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A phase field approach in the numerical study of the elastic bending energy for vesicle membranes

Authors:

Xiaoqiang WangAuthors Info & Claims

Journal of Computational Physics, Volume 198, Issue 2

Pages 450 - 468

https://doi.org/10.1016/j.jcp.2004.01.029

Published: 10 August 2004 Publication History

Abstract

In this paper, we compute the equilibrium configurations of a vesicle membrane under elastic bending energy, with prescribed volume and surface area. A variational phase field method is developed for such a problem. Discrete finite difference approximations and numerical simulations are carried out in the axial symmetrical cases. Different energetic bifurcation phenomena are discussed.

References

[1]

{1} N.D. Alikakos, P.W. Bates, X.F. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal. 128 (1994) 165-205.

[2]

{2} W. Bao, Q. Du, Computing the ground state of the BEC via normalized gradient flow, 2003 (preprint). Available from 〈http:// arxiv.org/abs/cond-mat/0303241〉.

[3]

{3} R.L. Bryant, Surfaces in conformal geometry, in: The Mathematical Heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math., vol. 48, American Mathematical Society, Providence, RI, 1988, pp. 227-240.

[4]

{4} G. Caginalp, X.F. Chen, Phase field equations in the singular limit of sharp interface problems, in: On the Evolution of Phase Boundaries (Minneapolis, MN, 1990-91), Springer, New York, 1992, pp. 1-27.

[5]

{5} J.W. Cahn, S.M. Allen, A microscopic theory for domain wall motion and its experimental verification in fe-al alloy domain growth kinetics, J. Phys. Colloque C7 (19778) C7-51.

[6]

{6} J.W. Cahn, J.E. Hillard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958) 258-267.

[7]

{7} R.S. Chadwick, Axisymmetric indentation of a thin incompressible elastic layer, SIAM J. Appl. Math. 62 (2002) 1520-1530.

[8]

{8} Y.C. Chang, T.Y. Hou, B. Merriman, S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys. 124 (1996) 449-464.

Digital Library

[9]

{9} X.F. Chen, Generation and propagation of interfaces in reaction-diffusion systems, Trans. Am. Math. Soc. 334 (1992).

[10]

{10} P.G. Ciarlet, Introduction to Linear Shell Theory, Series in Applied Mathematics (Paris), vol. 1, Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 1998.

[11]

{11} P.G. Ciarlet, Mathematical elasticity, vol. III, in: Studies in Mathematics and its Applications, vol. 29, North-Holland, Amsterdam, 2000, Theory of shells.

[12]

{12} V. Cristini, J. Blawzdziewicz, M. Loewenberg, Drop breakup in three-dimensional viscous flows, Phys. Fluids 10 (1998) 1781-1783.

[13]

{13} P.G. de Gennes, The Physics of Liquid Crystals, Oxford University Press, Oxford, 1974.

[14]

{14} Q. Du, Finite element methods for the time dependent Ginzburg-Landau model of superconductivity, Comput. Math. Appl. 27 (1994) 119-133.

[15]

{15} J.E. Dunn, J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal. 88 (1985) 95-133.

[16]

{16} C.M. Elliott, R. Schätzle, The limit of the fully anisotropic double-obstacle Allen-Cahn equation in the nonsmooth case, SIAM J. Math. Anal. 28 (1997) 274-303.

Digital Library

[17]

{17} Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissues, Springer, New York, 1981.

[18]

{18} J. Glimm, X.L. Li, Y. Liu, N. Zhao, Conservative front tracking and level set algorithms, Proc. Natl. Acad. Sci. USA 98 (2001) 14198-14201.

[19]

{19} J. Glimm, J.W. Grove, X.L. Li, D.C. Tan, Robust computational algorithms for dynamic interface tracking in three dimensions, SIAM J. Sci. Comput. 21 (2000) 2240-2256.

Digital Library

[20]

{20} F. Hélein, A Weierstrass representation for Willmore surfaces, in: Harmonic Morphisms, Harmonic Maps, and Related Topics (Brest, 1997), Chapman & Hall/CRC Res. Notes Math., vol. 413, Chapman & Hall/CRC, Boca Raton, FL, 2000, pp. 287-302.

[21]

{21} D. Hilbert, S. Cohn-Vossen, Geometry and the Imagination, Chelsea Publishing Company, New York, NY, 1952 (Transl.: P. Neményi).

[22]

{22} R.E. Khayat, Three-dimensional boundary-element analysis of drop deformation for Newtonian and viscoelastic systems, Int. J. Numer. Meth. Fluids 34 (2000) 241-275.

[23]

{23} D. Kinderlehrer, C. Liu, Revisiting the focal conic structure in smectic-A, in: R. Batra, M.F. Beatty (Eds.), Contemporary Research in the Mechanics and Mathematics of Materials, CIMNE, 1998.

