A finite element scheme for the evolution of orientational order in fluid membranes

Bartels, Sören; Dolzmann, Georg; Nochetto, Ricardo H.

ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 1, pp. 1-31.

We investigate the evolution of an almost flat membrane driven by competition of the homogeneous, Frank, and bending energies as well as the coupling of the local order of the constituent molecules of the membrane to its curvature. We propose an alternative to the model in [J.B. Fournier and P. Galatoa, J. Phys. II 7 (1997) 1509-1520; N. Uchida, Phys. Rev. E 66 (2002) 040902] which replaces a Ginzburg-Landau penalization for the length of the order parameter by a rigid constraint. We introduce a fully discrete scheme, consisting of piecewise linear finite elements, show that it is unconditionally stable for a large range of the elastic moduli in the model, and prove its convergence (up to subsequences) thereby proving the existence of a weak solution to the continuous model. Numerical simulations are included that examine typical model situations, confirm our theory, and explore numerical predictions beyond that theory.

Classification : 35K55, 74K15
Mots clés : biomembrane, orientational order, curvature

[1] F. Alouges, A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34 (1997) 1708-1726. | Zbl

[2] F. Alouges, A new finite element scheme for Landau-Lifchitz equations. Discrete Contin. Dyn. Syst. Ser. S 1 (2008) 187-196. | Zbl

[3] J.W. Barrett, S. Bartels, X. Feng and A. Prohl, A convergent and constraint-preserving finite element method for the $p$-harmonic flow into spheres. SIAM J. Numer. Anal. 45 (2007) 905-927. | Zbl

[4] J.W. Barrett, H. Garcke and R. Nürnberg, On the parametric finite element approximation of evolving hypersurfaces in ${ℝ}^3$. J. Comput. Phys. 227 (2008) 4281-4307. | Zbl

[5] S. Bartels, Stability and convergence of finite-element approximation schemes for harmonic maps. SIAM J. Numer. Anal. 43 (2005) 220-238 (electronic). | Zbl

[6] T. Baumgart, S.T. Hess and W.W. Webb, Imaging co-existing domains in biomembrane models coupling curvature and tension. Nature 425 (2003) 832-824.

[7] T. Biben and C. Misbah, An advected-field model for deformable entities under flow. Eur. Phys. J. B 29 (2002) 311-316.

[8] P. Biscari and E.M. Terentjev, Nematic membranes: Shape instabilities of closed achiral vesicles. Phys. Rev. E 73 (2006) 051706.

[9] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York, USA (1991). | Zbl

[10] P.B. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theort. Biol. 26 (1970) 61-81.

[11] Y.M. Chen, The weak solutions to the evolution problems of harmonic maps. Math. Z. 201 (1989) 69-74. | Zbl

[12] C.H.A. Cheng, D. Coutand and S. Shkoller, Navier-Stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal. 39 (2007) 742-800 (electronic). | Zbl

[13] P.G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (2002). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. | Zbl

[14] K. Deckelnick, G. Dziuk and C.M. Elliott, Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005) 139-232. | Zbl

[15] Q. Du, C. Liu and X. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198 (2004) 450-468. | Zbl

[16] Q. Du, C. Liu and X. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys. 212 (2006) 757-777. | Zbl

[17] G. Dziuk, Computational parametric Willmore flow. Numer. Math. 111 (2008) 55-80. | Zbl

[18] E. Evans, Bending resistance and chemically induced moments in membrane bilayers. Biophys. J. 14 (1974) 923-931.

[19] J.B. Fournier and P. Galatoa, Sponges, tubules and modulated phases of para-antinematic membranes. J. Phys. II 7 (1997) 1509-1520.

[20] A. Freire, S. Müller and M. Struwe, Weak compactness of wave maps and harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 725-754. | Numdam | Zbl

[21] M. Giaquinta and S. Hildebrandt, Calculus of variations I: The Lagrangian formalism, Grundlehren der Mathematischen Wissenschaften 310, [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, Germany (1996). | Zbl

[22] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman (Advanced Publishing Program), Boston, USA (1985). | Zbl

[23] C. Grossmann and H.-G. Roos, Numerical treatment of partial differential equations. Universitext, Springer, Berlin, Germany (2007). Translated and revised from the 3rd (2005) German edition by Martin Stynes. | Zbl

[24] W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C 28 (1973) 693-703.

[25] W. Helfrich and J. Prost, Intrinsic bending force in anisotropic membranes made of chiral molecules. Phys. Rev. A 38 (1988) 3065-3068.

[26] J.T. Jenkins, The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math. 32 (1977) 755-764. | Zbl

[27] M.A. Johnson and R.S. Decca, Dynamics of topological defects in the $l_{{\beta }^{\text{'}}}$ phase of 1,2-dipalmitoyl phosphatidylcholine bilayers. Opt. Commun. 281 (2008) 1870-1875.

[28] O.A. Ladyzhenskaya and N.N. Ural'Tseva, Linear and quasilinear elliptic equations. Academic Press, New York, USA (1968). Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. | Zbl

[29] T. C. Lubensky and F.C. Mackintosh, Theory of “ripple” phases of bilayers. Phys. Rev. Lett. 71 (1993) 1565-1568.

[30] F.C. Mackintosh and T.C. Lubensky, Orientational order, topology, and vesicle shapes. Phys. Rev. Lett. 67 (1991) 1169-1172. | Zbl

[31] S.J. Marrink, J. Risselada and A.E. Mark, Simulation of gel phase formation and melting in lipid bilayers using a coarse grained model. Chem. Phys. Lipids 135 (2005) 223-244.

[32] S.T.-N.J.F. Nagle, Structure of lipid bilayers. Biochim. Biophys. Acta 1469 (2000) 159-195.

[33] P. Nelson and T. Powers, Rigid chiral membranes. Phys. Rev. Lett. 69 (1992) 3409-3412.

[34] R. Oda, I. Huc, M. Schmutz and S.J. Candau, Tuning bilayer twist using chiral counterions. Nature 399 (1999) 566-569.

[35] M.S. Pauletti, Parametric AFEM for geometric evolution equations coupled fluid-membrane interaction. Ph.D. Thesis, University of Maryland, USA (2008).

[36] R.E. Rusu, An algorithm for the elastic flow of surfaces. Interfaces Free Bound. 7 (2005) 229-239.

[37] U. Seifert, Configurations of fluid membranes and vesicles. Adv. Phys. 46 (1997) 13-137.

[38] J.V. Selinger and J.M. Schnur, Theory of chiral lipid tubules. Phys. Rev. Lett. 71 (1993) 4091-4094.

[39] D. Steigmann, Fluid films with curvature elasticity. Arch. Ration. Mech. Anal. 150 (1999) 127-152. | Zbl

[40] M. Struwe, Geometric evolution problems, in Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), IAS/Park City Math. Ser. 2, Amer. Math. Soc., Providence, USA (1996) 257-339. | Zbl

[41] N. Uchida, Dynamics of orientational ordering in fluid membranes. Phys. Rev. E 66 (2002) 040902.

[42] E.G. Virga, Variational theories for liquid crystals, Appl. Math. Math. Comput. 8. Chapman & Hall, London, UK (1994). | Zbl

[43] T.J. Willmore, Riemannian geometry, Oxford Science Publications. The Clarendon Press Oxford University Press, New York, USA (1993). | Zbl