Forces on membranes with in-plane order

An ordered configuration of a vector field on an elastic curved surface induces a long-range interaction with the Gaussian curvature of the surface through the Green function of the Laplace–Beltrami operator of the surface. This in-plane orientational order configuration gives rise to a stress inducing a competition between this effective interaction and the surface tension. In this work we obtain the shape equation of the model from the surface tension term, that is a simpler way than those previously presented in the literature. We show that the stress and therefore the induced force, is a vector field tangent to the surface whose magnitude is given by the stiffness constant times the gaussian curvature of the membrane. We found that the shape of the membrane can be either a minimal surface, a developed surface or a hyperbolic one, depending on the particular regime. Explicit results are shown in the case of the catenoid and the helicoid. Finally, we find the self-adjoint stability operator of the model and obtain its eigenvalues for the nematic catenoid.


Introduction
Oriented configurations of vector fields on surfaces and its consequences have been of great interest over the years [1,2]. This is due to vector fields over surfaces are models of two-dimensional systems such as liquid crystals, superfluids, and fluid membranes [3][4][5][6]. Although the description of thermal fluctuations is interesting, there is also interest in elucidating the limit of these phenomena in the case of zero temperature. For instance, some classical configurations of fields are only possible if the surface is curved, as for the well known phenomenon of geometric frustration [7]. In order to reduce the energy cost of this geometric frustration, the field induces topological defects in its configuration. The presence of these topological defects has the effect of shielding the Gaussian curvature of the surface. There is a force on the surface that appears as a consequence of the curvature of the membrane and causes that the surface acquires between other morphologies, the shape of cylindrical micelles, of tetrahedral and ellipsoidal vesicles [8].
Fluctuations in membranes with hexatic order were considered in the classical work by Nelson and Peliti [9]. In [10], research about equilibrium of nematic membranes was considered. In [11] authors deal with alternative forms of free energies for nematic membranes. Hexatic undulations modes in curved geometries have been studied in [12]. Patterns on curved surfaces of lamellar structures have been studied in [13]. In [14] authors apply conformal mapping to analyze forces on curved surfaces. In [16] several very interesting topological results about non-orientable nematics membranes were presented. Recently, in [15] the authors provide explicit formulas for the curvature associated with splay and bend fields in two-dimensions and relate them with a geometric frustration problem.
In these models the energy associated with ordered fields can be split into two parts. The first part is a bulk term that involves Gaussian curvature mediated by the Green function of the Laplace-Beltrami operator on the surface. The second term involves topological defects and also depends on the global shape of the surface through the geometric potential evaluated at the point defects [7]. Far from the points where topological defects are located, the in-plane orientational order induce a force that goes against the elastic forces induced by the surface itself, either surface tension or elastic bending forces.
In this work, we obtain the shape equation of the membrane  s k + = ( ) K 0, which was previously obtained in [17]. σ is the surface tension, κ the stiffness constant, K and  the mean and scalar curvature of the surface respectively. Only normal deformations of the surface play a role, this is a consequence of the energy invariance under surface reparameterizations. Thus, the effect of the tangential deformations will be evident in the boundary terms Minimal surfaces K=0 remain as solutions of the shape equations if the Canham-Helfrich bending energy is added but if neglected, the corresponding solutions are hyperbolic-like surfaces  s k = -, [17,18]. In the hyperbolic case, the force coming from the in-plane order cancels out at every point on the membrane with the force coming from the surface tension. In the in-plane ordering regime we also found that developed surfaces  = 0, are solutions of the shape equations as well.
From the boundary terms, through translational invariance of the energy we can identify the conserved Noether charge [20]. We obtain that the associated stress can be written as A term proportional to the scalar curvature appears. We note that there is no component of the stress in the outward direction to the membrane, which is what one expects from purely intrinsic energy. Hence, in the case of hyperbolic-like membranes the effective surface tension vanishes. On minimal surfaces the scalar curvature is not a constant and the behavior of the stress does depend on the point where we evaluate. In a generic model, rotational symmetry generates an orbital and also an intrinsic torque. In such a case, as there is no component in the normal direction of the stress tensor, we do not have intrinsic torque either. In order to illustrate the results both the force and torque on the catenoid and the helicoid are studied.
The paper is organized as follows, in section 2 we give a brief introduction to the geometric description of curved surfaces. We also give a short presentation of the in-plane ordering energy starting with the Frank energy for liquid crystals on a curved surface. Although this information has been included in several previous works, this will be useful to establish definitions and notation along the paper. In section 3 we obtain the response of the energy to infinitesimal deformations of the embedding function. Then we identify the Euler-Lagrange derivative as well as the boundary terms The in-plane energy is written in isothermal coordinates so that the information of the curvature and its deformations is encoded in the conformal factor. The stress tensor and the torque are obtained in section 4. As a classical example of a minimal surface, in section 5 we present some results on a catenoid and the helicoid is briefly considered. The in-plane force takes its maximum value at the waist of the catenoid and decrease as we move away. Finally, we obtain the stability operator of the energy in section 6. In section 7 summary and some conclusions are presented.

