration of PDs is very similar, as long as β>1. In general, the wavelength of the vortices itself is not very sensitive to the field strength, but it is possible to observe an interesting behavior reported also by the experimentalists. In the interval R c R R m = 2.7, simulations provide a growing β up to a maximum β m = 1.52, while in the interval R m R R t the computed β decreases more and more rapidly. The existence of a particular field R m identifying a change of the regime for β was pointed out in Srajer et al. (1991), in commenting on the experimental data expressed in terms of the measured wavelength λ = 2d/β.Sra-

15 Fig. 4 Homogeneous distortions at R = 4.25, β = 0, for the numerical simulation of the nematic studied in Srajer et al. (1991). (a) Side view of the director orientation; (b) front view of the director orientation; (c) contour levels of the total energy distribution. The regions at the glass boundaries are dominated by splay energy; the region at the center is dominated by field interaction. Results of simulation M3 Fig. 5 Periodic distortions at R = 2.2, characterized by β = 1.49, for the numerical simulation of the nematic studied in Srajer et al. (1991). The twist dominated vortex is characterized by regularity and symmetry. (a) Side view of the director orientation; (b) front view of the director orientation; (c) contour levels of the dimensionless energy distribution. The regions at lower energy (lateral sides) are dominated by field interaction energy. The regions confining with the glasses are characterized by splay. Twist energy dominates elsewhere. Results of simulation M3 jer and collaborators justified the behavior of λ by arguing that, as long as H 1 d, the wavelength decreases with increasing field. This hypothesis is correct for an undulation pattern essentially sinusoidal, with amplitude increasing with H. In such case, the increment of the field causes a progressive increment of the orientation of the director towards the field direction, keeping the distortion on average in a twist form, dominated by the lower elastic modulus. Such a behavior is observed in our nu-

16 Fig. 6 Splay and twist dominated periodic distortions at R = 3.9, characterized by β = 1.0, for the numerical simulation of the nematic studied in Srajer et al. (1991). The twist vortex occupies most of the cell and shows a certain degree of asymmetry. On the left side, splay occupies a small portion of the cell. (a) Side view of the director orientation; (b) front view of the director orientation; (c) contour levels of the total energy distribution. Splay is identified by positive energies; low energies characterize the field interaction energy. Results of simulation M3 merical simulation, for strengths in the range R c R R m ; see Fig. 5. Significantly, the twist vortices do not show noticeable changes with the field strength. Our calculations show that PDs at field strengths closer to R t are characterized by a minor degree of symmetry. Figure 6, for example, shows a stable equilibrium state at R = 3.9 and β 1, where it is possible to observe the presence of a small splay region on the right side of the cell. This behavior was described in Srajer et al. (1991):... as H 1 becomes much less than d, saturation effects appear, with regions near the midplane of the sample becoming almost uniformly oriented parallel to H. With these nonsinusoidal distortions, the splay cancelation mechanism also breaks down in increasingly large volumes around the midplane of the sample, which now contains an energetically costly array of twist walls. The remaining splay cancelation occurs only in boundary layers that get thinner as field increases. Values 0 <β<1 are observed only for R slightly below the transition field R t. The structure of PDs associated to small values of β is considerably different from the one observed for β>1. In general, the vortices are characterized by very strong asymmetry; they are well defined on one side of the unit cell, while fade smoothly on the other side, where twist distortions are replaced progressively by pure splay. Figure 7 shows the side and front views of the director orientations for the case R = 4.1 and β = 0.8; the predominant splay region occupies the left side of the cell. The particular distribution of the director is an interesting outcome of the fully nonlinear calculations, and because of its complexity it cannot be captured analytically. Interestingly, Fig. 7 provides an excellent visualization of the way the first-order phase transition between PD and HD actually takes place. This behavior is in agreement with the comments of the experimentalists (Srajer et al. 1991):... As this happens, the period λ begins to increase with the field. One could imagine it diverging continuously, but that possibility is apparently cut

