Dimension reduction for the Landau-de Gennes model on curved nematic thin films
We use the method of -convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach in  where we considered a similar problem for a planar surface. Since the anchoring energy dominates when the thickness of the film is small, it is essential to understand its influence on the structure of the minimizers of the limiting energy. In particular, the anchoring energy dictates the class of admissible competitors and the structure of the limiting problem. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and strong Dirichlet boundary conditions on the lateral boundary of the film when the surface is not closed. We establish a general convergence result to an energy defined on the surface that involves a somewhat surprising remnant of the normal component of the tensor gradient. Then we exhibit one effect of curvature through an analysis of the behavior of minimizers to the limiting problem when the substrate is a frustum.
In this paper we expand our analysis of thin nematic liquid crystalline films, initiated in  for planar films, to include the setting of general smooth surfaces. The focus of the present work is on rigorous dimensional reduction of the Landau-de Gennes -tensor model to its surface analog, in particular, to justify asymptotic arguments in  (see also ). The Landau-de Gennes theory is based on the -tensor order parameter field that is related to the second moment of the local orientational probability distribution. The relevant variational model involves minimization of an energy functional consisting of elastic, bulk and weak anchoring surface contributions. The significance of weak anchoring energy terms within both -tensor and director theories has been highlighted in numerous recent contributions, including for example, [4, 5, 6, 7, 8, 9].
Having already established in  the dimension reduction for a planar film, we now wish to explore the possible influence of curvature on the limiting energy in the thin film limit. To achieve this goal we use the theory of -convergence that has proved successful in tackling problems of dimension reduction in other settings, such as elasticity  and Ginzburg-Landau theory .
In Section 2 we define the full three-dimensional energy, perform non-dimensionalization, and review some elementary facts from calculus on surfaces. In Section 3 we prove -convergence to a limiting energy , cf. Theorem 3.1. One feature of the -limit derived in Section 3 is that it includes within its definition a minimum of a certain scalar function defined over the set of traceless symmetric tensors. This minimization arises as a sort of remnant of the normal component of the -tensor gradient. In Section 4, we carry out this minimization thereby obtaining an explicit formula for the -limit. The formula demonstrates that the limiting energy density contains a number of previously unreported elastic “strange” terms coupling the surface gradient of the -tensor to the normal to the surface. Next, as an example, in Section 5 we compute the expression for the limiting energy in the geometry of a surface of revolution. Specializing further in Section 6, we analyze the case of a frustum. We discover a dichotomy between the behavior of minimizers for broad and for narrow cones when the nematic coherence length is small. When the angle of the frustum is small, the director field tends to follow the generators of the frustum. However, the director deviates from such a path significantly when the angle broadens and eventually approaches a constant state. As the result, we observe that the degree of the director along the boundary components depends on the angle of the frustum.
2 Statement of the problem
2.1 The -tensor
In the three-dimensional setting, one describes a nematic liquid crystal by a -tensor which takes the form of a symmetric, traceless matrix. Here models the second moment of the orientational distribution of the rod-like molecules near . The tensor has three real eigenvalues satisfying and a mutually orthonormal eigenframe . We refer the reader to  for more details but below we summarize the key elements of this theory that we will utilize.
Suppose that Then the liquid crystal is in a uniaxial nematic state and
where is the uniaxial nematic order parameter and is the nematic director and
If there are no repeated eigenvalues, the liquid crystal is in a biaxial nematic state and
where and are biaxial order parameters. Note that uniaxiality can also be described in terms of and , that is one of the following three cases occurs: but , but or When so that the nematic liquid crystal is said to be in an isotropic state associated, for instance, with a high temperature regime.
2.2 Geometry of the Domain
We will use to denote a point in .
We let denote a bounded, two-dimensional, orientable manifold embedded in , either closed or with smooth boundary, and we let denote a point on . Fixing an orientation, we write for the unit normal and due to the smoothness of we have that the mapping is on the compact set . It follows from the inverse function theorem that for some sufficiently small positive number , the map is one-to-one on
With this observation in hand, for we shall assume the nematic film occupies a thin neighborhood of given by
and we can unambiguously express each point in the form
for some unique pair and .
