Statistical Mechanics of Membranes and Surfaces

by Nelson, David R.; Piran, Tsvi; Weinberg, Steven

• ISBN13: 9789812387608
• ISBN10: 9812387609
• Edition: 2nd
• Format: Hardcover
• Copyright: 2004-09-01
• Publisher: World Scientific Pub Co Inc

Summary

This invaluable book explores the delicate interplay between geometry and statistical mechanics in materials such as microemulsions, wetting and growth interfaces, bulk lyotropic liquid crystals, chalcogenide glasses and sheet polymers, using tools from the fields of polymer physics, differential geometry, field theory and critical phenomena. Several chapters have been updated relative to the classic 1989 edition. Morever, there are now three entirely new chapters -- on effects of anisotropy and heterogeneity, on fixed connectivity membranes and on triangulated surface models of fluctuating membranes.

Table of Contents

Preface to the First Edition	v
Preface to the Second Edition	vii
Chapter 1 The Statistical Mechanics of Membranes and Interfaces	1	(18)
David R. Nelson
1 Flat Surfaces	1	(4)
1.1 The Roughening Transition	1	(3)
1.2 Wetting Transitions	4	(1)
2 Crumpled Membranes	5	(11)
2.1 Experimental Realizations	5	(3)
2.2 Plaquette Surfaces	8	(3)
2.3 Perturbation Theory for Tethered Surfaces	11	(5)
References	16	(3)
Chapter 2 Interfaces: Fluctuations, Interactions and Related Transitions	19	(30)
Michael E. Fisher
Introduction	19	(1)
1 Interface Models, Mean Field Theory, and Wetting	20	(6)
1.1 Levels of Theory	20	(1)
1.2 Mean Field Theory for Order Parameters	20	(1)
1.3 Derivation of Interface Models	21	(1)
1.4 External Forces	22	(1)
1.5 The Complete Wetting Transition	23	(1)
1.6 Wall Effects and the Interface Hamiltonian	23	(2)
1.7 Wetting Transitions with Short-range Forces: Mean Field Theory	25	(1)
2 Fluctuations and Steric Repulsions	26	(10)
2.1 The Wandering Exponent	26	(1)
2.2 Interfaces in Two-dimensions: Random Walks	27	(1)
2.3 Correlations and Correlation Lengths	28	(1)
2.4 The Stiffening or Roughening Transition	28	(1)
2.5 Random Media	29	(2)
2.6 Fluctuations at Complete Wetting under Long-range Forces	31	(1)
2.7 Constrained Interfaces and the Wall-interface Potential	31	(1)
2.8 Membranes	32	(1)
2.9 Checks of the Wall-interface Potential	33	(1)
2.10 Complete Wetting Revisited	33	(1)
2.11 Many Walls and the Shape of a Vicinal Crystal Face	34	(2)
3 Critical Wetting and Renormalization Groups for Interfaces	36	(13)
3.1 Critical Wetting in d = 2 Dimensions	37	(1)
3.2 Renormalization Groups for Critical Wetting	38	(1)
3.3 The Linearized Functional Renormalization Group	39	(1)
3.4 Critical Wetting in d = 3 Dimensions	40	(1)
3.5 Test of the d = 3 Critical Wetting Predictions	41	(1)
3.6 Approximate Nonlinear Renormalization Group	42	(1)
3.7 Numerical Studies	43	(1)
3.8 Approach to d = 3: Anomalous Bifurcation	43	(1)
3.9 Concluding Remarks	44	(1)
References	45	(4)
Chapter 3 Equilibrium Statistical Mechanics of Fluctuating Films and Membranes	49	(54)
Stanislas Leibler
Introduction	49	(2)
1 Cell Membranes as an Inspiration for Physics	51	(5)
