液晶相生物膜弹性理论的几何方法/高瞻系列/中外物理学精品书系

《液晶相生物膜弹性理论的几何方法》将膜的力 学平衡和形变作为微分几何中的表面问题而进行了综 合阐述。根据前人的理论，液体膜可以看成向列相或 近晶相的液晶薄膜，而且能量形式可以精确地利用液 晶的曲率弹性理论导出。在固定渗透压和表面张力时 利用表面变分和能量极小化能够得出完全新的几何上 的表面方程，这些方程具有数学上的潜在价值。作者 涂展春、欧阳钟灿、刘寄星、谢毓章对于这些方程的 严格解做过深入研究，本书就是这方面研究的总结。 本书适合本领域以及对这一领域有兴趣的科研工作者 阅读，也可以作为研究生的参考读物。

谢毓章 清华大学现代应用物理系教授。中国液晶物理学开拓者，中国物理学会液晶分会的主要创始人。曾任美国汉普顿学院物理系副教授、中国物理学会液晶分会第一届理事长。 刘寄星 中国科学院理论物理研究所研究员。研究方向为等离子体理论和软物质物理，近年兴趣在中国物理学史。曾因参与生物膜形状的液晶理论研究获中国科学院自然科学一等奖和国家自然科学二等奖。 欧阳钟灿 中国科学院理论物理研究所研究员、中国科学院院士、第三世界科学院院士，在脂质膜泡形状方程研究方面做出奠基性工作。曾获海外华人物理协会亚洲杰出成就奖、国家自然科学二等奖、何梁何利基金科学与技术进步奖。 涂展春 北京师范大学教授。研究方向为理论生物物理学和非平衡热力学，在开口膜泡的形状问题研究中做出重要工作。曾获全国百篇优秀博士学位论文奖，获批国家自然科学基金优秀青年科学基金项目。