Geometric preliminaries
Let us consider a two-dimensional surface embedded in the three-dimensional euclidean space 3 . We can write an explicit equation for the embedding as . Therefore, these three vectors form a right-hand base as can be seen in where K ab is a second order symmetric tensor called the extrinsic curvature of the surface (also known as the second fundamental form), G bc a are the Christoffel symbols needed to define the covariant derivative ∇ a compatible with the induced metric such that  = g 0 a bc . Using this derivative the Gauss equation can be rewritten simply as  = -K e n a b ab . As a consequence of the integrability conditions the extrinsic and intrinsic curvature of the surface are related by the well-known Gauss-Codazzi-Mainardi equations [21] The inverse of the induced metric g ab is used in these equations to rise tangential indices, then ac cb and so on. Also = K g K ab ab is the mean curvature of the surface and   = g ab ab is the scalar curvature. Thus, we can then obtain the scalar version of equation (2) as The Ricci tensor of the surface   = ab d adb is a contraction of the Riemann tensor defined as In the case of two-dimensional surfaces we know that   = g 2 ab ab . With the covariant derivative we can write the Laplace-Beltrami operator of the surface as follows Instead of using generalized coordinates x a we can parameterize the surface with the so-called isothermal coordinatesu a . The advantage of this coordinates is that the metric is conformally flat where w is the conformal factor. In such coordinates the mean curvature can be conveniently written as where ∇ 2 is the euclidean laplacian in the isothermal u a coordinates, and the Laplace-Beltrami operator ∇ 2 g is just  w 1 2 in these coordinates. It is not difficult to write the curvature scalar as . The splay and bending energy of the field are modeled by the Frank energy which on two-dimensional curved surfaces is given by Here the field m a describes the ordering direction of the nematic phase. When the coefficients satisfy k = = K K 1 3 , equation (10) simplifies to is the area element. On the other hand, the energy density  can be expanded in terms of the extrinsic curvature as follows [6] where the Gauss equation was used. The first term in equation (12) is a contribution intrinsic to the surface, while the second one has a quadratic coupling with the extrinsic curvature K ab of the surface. By using equation (2) we can write In the second line of equation (13) we used the fact that the Einstein tensor  , for two-dimensional surfaces. When integrated, the last term is a boundary term consequence of the Gauss-Bonnet theorem. When the surface is a sphere, we note that only the intrinsic energy becomes relevant since the extrinsic energy is a contribution to the surface tension. Some extrinsic effects in the Frank energy have also been studied [19,22,28].
If the direction of m a is along the normal vector to the layers then the second term in equation (10)  . That is, the layers are normal to the geodesic of the surface. The extrinsic energy density can be identified as n , where k n is the normal curvature of the geodesics [21]. It has a minimum along asymptotic curves on the surface where k = 0 n . On a cylindrical surface, for instance, this energy vanishes if the director is aligned with the cylinder's axis.
This field theory can be alternatively described in terms of the spin connection W = W e a a which is a vector function defined in the tangent space of the surface [23], and is related with gaussian curvature as follows By using the orthonormal frame  a with a = { } 1, 2 , the field m is written in terms of the angle Θ with  1 as sin . 15 1 2 The intrinsic energy can be written as a function of this angle For a cylindrical surface this energy reaches a minimal value for Q = Const., as horizontal loops.
As it is well known along in-plane ordering configuration, this energy can be written as a non-local interaction of the Gaussian curvature mediated by the Green function of the laplacian defined on the surface [6]. By adding this to the bending energy of the membrane and taking into account the topological charges q i , the energy is given by is the Green function of the operator on the surface. Without the topological charges this energy is known as the Liouville field theory in quantum gravity [24]. In the next section we obtain the infinitesimal deformations of this energy (neglecting the bending energy) and identify the shape equations.

