Morphology change in nematic membranes induced by defects/Mudanca de morfologia em membranas nematicas induzida por defeitos.
The biomembrane is one of the most intriguing structures in nature. The plasma membrane is made of lipids, having amphiphilic nature, just as liotropic liquid crystals (JAKLI; SAUPE, 2006). There are several functions attributed to it, such as permeable selectivity, protection, anchoring to the cytoskeleton and so many others (ALBERTS et al., 2002). All these functions can be greatly affected by external factors, which include changes in the membrane morphology, often described by geometry and topology (FRANK; KARDAR, 2008). In fact, shape change in membranes has been subject of great effort in condensed matter. Very often, these shapes are investigated by analyzing the coupling between geometrical shape and orientational order. These studies include orientional order and defects in deformable vesicles (LUBENSKY; PROST, 1992; HIRST et al., 2013; JIANG et al., 2007; NYUGEN et al., 2013; PARK et al., 1992; RAMAKRISHNAN et al., 2010) and cytoskeletal filaments (NEDELEC et al., 1997; SURREY et al., 2001). Furthermore, vector fields on a surface or nematic membranes, are considered simplified models to describe membranes that are more complex. These include any flexible sheet with ordered road-like constituents (SHALAGINOV, 1996; SPECTOR et al., 1993; YOUNG et al., 1978).
In this work, we investigate the problem of a flat membrane with coupled nematic order when defects of the +1 kind appear. In regular nematic cells, it is expected that the bulk will buckle into the third dimension when defects are present (CLADIS; KLEMAN, 1972; JAKLI; SAUPE, 2006; MEYER, 1973). By using a simple model composed of nematic order and bending rigidity in covariant form, we are able to determine that a second-order like transition may guide the buckling of the membrane depending on the ratio of the parameters involved. Furthermore, we obtain an analytical result for the shape of the membrane, which buckles into pseudo-spheres.
Material and methods
In order to model the membrane, the Monge parameterization (NELSON, 2004) was used, which defines a surface, mapped on a plane defined by the variables [[sigma].sub.1] and [[sigma].sub.2] with height h([[sigma].sub.1], [[sigma].sub.2]). The position vector then can be written as [??]([[sigma].sub.1], [[sigma].sub.2]) = [[[sigma].sub.1], [[sigma].sub.2], h([[sigma].sub.1], [[sigma].sub.2])], which defines the tangent vectors as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with i = 1, 2.
We consider the free energy of the system to come from two contributions: the field, which wants to make all the vectors on the surface of the membrane parallel one to each order, supposing initial flat membrane, and the bending rigidity term. In a configuration with a defect, the flat configuration cannot minimize the free energy of the system, and the membrane has to bend, changing its shape. However, it has to pay a price for bending. The total free energy of the system, in the absence of surface tension (RAMAKRISHNAN et al., 2011), can be written in covariant form as:
F = [integral] [K/2 [H.sup.2]] dS + [integral] [K.sub.A]/2 [square root of g] [g.sup.uv] [g.sub.py] [[nabla].sub.u] [n.sup.p] [[nabla].sub.v][n.sup.y], (1)
k is the bending rigidity, H is the mean curvature, [K.sub.A] is the nematic elastic constant (elastic isotropy approximation), [n.sup.y] represent the components of the director field and g is the metric tensor. The director field, in the presence of defects of +1 kind can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[beta] is a parameter used to distinguished defects asters ([beta] = 0) and vortices ([beta] = [pi]/2), which is shown in Figure 1.