17 Fig. 7 Splay and twist dominated periodic distortions at R = 4.1, characterized by β = 0.8, for the numerical simulation of the nematic studied in Srajer et al. (1991). The twist vortex is located at the left side of the cell and shows asymmetry. On the right side, splay progressively replaces twist defining a splay region. (a) Side view of the director orientation; (b) front view of the director orientation; (c) contour levels of the total energy distribution. Splay is characterized by high values of the energies; regions dominated by field interaction are characterized by negative values of the energy. Results of simulation M3 off by the first-order transition back to the homogeneous state. In contrast with this comment and with our results, in Krzyzanski and Derfel (2000) concluded that the transition from PD to HD is due to the divergence of the spatial period of the deformation to infinity, i.e. the periodic structure expands without substantial modification of its conformation. Such conclusions were supported by numerical results, unfortunately affected by the symmetry constraints and therefore probably unable to capture the lowest energy configurations. Curiously, the spatial distribution of the director as shown in Fig. 7 may also explain the reason why coarser meshes are not able to capture the behavior of the

18 nematic in proximity of the transition region. In fact, squeezed vortices cannot be modeled by large elements, and in that case the numerical solver finds the only configuration compatible with the discretization, i.e. the homogeneous splay. 4.2 Metastable or Higher Energy Configurations As mentioned before, our numerical procedure furnishes stationary points of the equilibrium path. Remarkably, this allows one to follow metastable branches which are not discoverable easily in experiments. We performed two additional analyses disregarding the search for stable configurations. To achieve this, at every variation of the strength we started the iterative search (26) from the solution of the previous calculation, without applying any perturbation of the solution. We performed two analyses, one from high fields (R = 4.5) to low fields R = 3.35, and the second from low fields (R = 2.1) to high fields (R = 3.65). The solver was able to track two alternative metastable equilibrium branches, reported in Fig. 8. As mentioned in the introduction, R t defines a first-order transition, in the sense that, below R t, HD are metastable with respect to PD. Vice versa, above R t,pdare metastable with respect to HD. The first metastable branch of our calculations is obtained as the prosecution towards low strength fields of the HD branch, and it results into pure splay configurations below the critical transition point R t. Metastable HD do not differ substantially from the stable HD discussed in the previous section, except for the lower energy level. It is interesting to mention that, when coarse meshes are used, the rough discretization makes it impossible to track the PD stable equilibrium branch below H t. Therefore, the only numerical solution attained by coarse meshes is this metastable HD state, see Fig. 2. The situation involving the second metastable branch is even more interesting, describing clearly a new configuration predicted by the numerical calculation. The configuration of the nematic is characterized by the formation of two vortices, associated to low values of β. An example of metastable PD as obtained from the numerical calculation is shown in Fig. 9. The two vortices are clearly asymmetric and, according to our model, define only half of the expected wavelength. The energy associated to such configuration is stationary, and therefore this is an equilibrium configuration. Nevertheless, the total energy of the system exceeds the energy related to the formation of a single vortex; thus, the configuration with two vortices must be classified as metastable. Obviously, in standard quasi-static laboratory experiments such configurations are not likely to manifest. We believe, though, that our finding is theoretically of interest; since it is a prelude to the existence of additional higher energy equilibrium states characterized by asymmetric vortices, which might be exploited in some technical application. 5 Discussion and Conclusions We developed a finite element code to study the interaction of NLC with an applied magnetic field, under the assumption that the reorientation of the molecules does

19 Fig. 8 Stable and metastable equilibrium branches obtained from numerical calculations. Plot of (a) dimensionless energies and (b) wavelength β as a function of the dimensionless field strength R, for the particular NLC considered in Lonberg and Meyer (1985). Results of simulation M4 not deviate the applied field itself. We believe that our assumption of the magnetic approximation is reasonable, in the light of the normal treatment of the physics of the problem where such a coupling is neglected (Srajer et al. 1991; Lonberg and Meyer 1985). It is well known that finite elements have noticeable advantages with respect to the more traditional finite difference approach. In particular, it is possible to adopt a non-uniform grid and to neatly reproduce high spatial gradients of the director with a limited computational effort. Another important advantage of the finite element approach is the possibility to incorporate naturally boundary conditions involving the derivatives of the discretized fields without introducing a fictitious extension of the