Shape equation and boundary conditions
To find the shape equations we track the response of the energy in equation (17) (neglecting the bending energy) to small deformations of the embedding functions, d  + X X X. We project the deformation in the Gauss frame as [27] d Under this deformation one can calculate the corresponding deformation of each geometric function defined on the surface. Particularly relevant are the perpendicular deformations that have been calculated in detail [27]. The deformation of the induced metric is With this result the area element of the surface deforms as follows d a a In isothermal coordinates we have that the deformation of the conformal factor is given by d The tangential deformation of the conformal factor is identified as whereas the normal deformation of the conformal factor can be written as We can now find the deformations of the scalar curvature. From its definition equation (9), we have and thus we can get With these elements we can find the shape equations of the model. Write the deformation of the in-plane order energy has been introduced. A technical problem for this calculation is to find the deformation of Green function. One way is to use the switch with the Laplacian to pass the deformation to the Green function. This will be addressed in an upcoming note. Here we use an idea based in a recent work by Giomi [17]. Let us write the energy in isothermal coordinates such that the area element turns to = A w u d d 2 , then for dF we have In these coordinates we can write such that d r ( ) w involves only deformations in equation (23). We then obtain the normal deformation of the energy that includes the surface tension as where the Euler-Lagrange derivative is given by far from the topological defect positions  r = , as was anticipated in [17]. From equation (29), we see that minimal surfaces K=0 and hyperbolic-like surfaces s k = -, are solutions of the shape equations  s k + = ( ) K 0, [17]. Different types of solutions appear by considering the regime where the surface tension is negligible. In such a case developable surfaces also appear as solutions. As a consequence of invariance under translations the tangential deformation of the energy is a boundary term. To see this explicitly we write In the first term we can integrate by parts to obtain ò ò ò s kr s kr k r Due to the relation we can write the second integral in equation (31) as For the case of a membrane without a boundary these terms vanish. Note that S 2 a involves the deformation of the membrane's area, that is This means that if the mass density of the membrane remains constant under deformations, then F +  F = K 0 a a , even at the boundary. In the presence of a boundary we can add the term where s b is the linear tension of the membrane's boundary. Deformation of the boundary term is found to be where k g is the geodesic curvature of the boundary and k = K t t

Stress
Let us take an infinitesimal translation on the energy in equation (34). Then d = X awhich implies that F = · a n and F = · g a e and therefore the only contribution comes from the S 1 a term. We have where the stress is given by the vector field s kr = - In this manner we can identify the force acting on the boundary C of the patch Σ, [20]: where = ℓ ℓ e a a is a vector field tangent to the membrane and in the normal direction to the loop C (see figure 3).
Additionally to the surface tension, a force due to the in-plane ordered phase that is proportional to the curvature of the membrane emerges. Depending on the curvature  this force increases or decreases the area of the patch Σ. On hyperbolic-like surfaces with scalar curvature  s k =the force coming of the in-plane ordered phase is cancelled with the surface tension.
An alternative route to find the stress tensor is using a general expression obtained introducing auxiliary variables [25]. In the case of the in-plane energy with surface tension σ, we see that it does not depend on the extrinsic curvature, so we have that  = 0 ab . We can write the deformation in isothermal coordinates After integrations by parts, neglecting boundary terms and using the fact that  r r  = -=w g 2 , the second term of the last equation can be written as In this way only the symmetric tensor T ab given by contributes to the stress tensor. In the surface tension regime, that is when  k s < < 0 1, the surface's shape tends to a minimal surface with s = -T g ab ab , as expected [17]. If the shape of the membrane is such that its scalar curvature is  s k = -, the T ab tensor vanishes at every point of the membrane (except at the defect points). Note that the contracted tensor ab ab also vanishes for both hyperbolic-like membranes and developable surfaces if the surface tension is negligible. Let us now take an infinitesimal rotation d =X b X, then we have that F =· b X n and F =· b X e a a . Thus, we see again that  F + F = K 0 . So that the energy deforms as where the vector field s kr is the torque about the origin =Ḿ X f .
In the next section the force and torque on the nematic catenoid, a minimal surface with in-plane order, are obtained.