1 Introduction to Liquid Crystal Biomembranes 1.1 Liquid Crystals 1.1.1 Mysterious Matter 1.1.2 Orientational Order 1.1.3 Classification of Thermotropics 1.1.4 Classification of Lyotropics 1.2 Amphiphiles and Lyotropic Liquid Crystals 1.2.1 Amphiphile 1.2.2 Monolayer 1.2.3 Micelle 1.2.4 Phase Diagram 1.3 Phase Transitions in Biomembranes 1.3.1 Fluid Mosaic Model 1.3.2 Lipid Bilayer 1.3.3 Phase Transitions in Bilayer 1.3.4 Classification of Lipid Bilayer Phase 1.4 Simple Biochemistry of LC Biomembranes 1.4.1 Effect of Chain Length 1.4.2 Double Bond Effect 1.4.3 Effect of Ionic Condition 1.4.4 Cholesterol Effect 1.4.5 Complexity in Biomembrane 1.5 Artificial Bilayers and Vesicles 1.5.1 Lipids for Artificial Vesicles 1.5.2 Multilamellar Vesicles 1.5.3 Single-Bilayered Vesicles References 2 Curvature Elasticity of Fluid Membranes 2.1 Shape Problem in Red Blood Cell 2.1.1 Membranes in Cell 2.1.2 High Deformability of Cell Membranes 2.1.3 Difficulty in the Explanation of Discocyte Shape 2.2 Classic Differential Geometry of Surface 2.2.1 Lipid-Bilayer Vesicle Viewed as a Closed Surface 2.2.2 Space Curve 2.2.3 Surface and Parametric Curves 2.2.4 First Fundamental Form 2.2.5 Area 2.2.6 The Normal and the Tangent Plane 2.2.7 Second Fundamental Form 2.2.8 Christoffel Symbols 2.2.9 Curves and Directions on a Surface 2.2.10 Normal Curvature of a Curve on a Surface 2.2.11 Principal Directions, Line of Curvature and Principal Curvatures 2.2.12 The Mean Curvature and the Gaussian Curvature 2.3 Differential Invariants on a Surface 2.3.1 Gradient of a Scalar Field 2.3.2 Divergence of a Vector Field 2.3.3 Laplace-Beltrami Operator on a Scalar Function 2.3.4 Two-Dimensional Curl of a Vector 2.3.5 Other Differential Invariants 2.4 Curvature Elasticity of Fluid Membranes in Liquid Crystal Phase 2.4.1 Fluid Membranes Viewed as Liquid Crystals 2.4.2 Helfrich's Approach 2.4.3 A Derivation by Way of 2D Differential Invariants 2.4.4 The Spontaneous Curvature Viewed from Landau-de Gennes Theory 2.4.5 Discussion of Helfrich Bending Energy on Liquid Crystal Point of View 2.4.6 Spontaneous Curvature and Flexoelectric Effect References 3 Shape Equation of Lipid Vesicles and Its Solutions 3.1 Mathematical Preliminary 3.2 General Shape Equation 3.3 Spherical Vesicles 3.4 Nearly Spherical Vesicles and Third Order Energy Variation 3.5 Circular Cylindrical Vesicles 3.6 Noncircular Cylindrical Vesicles 3.7 Clifford Torus 3.8 Dupin Cyclide 3.9 Shape Equation for Axisymmetric Vesicles 3.10 Circular Biconcave Discoid 3.11 Surfaces of Revolution with Constant Mean Curvature and Extended Surfaces 3.12 Challenge References 4 Governing Equations for Open Lipid Membranes and Their Solutions 4.1 Mathematical Preliminary 4.1.1 Surface Theory Based on Moving Frame Method 4.1.2 Hodge Star, Stokes Theorem and some Important Geometric Relations 4.1.3 Variational Theory Based on the Moving Frame Method 4.2 Governing Equations for Open Lipid Membranes 4.2.1 Governing Equations 4.2.2 Compatibility Conditions 4.3 Analytic Solutions 4.4 Quasi-Exact Solutions 4.5 Challenge 4.5.1 Neck Condition of Two-Phase Vesicles in the Budding State. 4.5.2 Minimal Surface with Boundary Curve References 5 Theory of Tilted Chiral Lipid Bilayers 5.1 Theory of Tilted Chiral Lipid Bilayers with Strong Chirality 5.1.1 Geometric Description and Free Energy 5.1.2 Tilt-Equilibrium and Surface-Equilibrium Equations in Case of Strong Chiral Effect 5.1.3 Wound Ribbon Helix 5.1.4 Twisted Ribbon 5.1.5 Spherical Vesicle 5.2 General Theory for TCLB 5.2.1 General Formula of Chiral Free Energy 5.2.2 The Effect of Other Elastic Constants 5.2.3 High- and Low-Pitch Helical Structures of TCLBs 5.3 Concise theory for Chiral Lipid Membranes 5.3.1 Brief Introduction to Moving Frame Method and Exterior Differential Forms 5.3.2 Constructing the Free Energy 5.3.3 Governing Equations to Describe Equilibrium Configurations of CLMs without Free Edges 5.3.4 Governing Equations to Describe Equilibrium Configurations of CLMs with Free Edges 5.3.5 Solutions to Governing Equations of CLMs without Free Edges 5.4 Challenge References 6 Some Untouched Topics 6.1 Nonlocal Theory of Membrane Elasticity 6.1.1 Area-difference Elasticity 6.1.2 Membrane with Nonlocal Interactions 6.2 Numerical Simulations 6.2.1 Surface Evolver 6.2.2 Phase Field 6.3 Effects of Membrane Skeleton 6.3.1 Composite Shell Model of Cell Membranes 6.3.2 Stability of Cell Membranes and the Function of Membrane Skeleton 6.4 Application in Elasticity of Low-Dimensional Carbon Materials 6.4.1 Revised Lenosky Model 6.4.2 Carbon Nanotubes References Appendix A Tensor Calculus Appendix B The Gradient Operator Reference Appendix C Elastic Theory of Membranes Viewed from Force Balance Appendix D A Different Viewpoint of Surface Tension Reference