Results and discussion
The first approach for solving this problem is to try some specific shape for the membrane that would minimize the energy of the membrane (SEUNG; NELSON, 1998). One could argue that the actually shape for minimizing a + 1 defect configuration would be a cone shape, and we can find how deep the cone goes by minimizing the free energy with respect to the angle a between the cone side and the z axis (in cylindrical coordinates), going from [alpha] = [pi]/2 for a flat membrane until [alpha] = 0. For this configuration, the tangent vectors may be defined everywhere in terms of a as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)
The element of area for a conic surface is given by dS = rdrd[phi]csc[[alpha]]. Furthermore, the mean curvature of a cone is given by H = cos[[alpha]]/2r. Therefore, the free energy's first term is described by the following equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
[F.sub.1] = 1/4 [kappa][pi] [[r.sub.max]/[r.sub.min]] cos[[[alpha]].sup.2]/sin[[alpha]]. (5)
In equation (5), [r.sub.min] and [r.sub.max] are the distances from the core of the defect to the size of the membrane. The second term of the free energy becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
[F.sub.2] = 1/2 [K.sub.A] [pi] ln [[r.sub.max]/[r.sub.min]](sin[[alpha]]cos[[[beta]].sup.2] + sin[[[beta]].sup.2]) (7)
Therefore, the total free energy is
F = [F.sub.1] + [F.sub.2] = 1/2 [pi] ln [[r.sub.max]/[r.sub.min]]
x (1/2 [kappa] cos[[[alpha]].sup.2]/sin[[alpha]] + [K.sub.A](sin[[alpha]]cos[[[beta]].sup.2] + sin[[[beta]].sup.2]/sin[[alpha]])) (8)
Let us first examine the case for a defect of the "aster" configuration, which means, [beta] = 0. In this case, the equilibrium situation is found by setting [partial derivative]F/[partial derivative][alpha] = 0. We find:
([K.sub.A]/[kappa] cos[[alpha]] - 1/4 cos[[alpha]] - 1/4 cot[[alpha]]csc[[alpha]]) = 0. (9)
Clearly, [alpha] = [pi]/2 is solution, which means a flat membrane. Nonetheless, the solution for [alpha] is
[alpha] = arccos [[square root of (1 - 1/4[K.sub.A]/[kappa] - 1)]] (10)
which is a minimum only when [K.sub.A]/[kappa] > 1/2. Therefore, there is a ratio between elastic constant and bend rigidity that determines the buckling of the membrane. If the ratio is smaller than 1/2 , than the stable configuration is the one where the membrane remains flat. However, as the ratio grows larger than 1/ 2, the system smoothly changes from flat to buckled, in a second order-like change transition. In Figure 2, we show the plot of F' = 2F/([pi]ln[[r.sub.max]/[r.sub.min]]) against the angle [alpha].
Notice that in Figure 2(a), we have [K.sub.A]/[kappa] = 1, resulting in a buckled configuration whose angle [alpha] = 35.3[degrees]. In Figure 2(b), [K.sub.A]/[kappa] = 0.2, so the membrane lies flat with [alpha] = 90.0[degrees].
Now, we can examine the vortex case, where [beta] = [pi]/2. In such case, the total free energy, equation (8), becomes:
F = 1/2 [pi] ln [[r.sub.max]/[r.sub.min]](1/2 [kappa] cos[[[alpha]].sup.2]/sin[[alpha]] + [K.sub.A] sin[[[beta]].sup.2]/sin[[alpha]]) (11)
By setting [partial derivative]F/[partial derivative][alpha] = 0 we can find the stable solution. In this case, the only [alpha] that satisfies the equation is [alpha] = [pi]/2, which means the membrane is always flat if defects of the vortex kind exists. Figure 3(a) shows a 3D curve of F' = 2F /([pi] ln[[r.sub.max]/[r.sub.min]]) against [alpha] and [beta] for [K.sub.A]/[kappa] = 3 plotted from [alpha] = 0 to [pi] and [beta] from 0 to [pi]/2. It is possible to observe that buckled configurations happen only for defects near the aster arrangement.
A more general solution
In the previous section, we have analyzed the morphology change of a membrane by assuming it would buckle into a cone. This means the function for the height was written as h = r cot[[alpha]].
Now, we shall look for profiles that are different from the conical shape. However, it is straightforward to foresee that the actual morphology of a buckled membrane in the presence of +1 defects should not differ much from the conical shape. This is indeed expected from the famous scape to the third dimension in bulk nematic (CLADIS; KLEMAN, 1972; JAKLI; SAUPE, 2006; MEYER, 1973). In fact, our first approach is to solve the problem imagining the following situation: h = [r.sup.[eta]] cot[[alpha]], where [eta] is a coefficient to be determined. By numerically integrating and finding the minimum of the free energy, equation (11), we encounter that the stable configuration of the buckled membrane happens for [eta] [approximately equal to] 0.95, assuming [K.sub.A]/[kappa] = 1, [r.sub.min] = 1 nm and [r.sub.max] = 1 [micro]m. The behavior of f against [eta] is shown in Figure 3(b).