We will also set
2.3 Landau-de Gennes model
We assume that the bulk elastic energy density of a nematic liquid crystal is given by
and that the bulk Landau-de Gennes energy density is
cf. . Here is the -th column of the matrix and is the dot product of two matrices Further, the coefficient is temperature-dependent and in particular is negative for sufficiently low temperatures, and . One readily checks that the form (7) of this potential implies that in fact depends only on the eigenvalues of , and due to the trace-free condition, therefore depends only on two eigenvalues. Equivalently, one can view as a function of the two degrees of orientation and appearing in (2). Furthermore, its form guarantees that the isotropic state (or equivalently ) yields a global minimum at high temperatures while a uniaxial state of the form (1) where either or gives the minimum when temperature (i.e. the parameter ) is reduced below a certain critical value, cf. [4, 12]. In this paper we fix the temperature to be low enough so that the minimizers of are uniaxial. We also remark for future use that is bounded from below and can be made nonnegative by adding an appropriate constant. In light of this, we will henceforth assume a minimum value of zero for
We now turn to the behavior of the nematic on the boundary of the sample. Here two alternatives are possible. First, the Dirichlet boundary conditions on are referred to as strong anchoring conditions in the physics literature: they impose specific preferred orientations on nematic molecules on surfaces bounding the liquid crystal. In the sequel we impose these conditions on the lateral part of the film whenever is not closed. An alternative is to specify the anchoring energy on the boundary of the sample; then orientations of the molecules on the boundary are determined as a part of the minimization procedure. We adopt this approach, referred to as weak anchoring, on the top and the bottom surfaces of the film. Following the discussion in Section 3 of , we assume that, up to an additive constant, the anchoring energy has the form
for any and , where , , and
This form of the anchoring energy requires that a minimizer of has as an eigenvector with corresponding eigenvalue equal to . From (3) it follows that . An alternative approach would be to extend the anchoring energy by including quartic terms  and even surface derivative terms .
Here represents surface measure, i.e. two-dimensional Hausdorff measure.
Again, we will consider surfaces that are either closed or have a smooth boundary. In the case when the boundary is nonempty, we set and for given uniaxial data we prescribe the lateral boundary condition of the form
Note that we assume that the boundary data does not vary in the direction normal to the surface . Some additional conditions on will be imposed later on in the text, cf. (32).
The admissible class of tensor-valued functions is then lying in the Sobolev space with , where is the set of three-by-three symmetric traceless matrices defined in (9). Throughout this work we assume that is uniaxial and is taken so that this set of admissible tensors is nonempty.
We non-dimensionalize the problem by scaling the spatial coordinates
where . Set and and introduce the small non-dimensional parameter representing the aspect ratio between the film thickness and the diameter of the closed surface. Then we define the non-dimensionalized elastic energy density and Landau-de Gennes potential by setting
respectively. Here the parameters and are all non-dimensional.
Finally, turning to the surface energy we let , and setting
we obtain an expression for the non-dimensionalized surface energy of the form
Now for convenience we drop all of the tildes and conclude that the total dimensionless energy is
Here the rescaled domain, denoted by , is given by
where now denotes the rescaled surface of diameter one.
2.5 -tensors on a fixed domain; Surface gradients and divergences
With an eye towards eventually passing to the limit via -convergence, we now find it convenient to re-express the -tensors, their gradients and their divergences in terms of tensors defined on the fixed domain rather than . To this end, we first recall some basic identities for the surface gradient and surface divergence, for which a good reference is , Chapter 2. For any scalar-valued function defined on we henceforth associate to it a function defined on via the formula
Then, for points and related via we readily compute that
and for any unit tangent vector to at we can calculate the directional derivative as
Hence, denoting by an orthonormal basis for the local tangent plane to at and invoking (19) and the summation convention on repeated indices we find that the surface gradient is given by
where we recognize as the shape operator. Consequently, if we introduce the matrix-valued mapping via the formula
then the previous calculation yields
Further, if we expand in as
and we use the properties resulting from the condition one sees that
We then obtain a corresponding formula for the divergence of in terms of and derivatives:
where is an orthonormal basis in . Defining the surface divergence of a vector field by , we again expand using (24) to find
3 -convergence to a surface energy defined on
In this section we pass to the limit in the energy given by (16). For convenience, we assume that an appropriate constant has been added to the Landau-de Gennes energy to guarantee that . Here we are assuming that the elastic constants satisfy the conditions stated in Theorem 3.1 that ensure the coercivity of . We wish to consider a range of asymptotic regimes corresponding to different magnitudes of and in the surface energy density given by (14). To this end, we will assume that and for some nonnegative constants . Then (14) can be written as
Here, we can assume that . Indeed, as will become evident later on, asymptotically vanishes at leading order in the thin film limit. Thus, if for instance, , the first term in is not present and we may take .
Note that we would like to capture the asymptotic behavior of for the range of parameter values, even when some of the material constants have magnitudes comparable to thickness. Mathematically, it then appears that these constants vary with thickness, even though this is not the case physically.
Next we define the spaces
for some uniaxial boundary data such that the set is nonempty. In the case where , this boundary condition is not present in these two definitions.
Now we are ready to define our candidate for the -limit of the sequence . We let be given by
where, recalling (12) we define
Here, as will become apparent later on, arises as a remnant of the normal component of the gradient of .
We wish to point out an omission in Theorem 5.1 of  where the -limit should have also been defined using (34). When this theorem is true as stated. Otherwise, the statement and the proof should be modified as in this paper. We note that the parameter studies in Section 6 of  are unaffected as they are conducted in the regime .