Minimal surfaces: the catenoid
An illustrative example here we present the surface of the catenoid. It is a minimal surface that can be parametrized as a surface of revolution as follows where a is the minimum radius at the waist, . In these coordinates the infinitesimal distance is given by a metric conformal to  2 : and the unit normal outward to the surface is The non-zero elements of the extrinsic curvature can be expressed in coordinates j { } l, and are the following The scalar curvature is found to be The force acting on a line segment does not depend on the angular variable and is given by On the waist of the catenoid l=0 and the force per unit length is just s k a 2 2 . It decreases as we move away from the waist. Along the parallel such that = l l 0 the unit tangent is complements the Darboux frame on this loop, [21]. When integrated, the radial component of the force is canceled and thus the total force on each horizontal loop is given by In the same way the torque acting on a segment of line is given by on the waist of the catenoid it is given by sin , cos , 0 . 70 2 In this example the resulting force due to the ordering implies an effective surface tension that up to leading order is s s k =a 2 eff 2 . To obtain experimentally a catenoid of height h 2 with a couple or rings with radius R each, we need a force that is tangential to the surface and at the same time perpendicular to the ring, whose magnitude is figure 4). Note that on a loop of radius < r R the force decreases as p s . This gives an upper limit to the force as a function of h expressed as . If we pull harder catenoid degenerates into two films on the rings. In order to have a better description one should take into account the bending force. Induced stress due to bending is given by [25] that on a minimal surfaces with K=0 becomes k = - On horizontal loops of the catenoid it is found that the bending induced force per unit length at a height z is has the effect of decreasing the area in the same way as the surface tension does. Moreover, the effect of the first term on the right-hand side of equation (72) depends on the sign of K 0 [26].
Once again, the radial forces are canceled out when integrating around the loop. Hence, the required force to build a nematic catenoid up to a loop of radius < r R and by considering (68) is  The force on meridians (red curves in figure 5) is , that vanishes when integrated over a period interval. The force on rulings (blue lines in figure 5) is given by the curvature is added to the surface tension to decrease the area of the helicoid.

Perturbations
We can now obtain some results about the stability operator of nematic membranes. Those are obtained from the second variation of the energy. By taking a variation in equation ( where the stability operator is given by It can be written in an explicit self-adjoint way as where Eigenvalues of this self-adjoint operator are real numbers. We can also check that a normal translation is a zero mode of . To see this take d = X a, a normal translation such that F = · a n and = · a e 0 a . Then we can write  F = · K a e a b a b . Taking a further derivative we see that   F = -· K K a n. We also have that Hence, substituting this results into equation (78) we find that   F = K , and therefore on equilibrium shapes it is a zero mode. On minimal surfaces the stability operator takes the form Solving the eigenvalue equation  = f Ef m helps us to analyze the stability of the surface. Let us look at the example of the catenoid given in previous section, that is a minimal surface of revolution with negative curvature equation (66), figure 4. In this example, taking advantage of the azimuthal symmetry, we express the solution as .. If we rename u = z/a, we can write the E=0 equation: The solution of this equation with m=0 is,  [29] . If we select the positive value then the conditions for the catenoid to exists, are given by < u u c and r r < c , which in turns implies that r r < u u c c and therefore we obtain the well known result, < h R 0.66. In this way, < z ais a condition that must be fulfilled for all < z h. In order to find solutions if ¹ E 0, we expand up to quadratic order in u. With this we are able to rewrite the eigenvalue equation as . Note that since ν is a real number, imposes the upper bound  < 2. The most general solution to equation (84), is therefore given by where C 1 and C 2 are constants and the parabolic cylinder function m n ( ) D appears, [30]. We take the boundary condition such that On the other hand if  -¥ u , the behavior of the function is n ñ - where = ¼ n 0, 1, 2, , and = | | m 0, 1. In table 1 we show some of the eigenvalues ò and in figure 6 some of the eigenfunctions. We see that eigenmodes with m=0 includes the negative value  = -2, that is, the nematic catenoid is unstable under these symmetric perturbations. On the other hand, the stability of the catenoid under m=1 modes is evident. Of course this happens in the surface tension regime since, as was discussed by Giomi [17], in the orientational regime, to be analyzed, the surface should have a constant negative curvature.

Concluding remarks
Taking into account in-plane ordering configuration of a vector field on a curved surface we have found the corresponding shape equations  s k + = ( ) K 0 of the surface. We find that in the regime where the bending energy is negligible not only minimal surfaces remains as solutions, but hyperbolic-like surfaces also appear among others such that developable surfaces. It should be noted that this is possible even without the presence of topological defects. Starting from the invariance of the energy under Euclidean transformations, we have found the stress as  x s k x = - b . The fact that this vector field is tangent to the surface is not surprising since it comes from a purely intrinsic energy. Because of this, the induced torque is only external. The stress is a vector field that vanishes at each point in a hyperbolic-like membranes,  s k = -. In the presence of topological defects with charge q i , the stress is modified as x s k rx = - In the case of the nematic catenoid the force takes its largest value at the waist of the surface and decreases as we move. Furthermore, we study the stability of the catenoid under azimuthal asymmetric perturbations, by showing the positiveness of the eigenvalues of the stability operator. In contrast, the catenoid is unstable under azimuthal symmetric perturbations. Several other geometries of interest can be studied. In the case of a closed surface one can take into account the volume inside through a Lagrange multiplier P. In this case one have to take into account that, due to the Poicaré-Hopf theorem, the total topological charge on the surface has the same value as the Euler invariant c = 2. Thereby, there are topological defects to take into account, [31]. We return to this issue elsewhere.