20 Fig. 9 Metastable periodic distortions at R = 2.5, characterized by β = 0.54, for the nematic studied in Srajer et al. (1991). Note the presence of a double asymmetric vortex dominated by twist. (a) Side view of the director orientation; (b) front view of the director orientation; (c) contour levels of the total energy distribution. The regions at lower energy (lateral sides) are dominated by field interaction energy. The regions confining with the glasses are characterized by splay. Twist energy dominates elsewhere. Results of simulation M3 computational grid. Therefore, difficulties such as the one mentioned in Golovaty et al. (2001) are ruled out; in particular, the detection of fragile structures on coarse meshes, or of non-physical high frequency microstructure solutions at the level of the mesh. We are not claiming to be the first to apply FEM to the study of nematics; in fact, interesting applications have been discussed in the introduction. At any rate, we did not find in the available literature an accurate discussion on the spatial orientation of the molecules and the transition between different phases; most of the previous finite element implementations have been done on linearized expression of the total energy. Our approach accounts for geometrical nonlinearities and allows for the evaluation of the magnitude of HD and PD distortions, which are stable at different

21 ranges of the applied field strength, and for the identification of the critical transition points between phases. Furthermore, the presence of the second derivative allows us to intercept and track unstable stationary points in their evolution. This represents a remarkable potential for full bifurcation analyses. In our calculations we evaluated the qualitative and quantitative behavior of the nematic at different strength fields, in particular between the two critical fields R c and R t identifying the formation of PD and HD, respectively. In the case of HD, the FE analyses provide the magnitude of the orientation of the director across the cell thickness. This magnitude is variable with the field strength R. In the case of PD, the code provides the wavelength of the cell and the magnitude of the two-dimensional twisted structures. When long molecules are considered, i.e. κ 2 < 0.303, in the range [R c,r t ] of the field strength the total energy associated to PD may be lower than the one associated to HD. Therefore, configurations dominated by twist may as a result be found to be energetically favorable for the nematic. The critical value R c of the field strength is lower than the theoretical value R c1 for the onset of periodic distortion, while the high critical field R t is larger than R c1. Therefore, the two critical values R c and R t identify the range where PDs manifest and in general dominate and are stable. HDs are stable only above the high critical value R t. The particular numerical response described above is associated to the small value of the parameter κ 2. Higher values of κ 2 will produce different responses of the system; a parametric analysis on constitutive parameters will be presented in a future work. The numerical calculations provide some additional data about the interplay between splay and twist energies. In our analyses it is possible to identify the presence of an intermediate strength R m, where the twist and splay energy are equivalent. For lower strengths, R c <R<R m, twist energy predominates, while for higher strengths, R m <R<R t, splay energy is bigger. In proximity of R t, the PD are characterized by slightly different structures, showing asymmetry and an important splay zone on a side. These observations suggest that in the high strength zones of the PD dominated domain there is competition between splay and twist, testified to by the decrease of β with R at high intensity fields; see Fig. 3. The present approach is suitable also for the analysis of different types of NLC and different periodic structures, as the ones observed by applying the external field in a different direction. The extension to a fully three-dimensional version is straightforward, and is already pursued, with the aim to study the spinodal point mentioned in Srajer et al. (1991). In the experiments, in fact, the thickness of the nematic cell was variable, an this expedient allowed the observation of a first-order transition between PD and HD. The achievement of this goal may require embedding the code with some time dependency, in order to describe the kinetics of the evolution of the metastable structures, as reported in Srajer et al. (1991). Finally, a more ambitious and long term task is to include more complex expressions of the free-energy, such as the Landau de Gennes, with a view to describe disclinations and the formation of other localized defects in LC characterized by positional order.