[24]

{24} J. Li, Y. Renardy, Numerical study of flows of two immiscible liquids at low Reynolds number, SIAM Rev. 42 (2000) 417-439.

Digital Library

[25]

{25} J. Li, Y. Renardy, Shear-induced rupturing of a viscous drop in a Bingham liquid, J. Non-Newtonian Fluid Mech. 95 (2000) 235-251.

[26]

{26} G. Lim, M. Wortis, A numerical study of the mechanics of red blood cell shapes and shape transformations. Available from 〈http://www.sfu.ca/ghlim/presentation/title.html〉.

[27]

{27} F.-H. Lin, Q. Du, Ginzburg-Landau vortices, dynamics, pinning and hysteresis, SIAM J. Math. Anal. 28 (1999) 1265-1293.

Digital Library

[28]

{28} C. Liu, J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D 179 (2003) 211-228.

[29]

{29} C. Liu, N.J. Walkington, An Eulerian description of fluids containing visco-hyperelastic particles, Arch. Rat. Mech. Anal. 159 (2001) 229-252.

[30]

{30} J. Lowengrub, L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A 454 (1998) 2617-2654.

[31]

{31} G.B. McFadden, A.A. Wheeler, R.J. Braun, S.R. Coriell, R.F. Sekerka, Phase-field models for anisotropic interfaces, Phys. Rev. E 48 (3) (1993) 2016-2024.

[32]

{32} L. Miao, U. Seifert, M. Wortis, H. Dobereiner, Budding transitions of fluid-bilayer vesicle: the effect of area-difference elasticity, Phys. Rev. E 49 (1994) 5389-5407.

[33]

{33} W.W. Mullins, R.F. Sekerka, On the thermodynamics of crystalline solids, J. Chem. Phys. 82 (1985).

[34]

{34} S. Osher, J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton Jacobi formulations, J. Comput. Phys. 79 (1988) 12-49.

Digital Library

[35]

{35} S. Osher, R. Fedkiw, The Level Set Method and Dynamic Implicit Surfaces, Springer, Berlin, 2002.

[36]

{36} Z. Ou-Yabg, W. Helfrich, Bending energy of vesicle membranes: general expressions for the first, second and third variation of the shape energy and applications to spheres and cylinders, Phys. Rev. E 39 (1989) 5280-5288.

[37]

{37} J. Rubinstein, P. Sternberg, J.B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math. 49 (1989) 116-133.

Digital Library

[38]

{38} U. Seifert, Adhesion of vesicles in two dimensions, Phys. Rev. A 43 (1991) 6803-6814.

[39]

{39} U. Seifert, Curvature-induced lateral phase segregation in two-component vesicles, Phys. Rev. Lett. 70 (1993) 1335-1338.

[40]

{40} H.M. Soner, Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling, in: Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids, Springer, Berlin, 1999, pp. 413-471.

[41]

{41} J.E. Taylor, J.W. Cahn, Linking anisotropic sharp and diffuse surface motion laws via gradient flows, J. Stat. Phys. 77 (1994) 183-197.

[42]

{42} E.M. Toose, B.J. Geurts, J.G.M. Kuerten, A boundary integral method for two-dimensional (non)-Newtonian drops in slow viscous flow, J. Non-Newtonian Fluid Mech. 60 (1995) 129-154.

[43]

{43} J. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous density variation, J. Stat. Phys. 20 (1893) 197-244.

[44]

{44} T.J. Willmore, Riemannian Geometry, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1993.

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A phase field approach in the numerical study of the elastic bending energy for vesicle membranes

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Published In

cover image Journal of Computational Physics

Journal of Computational Physics Volume 198, Issue 2

10 August 2004

382 pages

ISSN:0021-9991

Issue’s Table of Contents

Publisher

Academic Press Professional, Inc.

United States

Publication History

Published: 10 August 2004

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Cited By

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Deng DLin SWang Q(2024)A class of monotone and structure-preserving Du Fort-Frankel schemes for nonlinear Allen-Cahn equationComputers & Mathematics with Applications10.1016/j.camwa.2024.06.023170:C(1-24)Online publication date: 15-Sep-2024
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Zheng ZNi XHe J(2024)Lagrange multiplier structure-preserving algorithm for time-fractional Allen-Cahn equationComputers & Mathematics with Applications10.1016/j.camwa.2024.03.030164:C(67-78)Online publication date: 15-Jun-2024
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Liu SYu H(2024)Energetic spectral-element time marching methods for phase-field nonlinear gradient systemsApplied Numerical Mathematics10.1016/j.apnum.2024.06.021205:C(38-59)Online publication date: 1-Nov-2024
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