1.1 A History of the Discovery of Membrane Structure: A Few Basic Facts about Membranes	51	(4)
1.2 Some Physical Properties of Membranes and Amphiphilic Films	55	(1)
2 The Elastic Properties of Fluid Membranes and the Shapes of Vesicles	56	(10)
2.1 Curvature Energy and a Simple Elastic Model	56	(4)
2.2 Shapes and Fluctuations of Vesicles	60	(4)
2.3 Measuring of Elastic Constants	64	(2)
3 The Role of Thermal Fluctuations in the Behavior of (Fluid) Membranes and Films	66	(12)
3.1 Fluctuations of a Single Fluid Membrane	66	(3)
3.2 Perturbation Calculations and the Concept of Crumpling Transition	69	(4)
3.3 The Thermodynamic Behavior of an Ensemble of Fluid Membranes	73	(5)
4 Unbinding Transitions and the Swelling of Lamellar Phases	78	(10)
4.1 Molecular Forces between Membranes	78	(2)
4.2 Fluctuations-Induced Interactions	80	(2)
4.3 The Competition between Molecular and Fluctuation-Induced Interactions: Functional Renormalization	82	(2)
4.4 Complete versus Incomplete Unbinding and the Swelling of Lamellar Phases	84	(3)
4.5 The Critical Unbinding Transition	87	(1)
5 Membranes with Internal Degrees of Freedom	88	(10)
5.1 The "Membranology" of f-Membrane Systems	88	(1)
5.2 Curvature Instability in Fluid Membranes	88	(4)
5.3 The Polymorphism of Lipid/Water Systems: Different Kinds of f-Membranes	92	(1)
5.4 Towards a Mean-Field Theory of Lamellar Phases	93	(3)
5.5 Cubic Phases as Crystals of f-Membranes	96	(2)
References	98	(5)
Chapter 4 The Physics of Microemulsions and Amphiphilic Monolayers	103	(8)
David Andelman
References	108	(3)
Chapter 5 Properties of Tethered Surfaces	111	(20)
Yacov Kantor
1 Introduction	111	(3)
1.1 What is a "Tethered Surface"?	111	(2)
1.2 The Tethered Surface as a Polymer	113	(1)
2 Phantom Chains and Networks	114	(6)
2.1 Linear Polymers	114	(1)
2.2 Gaussian Networks and Surfaces	115	(4)
2.3 Properties of Phantom Tethered Surfaces	119	(1)
3 Excluded Volume Effects	120	(5)
3.1 Bounds on the Exponent ν	120	(2)
3.2 Analytic Estimates of ν	122	(1)
3.3 Monte Carlo Investigation of Tethered Surfaces	123	(2)
4 Crumpling Transition in Tethered Surfaces	125	(3)
4.1 Very Rigid and Very Flexible Surfaces	125	(3)
4.2 Excluded Volume Effects	128	(1)
5 Concluding Remarks	128	(2)
5.1 Summary and Discussion	128	(1)
5.2 What Next?	129	(1)
References	130	(1)
Chapter 6 Theory of the Crumpling Transition	131	(18)
David R. Nelson
1 Normal-Normal Correlation in Liquid Membranes	131	(3)
2 Tethered Surfaces with Bending Energy	134	(5)
3 Landau Theory of the Crumpling Transition	139	(4)
4 Defects and Hexatic Order in Membranes	143	(5)
References	148	(1)
Chapter 7 Geometry and Field Theory of Random Surfaces and Membranes	149	(62)
François David
1 Introduction	149	(1)
2 Differential Geometry for Surfaces	150	(20)
2.1 Surfaces, Tangent Vectors, Tensors	151	(5)
2.2 Geodesics, Parallel Transport, Covariant Derivatives	156	(3)
2.3 Integration, Stokes Formula	159	(1)
2.4 Extrinsic Curvature	159	(2)
2.5 The Riemann Curvature Tensor	161	(3)
2.6 The Gauss-Bonnet Theorem	164	(3)
2.7 Minimal Surfaces	167	(1)
2.8 Conformal (or Isothermal) Coordinates	168	(2)
