Abstract
The behavior of a lipid membrane on mesoscopic scales is captured unusually accurately by its geometrical degrees of freedom. Indeed, the membrane geometry is, very often, a direct reflection of the physical state of the membrane. In this chapter we will examine the intimate connection between the geometry and the physics of fluid membranes from a number of points of view. We begin with a review of the description of the surface geometry in terms of the metric and the extrinsic curvature, examining surface deformations in terms of the behavior of these two tensors. The shape equation describing membrane equilibrium is derived and the qualitative behavior of solutions described. We next look at the conservation laws implied by the Euclidean invariance of the energy, describing the remarkably simple relationship between the stress distributed in the membrane and its geometry. This relationship is used to examine membrane-mediated interactions. We show how this geometrical framework can be extended to accommodate constraints—both global and local—as well as additional material degrees of freedom coupling to the geometry. The conservation laws are applied to examine the response of an axially symmetric membrane to localized external forces and to characterize topologically nontrivial states. We wrap up by looking at the conformal invariance of the symmetric two-dimensional bending energy, and examine some of its consequences.
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Notes
- 1.
We abbreviate \(\partial _a = \partial _a\mathbf {X} /\partial u^a\).
- 2.
We are interested specifically in surface tensors and the scalars constructed out of them. Consider a surface reparametrization \((u^1,u^2) \rightarrow (\bar{u}^1(u^1,u^2),\bar{u}^2(u^1,u^2))\). Define \(J^{\bar{a}}{}_b= \partial \bar{u}^a/\partial u^b\), with inverse \(J_{\bar{a}}{}^{c}: J^{\bar{a}}{}_{c}J_{\bar{b}}{}^{c} = \delta ^{\bar{a}}{}_{\bar{b}}\). Tensor fields transform under reparametrization by matrix multiplication on each index with the Jacobian matrix of the reparametrization or its inverse. In particular, the metric transforms by \(\bar{g}_{\bar{a}\bar{b}} = J_{\bar{a}}{}^{c} J_{\bar{b}}{}^{d} g_{cd}\). Note that the three Cartesian embedding functions \(\mathbf {X}=(X^1,X^2,X^3)\) are each scalars under reparametrization: \(\bar{X}^1(\bar{u}^1,\bar{u}^2) = X^1(u^1,u^2)\), etc.
- 3.
Under reparametrization, \(\sqrt{\bar{g}} = \mathrm {det} J^{-1} \sqrt{g}\).
- 4.
It is simple to show that \(V^a\) transforms like a vector under reparametrization.
- 5.
It is straightforward to confirm that the two remaining symmetric quadratics, \(C_1^2+C_2^2\) and \((C_1-C_2)^2\), can be expressed as linear combinations of \(K^2\) and \(\mathcal {K}_G\).
- 6.
As we will show below, controlling area locally is equivalent, in equilibrium, to controlling it globally.
- 7.
For simplicity we will suppose that not only the surface height but also its normal vector are fixed on the boundary.
- 8.
The other half of helicoid is given by \(h = c (\pi + \varphi )\).
- 9.
Under a deformation \(h(\mathbf {r}) \rightarrow h(\mathbf {r}) + \delta h(\mathbf {r}) \), fixed on the boundary, the change in area A (6), is given by
$$\begin{aligned} \delta A = \int d\mathbf {r}\, \nabla _0\cdot \mathbf {J} \, \delta h \qquad {(13)} \end{aligned}$$where
$$\begin{aligned} \mathbf {J} = -\frac{ \nabla _0 h}{(1 +|\nabla _0 h|^2)^{1/2}} \,, \qquad {(14)} \end{aligned}$$so that \(\nabla _0\cdot \mathbf {J}= 0\) in equilibrium. One can evaluate \(|\nabla _0h|^2 = p^2/r^2\), so that \(\mathbf {J} = p(-\sin \theta ,\cos \theta ) /r (1+ p^2/r^2)^{1/2}\) and \(\nabla _0\cdot \mathbf {J}= 0\).
- 10.
Just as \(\ln |\mathbf {r}-\mathbf {r}'|\) is proportional to the Green’s function for the Laplacian, \(-|\mathbf {r}-\mathbf {r}'|^2 \ln |\mathbf {r}-\mathbf {r}'|\) is its counterpart for the bilaplacian.
- 11.
The Green’s function of the Helmholtz operator is proportional to \(K_0(|\mathbf {r}-\mathbf {r}'|)\).
- 12.
\(\nabla _a \Phi ^a =\partial _a (\sqrt{g} \Phi ^a)/\sqrt{g}\).
- 13.