In order to seek for a more general solution of the problem, we assume that height of the surface depends only on the radial distance from the core, or, h [right arrow] h(r). In this situation, the tangent vectors, as well as the normal vector [??] are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[h.sub.r] = dh(r)/dr. The total free energy, equation (1), is then written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)
Now, we perform the following change of variables
[h.sub.r] = tan[[delta](r)],
[delta](r) is an unknown function of the variable r. Therefore, it follows that [h.sub.rr] = sec[[[delta]].sup.2][[delta].sub.r] (we shall now drop the (r) indicating it is a function of r). Furthermore, based on previous results, we look for solutions in the case where [beta] = 0. Then, equation (11) becomes:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)
The profile that minimizes the free energy is the one that satisfies the Euler-Lagrange equation
[partial derivative]f/[partial derivative][delta] - d/dr [partial derivative]f/[partial derivative][[delta].sub.r] = 0, (14)
f is the integrand of the total free energy in equation (12). Hence
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)
First, we notice that if [[delta].sub.r] = [[delta].sub.rr] = 0, we obtain
(4[K.sub.A] - [kappa]) sin[2[delta]] - 2[kappa] tan[[delta]]/16r cos [[delta]] = 0, (16)
whose solution is
h(r) = r tan [arccos [[square root of (1 - 1/4[K.sub.A]/[kappa] - 1)]], (17)
which corresponds to the conical solution. The next step requires solving equation (14). Unfortunately, equation (14) is very complicated and has no analytical solution. Nonetheless, we can use our previous results to infer about solving it. The numerical minimization showed that the actual minimum of energy is slightly different from the conical configuration. Therefore, we assume the solution for equation (16) can be written as
h(r) = r tan[arccos [[square root of (1 - 1/4[K.sub.A]/[kappa] - 1)]] + m(r), (18)
m(r) is a function to be determined, considered small. By replacing equation (17) in (14), we get an equation depending only on m(r). Since it is small, we can expand this equation and take only linear terms on m(r). The usual minimization procedure allows the following equation for m(r):
[[m.sub.r] + r[m.sub.rr]](16[K.sub.A.sup.2] - 4[K.sub.A][kappa] + [[kappa].sup.2])r
- 2m(8[K.sub.A.sup.2] - 6[K.sub.A][kappa] + [[kappa].sup.2]) = 0, (19)
[m.sub.r] = dm/dr. Equation (18) is solved by setting m([r.sub.min]) = -r tan[[alpha]], where [alpha] is given by equation (10), so near the core the membrane remains flat; and m([r.sub.max]) = 0, meaning that far away from the core the configuration is basically the same as a cone. Therefore, we find that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)
[[theta].sub.m] = [square root of (2(6[K.sub.A] [kappa] - [[kappa].sup.2] - 8[K.sub.A.sup.2]))] (ln[[r.sub.max]/[r.sub.min]])/[square root of (-4[K.sub.A][kappa] + [[kappa].sup.2] + 16[K.sub.A.sup.2])], (21)
[[theta].sub.m] = [square root of (2(6[K.sub.A] [kappa] - [[kappa].sup.2] - 8[K.sub.A.sup.2]))] (ln[r/[r.sub.min]])/[square root of (-4[K.sub.A][kappa] + [[kappa].sup.2] + 16[K.sub.A.sup.2])]. (22)
Figure 4 shows a graphic of the height profile, equation (20) as a function of r for [K.sub.A]/[kappa] = 1, [r.sub.min] = 1 [sup.nm] and [r.sub.max] = 1 [micro]m. The profile is known as pseudosphere, with negative Gaussian curvature, in agreement with results reported before (FRANK; KARDAR, 2008).
In conclusion, the problem of a membrane with bend rigidity and a nematic vector field (nematic membrane) has been studied when defects of + 1 kind exist in the director. We were able to show that the conical shape minimizes, in a first approximation, the free energy of the system when the ratio between the bend rigidity and elastic constant are in the appropriate range and that the transition between the flat and buckled state is second-order like. Further analyzes have shown that without any approximation the lowest energy is slightly different than the cone, which was first numerically calculated. Then, by using a perturbation method, we were able to analytically calculate the shape of the buckled membrane and show that it is a pseudosphere.
The authors are thankful to J. Selinger for advising and discussions and to the first I Workshop Norte-Paranaense de Fluidos Complexos (UEM). R. S. Z. thanks CNPq for financial support.
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Received on April 30, 2014.
Accepted on July 14, 2014.
License information: This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Maiara Mitiko Taniguchi (1), Fernanda Cavicchioli Zola (2), Veridiana Garcia Guimaraes (3), Giane Goncalves (4), Roberto Rossato (1) and Rafael Soares Zola (1,3) *
(1) Universidade Tecnologica Federal do Parana, Rua Marcilio Dias, 635, 86812-460, Apucarana, Parana, Brazil. (2) Departamento de Engenharia de Producao, Universidade Estadual de Maringa, Maringa, Parana, Brazil. (3) Departamento de Fisica, Universidade Estadual de Maringa, Av. Colombo, 5790, 87020-900, Maringa, Parana, Brazil. (4) Universidade Tecnologica Federal do Parana, Ponta Grossa, Parana, Brazil.
* Author for correspondence. E-mail: firstname.lastname@example.org