In order to phrase our -convergence result we must deal with the issue that and are defined on very different spaces, a situation common to dimension-reduction analyses involving -convergence. To address this, we recall the association introduced earlier between any mapping, say , defined on and the mapping defined on , cf. (18). Then we will define the topology of the -convergence as weak -convergence in the following sense:
for any sequence (cf. (17)) and any limit
We now state our main theorem on dimension reduction via -convergence. For those unfamiliar with the notion, we refer, for example, to .
we find that the energy can be written in terms of as follows:
First, we demonstrate how one can choose a recovery sequence. If , so that either or else on a set of positive measure on , then choosing such that , from (36) we readily check that
Notice in particular that -derivatives enter at and contributes at in the energy. If, on the other hand, , then of course all -derivatives drop in (36), as do the terms from in the last integral and in the limit, one immediately arrives at
Now given any , we set
where solves (34) with playing the role of . One technicality we must confront with this proposed recovery sequence, however, is that since and since from (34) we see that depends on (cf. (93)), we are in the position of taking second derivatives of the tensor . Let us first establish the success of this recovery sequence under the additional assumption that is smoother than , say , and then we will treat the more general case at the end of the argument through a mollification procedure.
Note also that when the surface is not closed, this proposed recovery sequence must be modified via multiplication by a cutoff function so as to maintain -valued boundary data. As long as the width of the boundary layer associated with the cutoff function is of order lower than , say , the contribution to the energy of that layer will be negligible. We will leave out the details of this alteration.
Now consider the expression (36), evaluated using (37) as the proposed recovery sequence. Clearly, the Landau-de Gennes contribution to trivially converges to its limiting value and we only need to establish convergence of elastic and surface contributions. Next we observe that the elastic energy along the recovery sequence approaches
Turning our attention to the surface energy term, we have from (36) that the energy contribution due to is given by
on , the integral in (39) approaches zero. Thus the limiting contribution to the surface energy is simply
We conclude that the energy of the recovery sequence given by (37) approaches the -limit .
It remains for us to construct a recovery sequence in the general case where is in but no smoother. An obvious approach is to mollify but this mollification must be done with some care. Recall that in addition to satisfying the boundary data , the tensor is required to satisfy the condition on . Simply convolving with a standard mollifier will clearly violate both of these requirements. Maintaining the boundary condition can be handled simply enough through the straight-forward use of a smooth interpolation in a boundary layer, just as we described above for adjusting the tensor near the boundary. However, obtaining a smoother version of that still gives zero contribution to the leading order surface density is not as immediate. Recall, for example, that if the constants and in the definition of are both positive then admissible tensors must maintain the normal vector to as an eigenvector with corresponding eigenvalue at each on the curved surface .
To this end, we partition into finitely many smooth pieces, so that say, and on each piece we introduce a smoothly varying orthonormal frame where is an orthonormal frame in a plane tangent to at a given point. Such a smooth frame will not exist globally on if for instance is a topological sphere, hence the need for the decomposition. Then in the case , for example, we can introduce the scalar quantities and on each by expressing as
so that relative to this orthonormal basis one has a representation of on given by
This is a change of variables invoked, for example, in , motivated by simulations in . In this way, the tensor is characterized by just and and by mollifying these two quantities on each patch we obtain a smooth approximation to the original on that patch, maintaining the desired conditions that is always an eigenvector with corresponding eigenvalue . Now using the partition of unity to glue together the smooth approximation on individual patches and employing the fact that all of these approximations have the common eigenpair , we arrive at a global smooth approximation of in .
If, to describe another possibility, one is working in the case where but so that must be an eigenvector but the corresponding eigenvalue is free, one can again use the representation (40)-(41) but the constant is replaced by a third scalar unknown, say . Again mollification of , and produces a smooth approximation to on each that preserves the condition
Denoting the mollification of the original tensor by the smooth sequence with denoting the mollification parameter, the previously presented argument goes to show that the sequence of tensors defined on characterized by
satisfies the required property of a recovery sequence, namely
Here minimizes (34) for . Since the proposed -limit is clearly continuous under -convergence and since in , we have that , and so the existence of a recovery sequence for follows by a standard diagonalization argument applied to .
For the lower semicontinuity part of -convergence, consider an arbitrary sequence such that in for some Clearly we may assume
and so from (36), collecting the leading order terms, it is apparent that necessarily
Thus, only. Similarly, from the strong convergence of traces under weak -convergence, the last integral in (36) will only stay finite in the limit if as well. Hence, we may assume that . Furthermore, by (42) we also have that up to a subsequence, as for some in . It follows from the assumption in that
It has been established in (, Lemma 4.2) and  that when the elastic constants satisfy the conditions , and , then the elastic energy density is convex and consequently weakly lower semicontinuous in . Hence, using (34) and (36), we obtain