Note that the Ricci identify (4) implies \(R_{abcd}=-R_{bacd}\); whereas its application to the metric tensor implies \(R_{abcd}=-R_{abdc}\):
$$\begin{aligned} 0= [\nabla _a,\nabla _b] g_{cd} = R_{abcd}+ R_{abdc}\,. \qquad {(34)} \end{aligned}$$These account for all the independent constraints on \(R_{abcd}\) on a two-dimensional surface.
- 14.
The identity \(\mathrm {det}\, K^a{}_b = (K^2 - K_{ab} K^{ab})/2\) is true for the determinant of any two-dimensional symmetric matrix.
- 15.
\(V_\perp ^a= \epsilon ^{ab} V_b\) is orthogonal to \(V^a\).
- 16.
A later derivation accommodating the finite thickness of the membrane is presented in Lomholt and Miao (2006).
- 17.
In this approach, the deformation vector \(\delta \mathbf {X}\) is never disassembled into normal and tangential parts, so that its reassembly is never necessary.
- 18.
If \(\mathbf {t}=t^a\mathbf {e}_a\) is the unit tangent vector to the curve, \(\mathbf {l}\cdot \mathbf {t}=0\) or \(g_{ab} l^a t^b=0\) or \(l_a t^a=0\).
- 19.
The local parametrization is fixed.
- 20.
On a surface with boundary, this identity yields the volume of the cone standing on the surface patch, with its apex located at the origin.
- 21.
Intriguingly, the quadratic contribution to \(T^B_{ij}\) is trace-free in this approximation, a property we would associate with scale invariance. Yet the area itself is clearly not scale invariant. The source of this peculiarity is that, in the quadratic approximation in gradients of h, the area is represented by a massless two-dimensional scalar field on the plane which is scale invariant if the plane is scaled, but not if h is. On the other hand, \(T^B_{ji}\) is not trace-free but should not have been expected to be.
- 22.
Unlike the prolate, this geometry is stable with respect to membrane slippage under the ring.
- 23.
This is well known in the context of global constraints. In a symmetric closed fluid membrane subject to area and volume constraints, the identity \(2\sigma A - 3 P V=0\) is a consequence of the scale invariance of the bending energy, Svetina and Žekž (1989).
- 24.
\(K_{ab}\rightarrow K_{ab} = -|\mathbf { X}|^{2} \left( K_{ab} - 2 \,(\mathbf { X}\cdot \mathbf { n}) g_{ab} / |\mathbf { X}|^{-2}\right) \).
- 25.
Fixing the discocyte area at \(4 \pi r_0^2\) determines \(R_S = 1.089 \, r_0\).
- 26.
The bilateral symmetry (not necessarily in the original XZ plane) is preserved if the point strays off this axis; however, the up-down symmetry is broken, just as it was in the axially symmetric family.
- 27.
This is the shortest distance between the two points on the surface. They are, of course, in contact in space.
- 28.
An unexpected duality between the weak field behavior in one geometry and the strong field behavior in the other is evident: asymptotically, the catenoid is accurately described by the height function \(h \sim \ln r\), \(r\gg r_0\); this asymptotic region is mapped into the neighborhood of the poles described by \(h\sim -r^2 \ln r\), \(r\ll s\). Inversion provides a connection between the harmonic behavior in the former and the biharmonic behavior in the latter, Guven and Vázquez-Montejo (2013b). To understand this duality between harmonic and biharmonic function, look at the inversion in the origin \(\mathbf { x}\rightarrow \mathbf { x}/|\mathbf { x}|^2\) (for transparency set the scale to one), described in the height function representation by \((r, h) \rightarrow (r, h)/ (({r^2 +h^2})\), so that \(h \approx \ln r\) \(\rightarrow \) \(\frac{h}{r^2 + h^2}= \ln [{r}/({r^2 +h^2})]\). Now, if \(h\ll r\), then \(\frac{h}{r^2} \approx - \ln r \), and as claimed the Green function of the Laplacian is mapped to its biharmonic counterpart.
- 29.
In this context, note also that the symmetric saddle with \(h\sim r^2 \cos 2\theta \) maps to the biharmonic dipole \(h\sim \cos 2\theta \).
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Acknowledgements
JG would like to thank David Steigmann for the invitation to lecture at the CISM Summer School held in Udine, Italy during July of 2016. This chapter is based on these lectures. We would like also to thank Markus Deserno, Martin Müller and Saša Svetina for their valuable input. This work was partially supported by CONACyT grant 180901.
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Guven, J., Vázquez-Montejo, P. (2018). The Geometry of Fluid Membranes: Variational Principles, Symmetries and Conservation Laws. In: Steigmann, D. (eds) The Role of Mechanics in the Study of Lipid Bilayers. CISM International Centre for Mechanical Sciences, vol 577. Springer, Cham. https://doi.org/10.1007/978-3-319-56348-0_4
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