3 Fields on Surfaces	170	(4)
3.1 Free Field	170	(2)
3.2 The Heat Kernel Regularization	172	(1)
3.3 The Conformal Anomaly and the Liouville Action	173	(1)
4 Fluid Membranes Models	174	(11)
4.1 Continuous Model for Fluid Membranes	175	(1)
4.2 Partition Function, Gauge Fixing	176	(4)
4.3 Effective Action and the Background Field Method	180	(1)
4.4 Renormalization of the Bending and Gaussian Rigidity	181	(3)
4.5 Renormalization of the Surface Tension	184	(1)
4.6 Effect of Tangential Flows	184	(1)
5 Fluid Membranes: Non-Perturbative Issues and the Large d Limit	185	(9)
5.1 The Large d Limit	185	(1)
5.2 Planar Configuration	186	(3)
5.3 Renormalization Group Behavior	189	(2)
5.4 Conformal Fluctuations and Instabilities	191	(3)
6 Effective Models for Fluid Membranes and Strings	194	(3)
6.1 The Polyakov String Model	194	(1)
6.2 The Liouville Model	194	(2)
6.3 Discretized Models for Surfaces	196	(1)
7 Hexatic Membranes	197	(7)
7.1 Hexatic Membranes: Continuous Model	198	(3)
7.2 Hexatic Membranes: Renormalization Group Behavior	201	(3)
8 Crystalline Membranes	204	(7)
References	208	(3)
Chapter 8 Statistical Mechanics of Self-Avoiding Crumpled Manifolds - Part I	211	(34)
Bertrand Duplantier
1 Continuum Model of Self-Avoiding Manifolds	211	(7)
1.1 Edwards Model	211	(2)
1.2 Gaussian D-Dimensional Manifold	213	(1)
1.3 Dimensional Analysis	214	(2)
1.4 Higher Order Interactions	216	(1)
1.5 Analytical Continuation in Dimension and Regularization	216	(1)
1.6 Extension to Negative Dimensions	217	(1)
2 Perturbation Expansion	218	(10)
2.1 Rules	218	(2)
2.2 Divergences	220	(1)
2.2.1 End-to-end Distance	220	(1)
2.2.2 Partition Function	221	(2)
2.2.3 Dimensional Regularization	223	(5)
3 Direct Renormalization	228	(5)
3.1 Scaling Functions	228	(1)
3.2 Second Virial Coefficient	229	(3)
3.3 Critical Exponents	232	(1)
4 Contact Exponents	233	(5)
5 On the Nonuniversality of Exponent γ	238	(3)
6 Conclusion	241	(1)
References	242	(3)
Chapter 9 Statistical Mechanics of Self-Avoiding Crumpled Manifolds - Part II	245	(30)
Bertrand Duplantier
1 Interacting Manifold Renormalization: A Brief History	245	(3)
2 Manifold Model with Local 6 Interaction	248	(12)
2.1 Perturbative Expansion	248	(2)
2.2 Second Virial Coefficient	250	(1)
2.3 Resummation of Leading Divergences	251	(2)
2.4 Comparison to One-Loop Renormalization	253	(1)
2.5 Analytic Continuation in D of the Euclidean Measure	254	(2)
2.6 Analysis of Divergences	256	(1)
2.7 Factorizations	257	(1)
2.8 Renormalization	258	(2)
3 Self-Avoiding Manifolds and Edwards Models	260	(12)
3.1 Introduction	260	(1)
3.2 Renormalizability to First Order	261	(1)
3.3 Renormalizability to All Orders	262	(1)
3.4 Perturbation Theory and Dipole Representation	263	(3)
3.5 Singular Configurations and Electrostatics in RD	266	(1)
3.6 Multi-local Operator Product Expansion	267	(1)
3.7 Power Counting and Renormalization	268	(2)
3.8 Finite Size Scaling and Direct Renormalization	270	(1)
3.9 Hyperscaling	271	(1)
3.10 Θ-Point and Long-Range Interactions	271	(1)
References	272	(3)
Chapter 10 Anisotropic and Heterogeneous Polymerized Membranes	275	(48)
