1.3 Outline of thesis
2.1.1 Oseen-Frank static theory
The uniaxiality assumption results in rotational symmetry. The assumption is commonly made as it simplifies the analysis and contains the most important class of liquid crystals. If we assume that once the average molecular alignment is known at a point x, it varies slowly from point to point, then we can describe the response of a liquid crystal to deformations using continuum elasticity theory. A common starting point for the continuum description for nematic liquid crystals is to assume that there is a free energy density,w, representing the locally stored
energy associated with distortions of the uniform equilibrium alignment of the director. This takes the form
w=w(n,∇n).
Fluctuations in the director field nare mainly due to thermodynamical forces caused by elastic deformations in the form of bending, twisting and splay. These effects are accounted for in the famous Oseen-Frank energy functional as we will see. The total elastic/potential energy in a sample volume V ⊆Ω of the liquid crystal is then given by the elastic energy functional
W(n) :=Z
V
w(n(~x, t),∇n(~x, t))d~x. (2.1) A liquid crystal in a completely relaxed configuration, i.e., in the absence of forces is said to be in natural orientation. The elastic energy is determined up to an arbitrary constant c∈R.
We choose this constant in such a way, that the elastic energy density is zero in a natural orientation. The elastic energy density w(·,·) is minimal in such an orientation, as a physical system takes on equilibrium positions where the potential energy is minimized. Therefore we require wto be a positive semidefinite function, that is,
w(n,∇n)≥0, (2.2)
for all possible molecular alignmentsn∈S2. Moreover as nematic liquid crystals lack polarity, we have the invariancen→ −n, hence we require
w(n,∇n) =w(−n,−∇n), (2.3)
i.e.,w is an even function. Moreover we have a third constraint which needs to be satisfied by the free energy density. We require frame indifference, that is the free energy density must be the same when described in any two reference frames. This is more commonly referred to as Galilean invariance. One can also separate into a constraint of material symmetry, such thatw is required to satisfy 4 conditions as in [HP18]. Either way we require that
w(n,∇n) =w(Qn, Q∇nQT) (2.4)
for any orthogonal matrix with det(Q) =±1.
In a given microscopic region of a liquid crystal there is a preferred axis along which molecules orient themselves as shown in Figure 4. We want to determine how much energy it will take to deform this orientation. We assume that the free energy density is a quadratic function of the curvature strains that can occur. This leads to the so-called Oseen-Frank energy. Deformations relative to orientations of molecules away from equilibrium positions are called curvature strains, while the restoring forces which arise to oppose the deformations, are called curvature stresses.
The curvature strains can be mainly split into splay, twist, and bend strains as mentioned. The different geometrical effects these have are shown in Figure 5. Splay is strain that causes a fan-shaped outspreading of the molecules from the original direction, bending is a change in the molecular direction, while twisting corresponds to a rotation of the director in a plane parallel to the rotation axis. The Oseen-Frank free energy density for nematics and cholestrics, takes the form
w(n,∇n) =α
2 (∇ ·n)2
| {z }
Splay
+β
2 (n· ∇ ×n)2
| {z }
Twist
+γ
2 |n×(∇ ×n)|2
| {z }
Bend
+(β+η) 2
(Tr(∇n)2−(∇ ·n)2
| {z }
Saddle-Splay
. (2.5)
Hereα, βandγare coefficients which correlate to the splay, twist, and bend of the director field, respectively. The coefficients α, β, γ, andη sometimes go under the name Frank’s elasticity coefficients in the literature. Their values are tabulated for many liquid crystals. We say the
∇ ·n6= 0 α
(a)Illustration of splay
n× ∇ ×n6= 0 γ
(b)Illustration of bending
∇ · ∇ ×n6= 0 β
(c)Illustration of twist
Figure 5: The three types of elastic distortions (curvature strains) of the director field considered in the Oseen-Frank density.
motion consist of pure splay waves if the term involving the splay is the only nonzero term, and similar for twist and bend.
