Stresses in curved elastic membranes with in-plane order
Ordering configurations of a director field on a curved membrane induce stress. In this work, we present a theoretical framework to calculate the stress tensor and the torque as a consequence of the nematic ordering; we use the variational principle and invariance of the energy under Euclidean motions. Euler-Lagrange equations of the membrane as well as the corresponding boundary conditions also appear as natural results. The stress tensor found includes forces of attraction-repulsion between defects; likewise, defects are attracted(repelled) to patches with the same sign (opposite) of gaussian curvature. These forces are mediated by the Green function of Laplace-Beltrami operator of the surface. In addition, we find non-isotropic forces that involve derivatives of the Green function and the gaussian curvature, even in the normal direction to the membrane. We examine the case of axial membranes to analyse the spherical one. For the sphere, we find the Young-Laplace law and how the nematic texture, with defect at the poles, modifies the surface tension respect to soap bubbles. In the case of spherical cap with defect at the north pole, we find that the force is repulsive respect to the north pole, indicating that it is an unstable equilibrium point.
When extrinsic couplings of Frank’s energy describing liquid crystals on curved membranes are neglected, one finds that
defects interact with each other through the Green function of the Laplace-Beltrami operator of the surface(1); (2);
they also have interactions with the membrane itself and a bulk term appears that describes the interaction of the gaussian curvature
of the membrane mediated by the Green function.
Clearly, these interactions induce stresses along the membrane which in turn responds by modifying its shape:
the interest in determining the shape of biological membranes is in part because
it is related to specific functions of the cell(3); (4).
The distribution of stress along the membrane plays a relevant role, whether its shape can change or remain fixed.
If the shape of the membrane is frozen, the amount of topological charge is determined precisely by the topology of the membrane
through the Hopf-Poincaré and Gauss-Bonnet theorems(5); (6).
The texture of the nematic with defects on the membrane, determines how the stress is distributed on it.
As the main result of this paper, the covariant expression of the stress tensor is obtained. Since the divergence of the stress tensor vanishes on the surface, it is a conserved quantity; the distribution of stresses along the membrane is encoded in this object. The stress tensor found exhibit the forces in the nematic membrane: Defects with the same (opposite) sign attract (repel) each other; defects are attracted (repelled) to points with the same (opposite) gaussian curvature, the interaction being through the Green function of the surface. We also find non-isotropic forces that involves derivatives of the Green function and the gaussian curvature, a result that exhibits a more complex non-isotropic forces than those described above.
Using this theoretical framework, we also find the covariant Euler-Lagrange equation of the nematic energy. This equation describes the shape of the membrane that is coupled with the configuration of the director field. It is the covariant form of the von Karam equation(7), to which it is reduced when we use the Monge approach. In the calculation of deformations of the nematic energy, we have found that the tangential deformations do not imply only a boundary term, that is because this energy is not invariant under reparametrizations: the presence of the nematic texture implies elastic stresses tangent to the membrane. Moreover, when the variational principle is implemented, the boundary conditions for a free edge appear naturally. We write these conditions in terms of geometrical information of the edge curve.
As a relevant example, we obtain the stress tensor in the case of axially symmetric membranes. If the membrane is closed, we find the corresponding Young-Laplace law, which gives us the relationship with the pressure difference between inside and outside. In the spherical case and placing defect at each pole of the sphere, we find the relationship of with the radius of the membrane, the surface tension and the nematic constant .
The Green function on the sphere is obtained by using isothermal coordinates(8). The spherical case is relevant and has been studied not only from a theoretical point of view, but also experimentally(9); (10); (11); (12). Here, in the case of a spherical cap, we choose Dirichlet boundary conditions; by placing a defect in the north pole we obtain the force on a horizontal loop. We find that the total force on any point of the spherical cap is repulsive with respect to the point defect at the pole, indicating that it is an unstable equilibrium point.
The rest of the paper is organized as follows, in section II we give a briefly review to describe the Frank energy on a curved surface, in the limit of one constant. In section III we obtain the response of the energy to small deformations of the embedding function. To avoid confusion in the reading, we have separated the calculation of the normal and tangential deformation. In section IV the boundary conditions are obtained. The key point here is to project the edge deformations along the Darboux basis. By using the invariance of the energy under translations and rotations, in section V we find the stress tensor and the torque. In section VI, the case of membranes with axial symmetry is examined, and then some results for the spherical case are obtained, We finished the article with a brief summary in section VII. The most of the long calculations have been written in several appendices at the end of the paper.
Ii Nematic energy
Consider us a surface in of coordinates . The surface is parametrized by , through the embedding functions . The induced metric on the surface is given by , the euclidean inner product in of the tangent vectors to the surface. The unit normal vector to the surface is defined as where . The covariant derivative compatible with the induced metric will be denoted .
