Elasticity Tensors in Nematic Liquid Crystals

Abstract: The paper is discussing the contributions to the free energy of elastic distortions in a nematic liquid crystal. These contributions are here given by tensors, which are represented by means of the components of the director, the unit vector indicating the local average alignment of molecules, and by Kronecker and Levi-Civita symbols. The paper is also discussing the elasticity of the second order and its contribution in threshold phenomena.


Introduction
The equation of Oseen-Frank, which is providing the free energy density in nematic liquid crystals [1,2], is representing the part of that energy density coming from an elastic deformation of the bulk of the material. In such approach, the terms depending on the deformations are multiplied by the three elastic constants (K11, K22, K33) of "splay", "twist" and "bend". After, to this energy density, Nehring and Saupe added two terms having elastic constants of "mixed splay-bend" (K13) and of "saddlesplay" (K24) [3] . These terms, which have K13, K24 as elastic constants, are equivalent to contributions to the surface free energy of the material, because they are divergences of some vector functions of the director. These contributions to surface energy are relevant for nematic cells subjected to weak anchoring conditions [4][5][6].
The elastic constants previously mentioned are constants that multiply some scalar functions obtained from the director, the unit vector giving the local average alignment of the molecules, and from its derivatives, in the form of rotors and divergences. In this paper, we will discuss how the terms of the free energy density in nematic liquid crystals can be described by tensors, and how these tensors can be represented using the director components and the Kronecker and Levi-Civita symbols. This paper will also discuss the terms of the second order and their role in threshold phenomena [7,8].

The elastic deformation of a nematic
In an ideal nematic, the molecules are oriented, in average, along the director, the macroscopic unit vector n [9]. This alignment of molecules along a common direction is proper of the nematic phase, which, in this manner, is characterized by an orientational order. This order disappears in the isotropic phase. To describe the nematic phase, a tensor order parameter is given as: In (1), i and j are indices of the Cartesian frame of reference; ni are the director components and δij the Kronecker symbol. Scalar Q depends on the temperature and goes to zero in the isotropic phase. Tensor Qij can be a function of the position vector r. Let us remember that in the macroscopic approach of the continuum mechanics, this position vector does not describe the positions of the single microscopic molecules.
If the order parameter is depending on position r, then we need to assume this dependence through the director as follow: r r r (2) In the continuum approach, the deformation of the bulk of the material is described by means of three fundamental deformations [9]. They are the deformations of splay, twist and bend. The features of them are the following. In the splay, we have div n ≠ 0. In the bend, we have that rot n is perpendicular to n. Then, in the twist deformation, rot n is parallel to n.
Oseen and Frank demonstrated that the density of the free energy given by distortions in the second order of n is represented by the three deformations: In (3), K1 is the elastic constant of splay, K2 of twist and K3 of bend. Since it is possible to have deformations of pure splay, or of twist or bend, each of these elastic constants must be positive. If it were not so, the undistorted deformation would not be that having the minimum energy.
To determine the contribution to the free energy coming from the elastic deformation, Oseen and Frank used the following approach [9]. Let us assume the z-axis along the direction of director n, the x-axis perpendicular to the director and y-axis perpendicular to x-and z-axes, oriented according the right-hand rule.
We can distinguish six elemental distortions, linked to the director variation. Therefore, the deformations of splay, bend and twist can be described by means of six derivatives: For small deformations, we have: z a y a x a n z a y a x a n n y x z 6 5 4 Then, we can write the density of the free energy coming from the distortion of the nematic according to the Hooke law, as a quadratic function of the deformation: In (8), there are six elastic parameters Ki and 36 elastic parameters Kij. Anyway, using symmetries these numbers of parameter are reduced. For instance, the cylindrical symmetry of the nematic liquid crystal implies invariance for rotation about z-axis. So the energy is invariant during such rotation. Then, because of invariance for transformation x'= y, y'=x, z'= z, we have only two independent Ki parameters (K1, K2) and five independent Kij parameters (K11 ,K22 , K33, K 24, K 12 ).
Moreover, we have the condition n = -n, which is telling that the director direction does not influence the features of the medium. In this case, it is necessary that transformation z' =-z, y' =-y, x' =-x, does not change the deformation of the threedimensional structure. So we need K1 and K13 being zero. Also the mirror symmetry exists, and then, according to transformation x' = x, y' = -y, z' = z, we find K2 and K12 being zero.
In this manner, in the elastic energy density of nematics we have just four coefficients: K11, K22, K33 and K24 . However, the term multiplied by coefficient K24 is not proper of the bulk energy, because it is a contribution to the surface energy of the material [4]. Consequently, the Frank equation for the density of the elastic energy is given by the following elemental deformations [ In fact, squaring the second term, the contribution of (K2 /K22) 2 is not relevant, because it does not depend on the deformation. The ratio q0 = K2 /K22 is the modulus of the wave-vector of the cholesteric helical structure, having pitch P0 =2π/q0. In the case 0 2  K , the equilibrium configuration of the cholesteric nematic is that having a spontaneous twisted deformation.