Considering in particular the integral of the last term and rewriting it using the identity (Tr(∇n)2)−(∇ ·n)2=∇ ·
(∇n)n−(∇ ·n)n
, it becomes, by applying the divergence theorem,
(β+η)Z
Ω
∇ ·
(∇n)n−(∇ ·n)n
dx=Z
∂Ω
(∇n)n−(·∇)n
·ν0dx,
and we see its value only depends on the director field at the boundary ∂Ω. Hereν0 is the outward pointing unit normal on the boundary. Hence if the tracen|∂Ω is predescribed this term can be completely neglected, as this corresponds to a null Lagrangian. That means that the corresponding Euler-Lagrange equations for the functional consisting solely of this term vanish identically, so it yields no contribution to the Euler-Lagrange equations. The form taken by the Oseen-Frank free energy density is not arbitrary, it has been shown as noted in [HP18]
that if one restricts the free energy densitywto at most be a quadratic function of the gradient
∇nandn∈C1(Ω;S2), thenwcan only be frame indifferent if it takes on the particular form (2.5).
Now if we impose the nonnegativity constraint (2.2) on the Oseen-Frank free energy (2.5), we can derive the Ericksen inequalities stating that
α≥0, β≥0, γ≥0, β ≥ |η|, 2α≥β+η≥0.
These are necessary and sufficient to ensure a lower bound on the free energy density, in particular we have that w(n,∇n)≥c0|∇n|2 for all director fieldsnand some positive constant c0>0. If the elasticity coefficients are unknown one often resorts to the so-called one constant approximation. Under very special physical situations it may occur that the elasticity is isotropic, then this one-constant approximation reflects the physical situation. In general however the elasticity coefficients differ significantly. Either way one sets the coefficient for the splay, twist, and bend to be equal and η= 0. Combining this with the identity
|∇n|2= (∇ ·n)2+ (n· ∇ ×n)2+|n×(∇ ×n)|2+ ([Tr(∇n)]2−(∇ ·n)2), one obtains that the total free energy within the liquid crystal reduces to
W(n) =α 2 Z
Ω
|∇n|2dx,
which is the energy functional for harmonic maps, i.e., the common Dirichlet energy. It is independent ofnand only depends on the gradient.
A well-known principle from physics is the principle of minimum energy. This is essentially a restatement of the second law of thermodynamics, and states that for a closed system, the
internal energy will decrease and approach a minimum at equilibrium. Therefore it is of interest to consider the minimization of the energy functional (2.1). That is, one considers the following minimization problem
minn W(n) =Z
Ω
w(n,∇n)dx, n|∂Ω=nD,
subject to the constraint|n|= 1, wherenDis some given boundary data for the director field. A review of some results are found in [Bal17]. Here we assume that the director is known a priori at the boundary. This is called strong anchoring. We have neglected other common external forces such as electric and magnetic fields, which are present in common applications of liquid crystals. The most common modern application of liquid crystals is in LCD (liquid-crystal display) screens. In such cases strong anchoring is often inappropriate, since strong applied electromagnetic fields result in torques which typically overcome the boundary anchoring. Then one needs to introduce weak anchoring in the form of a penalty term of the free energy at the boundary. The total free energy functional then takes the form
Wˆ[n] =Z
Ω
WOF+WE+WM
dx+Z
∂Ω
WPds, (2.6)
whereWOF is the Oseen-Frank energy,WE andWM are the bulk energy density due to electric and magnetic fields, respectively, andWP is the penalty term. A nice account for these effects is given in [Aur15]. One looks for equilibrium solutions for the director fieldnby looking for stationary points of the functional (2.6). This amounts to considering variations of the director configuration of the form
n=n+φ,
for a smooth vector φand a small parameter. This vector is chosen in such a way, that if we have strong anchoring, n satisfies the boundary condition for any . For a given φ, n yields a path of configurations of the director parameterized by. We saynis an equilibrium configuration with respect to the functional if the first variation vanishes for all smooth vectors φ. That is, if
∂
∂Wˆ[n+φ]|=0= 0, for allφ.