Frank’s energy describes the ordering of a director field of unit magnitude. This energy include the effect of splay, twist and bend the field along the surface. In the limit of one coupling constant the Frank energy can be written as (13)
The integral involves the infinitesimal area element on the surface , and the coupling with the extrinsic curvature has been neglected; nevertheless by using theoretical and numerical simulations methods, some recent works have taken into account extrinsic effects(14); (15); (16).
A convenient alternative route to describe this field theory, is in terms of the spin connection , a vector valued function defined in the tangent space of the surface (17), whose fundamental property is its relationship with the gaussian curvature
We define an orthonormal basis , , such that the field can be written in terms of the angle with :
Euler-Lagrange equation of the field , implies that a scalar field exists such that , where . The presence of topological defects screening by the gaussian curvature of the membrane is the source of this field:
where is the charge density. A formal solution of (5) can be written as
where denotes the Green function associated with the Laplace-Beltrami operator on the surface such that
defines the geometric potential. The energy can thus be written as
The first integral in (9) is a boundary term and the second one is the bulk term that can be developed as
From this we see that defects interact with each other through the Green function, we also see that
the geometric potential plays the role of an external electric field.
The last term is the interaction energy between the gaussian curvature mediated by the Green function.
In the next section, the shape equation and boundary conditions of the functional energy
will be obtained, is the surface tension of the membrane and the linear tension of its boundary .
Iii Shape equations and Noether charges
To find the shape equation, we obtain the response of the energy (11), to small deformations of the embedding functions, . We project the deformation into its tangential and normal to the surface
As a first step, we get from eq.(5):
Now, when the area of the surface is modified, the total defects can also be modified. Nevertheless, if the total area remains fixed, local deformations of the surface implies deformations of the charge density without further changes in the total defects. Thus, since the total charge is preserved, we have that , in such a way that locally
where we used the area deformation, .
Let us first get the the normal variation of the nematic energy. This deformation can be obtained by using the commutator where , see (18), so that we can write and deformation of the energy gets
After some algebra and several integrations by parts we have
where the Euler-Lagrange derivative of the nematic energy and the Noether charge are given by
This expression for the Noether charge has not been completed;
tangential deformation is needed and as we shall see, it is not just a boundary term.
Let’s now get the tangential deformation. For the scalar curvature we have (see appendix)
Notice that the tangential deformation is not only a boundary term, this happens because the nematic energy is not reparameterization invariant. The presence of the director field breaks out this property of the bending energy. To prove this, we see that the commutator with the laplacian is given by where now,
By using this commutator we have that and thus the tangential deformation does depend on the Green function. By using that and proceeding as in the case of the normal deformation we have
where we have identified
In order to obtain the Euler-Lagrange equation of the energy (11), we write its bulk deformation
where the Euler-Lagrange derivative
An interesting fact occurs if there are no defects on the membrane, in such a case we have that and implies that , so that the Euler-Lagrange equation simplifies to
and therefore, minimal surfaces or hyperbolic-like surfaces
are solutions to the Euler-Lagrange equation (19); (20); (21).
As we will see below, from the the Noether charge , we can find both, the stress tensor and the torque; these can be found when writing explicitly a translation and rotation of the embedding function. Before that, let us find the boundary conditions that appear naturally in the variational principle.
Iv Boundary conditions
According to the previous section, in equilibrium shapes, deformation of energy (11) including the boundary terms, is given by
and thereby the boundary conditions will be obtained by doing .
The calculation involves the Darboux basis adapted to the boundary parametrized by arc length (22). Deformation of the boundary can be projected as
where we have defined the scalar funcions and . Therefore, deformation of the unit tangent can be written as
we have the boundary conditions, see Appendix(E)
where we have used that on the boundary and and the fact that on the boundary, the independent deformations are given by the scalar functions .
V Stress and torque
If the energy is invariant under reparametrizations, then its tangential deformation is a boundary term and vanish identically; however, if the energy does not have this invariance, as in the case of the nematic energy, these terms are not trivial as we see in eq.(20).
On the other hand, invariance of energy under translations implies that , so that locally we have
where is the stress tensor.
In equilibrium, the conservation law of the stress is fulfilled and
thus , is a conserved vector field along the surface; it is identified as
the force acting on the curve parametrized by arc lenght with normal .
The tangential derivatives will be relevant when coupled with
crystalline order through the strain deformation(7); (26).
In the case of a membrane that encloses a certain volume , we must add the term to the energy, where is the pressure difference between the interior and the exterior. In that case the stress tensor is not conserved but , in such a way that
where the coefficients are given by
We have verified that the relationship (33) with the Euler-Lagrange derivatives is fulfilled, this guarantees that both, the expression for the
stress tensor and the shape equation are self-consistent.