ni,j = ∂ni / ∂xj
Following the Oseen -Frank approach, let us being more general. If the director n does not depend on position, the liquid crystal is undistorted and the free energy density is supposed equal to a certain quantity represented by f0, which is a quantity that does not change if the liquid crystals is subjected to a deformation.
If n = n(r), the nematic is deformed. Let f being the density of elastic energy which is created in the material. We have then that ni,j=∂ni /∂xj are different from zero. Let us suppose that these derivatives of n are enough for describing the distorted nematic, and then: If these derivatives are small, it is possible to consider a series in ni,j, so that: In (129, the components of tensors E and K are given by: ; (12') These quantities are evaluated with respect the undistorted configuration, which is also that having the minimum energy. In (12), we have used Einstein notation which implies summation on a set of indices; Let us write E and K, using combinations made by n, and the Kronecker δij and Levi-Civita εijk symbols.
Since in a nematic, n and ─n are equivalent, each term in (2) must be even in n. Let us remember that, for what is concerning the Kronecker symbol, it is Let us consider tensor E. We can represent its components as: But E1 = E2 = 0, because the nematic is not polar. Moreover: Elasticity Tensors in Nematic Liquid Crystals http://www.ijSciences.com Volume 5 -July 2016 (07) This is a pseudoscalar, being the nematic helicity [11,12]. Since the energy is a scalar, n n rot  can be present in it, when this contribution is squared or if the nematic is a cholesteric nematic. In this case, the material is spontaneously showing a deformed configuration of the fundamental state having the minimum energy. For tensor K we have Kijkl=Kklij, because it is present in the term ; so we can write: In (16), we have: We used the fact that the director is a unit vector and then the derivative of its modulus is zero. (II)   The surviving terms which are contributing to the elastic energy are: We have [7]: We can change the symbols for the coefficients, to see that this expression is that of the free energy density proposed by Oseen -Frank, with the saddle-splay term too.
As previously told, K11, K22, K33 e K24 are the elastic constants of splay, twist, bend, and saddle-splay. The last term in (21), if we consider the Gauss theorem, is a contribution to the surface energy. Then, the density energy of the bulk is depending on the three elastic constants of splay, twist and bend. If we consider also the helicity, the free energy becomes: If we have not splay and bend, and only twist is present, Equation (22) is minimized by: Then, in the case we have helicity (E ≠ 0), the configuration of the director which is minimizing the energy is a distorted one. As previously told, this happens in the cholesteric liquid crystals. From now on, we assume a nematic having E = 0.

The analysis of Nehring and Saupe
Let n be the director and ni,j , ni,jk the derivatives of first and second order. Let us suppose a free energy density depending on these derivatives   Being tensor L multiplying derivatives of the second order, it must be: As we have done before, let us represent the components of tensors L, M and N, using n, δij and εijk. Let us start from L and determine its contribution to energy. Being n and ─n equivalent, the terms different from zero are: Then: We have already found the first three terms of (28), when we have discussed the tensor K. These terms can be added to those previously given. But in (28) we have a new term which is contributing to the surface term, like that having as elastic constant K24. This term is usually written with the constant K13, defined as the elastic constant of splaybend [13,14]. Supposing M = 0 and N = 0 , (24) becomes: Therefore, after renormalizing the constants: This is the free energy density of Nehring-Saupe [3]. However, we have also the terms coming from tensors M and N , to consider as sources of elastic constants, it they are different from zero. To dealt with them we need a second order analysis.