Let , be a curve parametrized by arc length on the surface (blue curve in Fig.(1)); as before, we identify the Darboux basis adapted to it: its tangent vector (black arrow) and the outward pointing unit vector (red arrow in Fig.(1)), such that . The force per unit of length can be written as
where , , and . We get
Note that includes . This force can be written explicitly
The second term is the force on the charge due to , it is given by , this force is repulsive(attractive) between defects with the same (opposite) sign charge. Similarly, the third term is the force on the point (of gaussian curvature ), caused by the presence of at the point : defects are attracted(repelled) to points with the same (opposite) sign of gaussian curvature. These interactions are mediated by the Green function. The fourth term is a self-force at the point with the gaussian curvature at the same point.
The total force along includes the anisotropic stress , along and . Finally, there is also a force along the unit normal to the surface as given in eq.(38). None of these forces has been reported so far.
Taking now an infinitesimal rotation , we have that and . Therefore, we can write
Similarly we have
where now . Deformation of the energy under a rotation is then given by (24)
being the stress tensor (35), and
In equilibrium we have so that is conserved as a consequence of invariance under rotations. The first term in eq.(43) is the orbital torque while can be seen as an intrinsic torque. If we use the fact that , then we obtain the intrinsic torque in the Darboux basis along a curve on the membrane.
Vi Axial nematic membranes
Let us see the case of axial surfaces parametrized as
where is a unit radial vector field, and . The tangent vectors to the surface can be found to be
where is the unit azimuthal vector and denotes derivative respect to . The induced metric on the surface can be written as
where we have taken the parameter along the meridians to be the arc length such that . The unit normal to the surface , is given by
The second fundamental form can be written as
whereas the mean curvature and the gaussian curvature, Let and be the unit basis so that the components of the spin connection are given by and . Along a horizontal curve we have and so that in these coordinates the coefficients (37) of the force per unit length on a horizontal loop can also be written as
where we have
We note that although by the axial symmetry, in a general setting, the presence of the nematic texture implies that the coefficients depend on both variables on the surface, through the function . This force has radial and vertical components. The total vertical force on the loop is then
where we have denoted . If the membrane is a closed surface, we must to take into account the pressure difference between the inside and outside to the nematic membrane. The equation (34) is then
where we have taken . This equation must be satisfied for each value of in the domain considered; it is the corresponding Young-Laplace law.
vi.1 Spherical particles
Without nematic texture in the membrane such that and , eq.(53) reduces to . By using that , and taking the simplest case such that is a constant we obtain
which is the representation of a sphere with radius , an expression known as the Young-Laplace equation, which relates the surface tension , the pressure and the radius of the sphere . Let’s see the equivalent result in the presence of the nematic texture. To this, write the metric in isothermal coordinates
That is, . Let and and write the Green function that satisfies the equation
replacing with isothermal coordinates , gets into
The last equality in eq.(58) implies the Green function in isothermal coordinates
If the surface is closed, the singularities that appear into the Green function can be eliminated if we subtract both and . Let us look explicitly the example of the sphere; parametrize it as
where . If we choose then we have
and we can obtain
where refers to the larger value between and . The Green function can then be written as
Therefore, as shown in appendix (F), the geometric potential is simply given by Thus, with a charge at each pole, the function can be written as
Now, since eq.(53) is fulfilled for , where is related with the core of defects, it can be rewritten as
where is a fixed number , that is obtained from
where the dot means derivative respect to . If we identify the effective surface tension as
then eq.(65) can be written as the corresponding Young-Laplace law . We see that the surface tension has been modified by the presence of the nematic texture with defects at the poles.
Some recent results of numerical simulations show for water whereas experimentally it has been found that for certain liquid crystals(27),while for those liquid crystals have been measured . For these values we find that for the nematic-coated sphere .
In the case of a spherical cap , Gauss-Bonnet implies that
where is the gaussian curvature of the boundary curve parametrized by arc length . The sum of these integrals is equivalent to the charge of defects into the surface. Integration of the gaussian curvature gives
If the boundary is the parallel , then we find
and therefore, the total charge on the spherical cap is given by
For a half sphere , we have , in such a case, the boundary is a geodesic curve with ; a cap with as boundary point, has a nematic texture with . If , then we have . Notice that if and there is not such that . Two of these caps with their nematic texture are shown in fig.(4)
For each of these spherical shells the Green function is given by eq.(59) while by eq.(61), but now we must to impose boundary conditions on the Green function at . Under Dirichlet boundary conditions it reads
where . After making some integrations we can find the geometric potential as
For a half spherical cap, and , and thus we get
If the boundary is at the point , we obtain the geometric potential as
Fig.(5) shows these geometric potentials: in order to minimize the energy, defects must to be at ; nevertheless as we shall see, it is an unstable equilibrium point. For a half sphere such that