Second order analysis
We have seen in the previous section, that the free energy density, as given by Nehring and Saupe, including the term with elastic constant K13, is originated from tensors K and L. However, other terms exist that we have not yet discussed. To analyse them let us follow the approach given in [7]. In this reference, the free energy density is supposed a function of deformations as: That is, we assume as deformation sources, the firstand second-order derivatives of the director, generalizing the approach of the previous section.
The virtual variation δf of the density of the free energy f , close to equilibrium, can be described to the second order as: In (31), we find the variations of the first-and second-order derivatives of director n. In the linear theory of elasticity, in (31) we have to consider just the derivatives of the first-order, neglecting those of higher order. If the second-order is involved too, tensors λ, μ must be expanded in terms of the sources of deformation, ni,j and ni,jk. Therefore :  mp  l  jk  i  ijklmp  r  q  p  m  l  k  j  i  ijklmpqr   q  p  lm  k  j  i  ijklmpq  l  k  j  i  ijkl  jk  i  ijk   n  n  N  n  n  n  n  H   n  n  n  D  n  n  A  n  f   ,  ,  ,  ,  , Let us expand tensors in (35'), using (ni,δij,εijk). After expanding the tensors, the terms that survive are (given in covariant form, that in using divergence and rotor): If we have only ni,j as source of deformation, (35') becomes: In this case, the term with K13 does not appear, in agreement with the elastic theory of the first order as given by Oseen-Frank. Therefore, if we consider K13, and also K24, we can ask ourselves what are the other terms we need to consider too, to have the expression of f coherent to the second order approach given in [7]. In we assume (36a) and (36b), we have the wellknown splay, twist and bend distortions coming from terms due to tensors μ° and A. However, we have many terms from tensors N, D and H (many of them are null because of the parity of n and pseudoscalarity of (n · rot n)). Let us note also that some terms are coming from both N and D, or from D and H, or from the three tensors at the same time.
In the three Tables I-III given after the References we show the contributions to the density of the free energy of a nematic liquid crystal coming from N, D and H. In the first of the three tables, the terms are coming from N. In the second table we have contributions coming from D, and those marked by * are shared with N. In the third table we have the contributions to the density of the free energy coming from H (terms marked by * are shared with D, whereas those marked by ** are shared with N and D).

Second-order elasticity close to a threshold
If we use for a nematic liquid crystal the theory of the first-order elasticity, we have only three constants to deal with. Whereas, if we use the theory of the second order, we have a quite larger number of elastic constant. Then, the use of such theory turns out to be unworkable. In fact, the elasticity of the second order can become quite simple, when it is involved in a threshold phenomenon, when the nematic is changing its configuration from an undistorted to a distorted one.
Let us remember that the most common experimental condition to study the nematic is that of creating a cell composed by two plane-parallel glass slides, spaced a few microns, in which the nematic is inserted by capillarity. The two inner surfaces of the cell may have the same or different treatments, in order to induce the desired nematic configuration (planar, homeotropic or hybrid). The cell can be placed in a thermostat with appropriate optical windows for the observation under polarized microscopy of the phases or of the material configurations. Let us consider a flat deformation of the nematic, that is, one which is obtained for example in a cell with opposite conditions, homeotropic on one of the walls and planar on to the other (this is the so-called hybrid HAN-cell. We can use a frame of reference (x,z) with the origin on the wall with homeotropic anchoring, the x-axis parallel to that wall, and the z-axis perpendicular to it.  Locally, the director n is given by: In (37), Ө(z) is the angle shown in the Figure 1.
Vectors i and k are the unit vectors of axes.
If we imagine to decrease the cell thickness d, the deformed configuration becomes unstable and the cell, if we suppose that the planar anchoring is stronger than the homeotropic one, assumes the planar configuration. Or, if we assume the homeotropic anchoring stronger, the cell assumes the homeotropic configuration. The transition that occurs in the HAN cell, when we are reducing the thickness of the liquid crystal, is from a distorted configuration to the undistorted planar (P) or homeotropic (H) configuration (see Figure  2). In order to describe the occurrence of a limit for the mechanical stability of the HAN cell, due to the decrease of the cell thickness d, the free energy density must be given as a function of angle Ө. Close the threshold of the passage from the HAN configuration to the undistorted P or H configurations, we have Ө changing its value. However, the value of this angle depends on the positon of the director in the cell, that is, on z. So let us take as the parameter for studying the problem, the maximum angle assumed by the director with respect to z-axis and call it Өmax. If the planar anchoring is strong, this angle can also be of 90°. In fact, we can think the angle Ө as a function of z, and therefore as a function Ө = Өmax g(z/d).
Now, let us consider some terms from Tables I-III, and  is of the second order in Өmax, whereas the other contributions vanish more rapidly, when Өmax goes to zero. As a consequence, we have an additional term to the free energy density which can be written as: In the framework of the theory of elasticity here discussed, only K* is the new elastic constant of the second order which survives in a threshold phenomenon. In this case, the free energy density becomes:   Let us note that the first five elastic constants have dimensions of energy divided by length; the elastic constant K* is an energy times a length. Let us evaluate its order as made for (4) Tables I-III: These three Tables are giving the contributions to the density of the free energy of a nematic liquid crystal coming from tensors N, D and H, respectively. In the second Table, contributions coming from D marked by * are those shared with N. In the third Table, contributions to the free energy density coming from H, marked by *, are shared with D, whereas those marked by ** are shared with N and D.