Leo Radzihovsky
1 Preamble	275	(1)
2 Anisotropic Polymerized Membranes	276	(27)
2.1 Motivation and Introduction	276	(3)
2.2 Model	279	(1)
2.3 Mean-field theory	279	(1)
2.4 Fluctuations and Self-avoidance in the Crumpled and Flat Phases	280	(1)
2.4.1 Anomalous Elasticity of the Flat Phase	281	(1)
2.4.2 SCSA of the Flat Phase	282	(3)
2.5 Fluctuations in "Phantom" Tubules	285	(1)
2.5.1 Anomalous Elasticity of the Tubule Phase	286	(2)
2.5.2 Zero-modes and Tubule Shape Correlation	288	(2)
2.6 Self-avoidance in the Tubule Phase	290	(1)
2.6.1 Flory Theory	290	(1)
2.6.2 Renormalization Group and Scaling Relations	291	(7)
2.7 Phase Transitions	298	(1)
2.7.1 Renormalization Group Analysis of Crumpled-To-Tubule Transition	298	(2)
2.7.2 Scaling Theory of Crumpled-To-Tubule and Tubule-To-Flat Transitions	300	(3)
3 Random Heterogeneity in Polymerized Membranes	303	(11)
3.1 Motivation	303	(1)
3.2 Model of a Heterogeneous Polymerized Membrane	304	(2)
3.3 Weak Quenched Disorder: "Flat-glass"	306	(1)
3.3.1 Short-range Disorder	306	(3)
3.3.2 Long-range Disorder	309	(1)
3.4 Strong Quenched Disorder: "Crumpled-glass"	310	(4)
4 Interplay of Anisotropy and Heterogeneity: Nematic Elastomer Membranes	314	(4)
5 Summary	318	(1)
6 Acknowledgments	318	(1)
References	318	(5)
Chapter 11 Fixed-Connectivity Membranes	323	(36)
Mark J. Bowick
1 Introduction	323	(1)
2 Physical Examples of Membranes	324	(2)
3 Phase Diagrams	326	(12)
3.1 Phantom Membranes	328	(1)
3.1.1 The Crumpled Phase	329	(1)
3.1.2 The Crumpling Transition	330	(2)
3.1.3 The Flat Phase	332	(2)
3.1.4 The Properties of the Flat Phase	334	(2)
3.2 Self-avoiding Membranes	336	(1)
3.2.1 Numerical Simulations	336	(1)
3.2.2 The Properties of the Self-avoiding Fixed Point	337	(1)
4 Poisson Ratio and Auxetics	338	(2)
5 Anisotropic Membranes	340	(6)
5.1 Phantom Tubular Phase	341	(1)
5.1.1 The Phase Diagram	341	(2)
5.1.2 The Crumpled Anisotropic Phase	343	(1)
5.1.3 The Flat Phase	343	(1)
5.2 The Tubular Phase	343	(3)
6 Order on Curved Surfaces	346	(7)
References	353	(6)
Chapter 12 Triangulated-Surface Models of Fluctuating Membranes	359
G. Gompper and D.M. Kroll
1 Introduction	359	(2)
2 Polymerized Membranes	361	(20)
2.1 Elastic Free Energy and Flory Theory	361	(1)
2.1.1 The Crumpled Phase	362	(1)
2.1.2 The Flat Phase	363	(1)
2.1.3 The Crumpling Transition	363	(1)
2.1.4 Phantom Networks and the Influence of Self-Avoidance	363	(1)
2.2 Tethered Networks	364	(1)
2.2.1 Tether-and-Bead and Gaussian Models	365	(1)
2.2.2 Self-Avoidance and Bending Energy	366	(1)
2.3 Simulation Methods	367	(1)
2.3.1 Basic Algorithm	367	(1)
2.3.2 Periodic Boundary Conditions	367	(1)
2.3.3 Determination of Elastic Constants	368	(2)
2.4 Fluctuations About the Flat Phase	370	(1)
2.4.1 Free Energy and Renormalization Group Results	370	(1)
2.4.2 Simulations of the Flat Phase	371	(1)
2.4.3 Effect of Self-Avoidance	372	(2)
2.5 Heterogeneous Polymer-Fluid Networks	374	(1)
2.6 Shape of Spherical Shells and Forced Crumpling	375	(1)
2.6.1 Scaling Theory of Stretching Ridges	375	(3)
2.6.2 Simulated Shapes of Spherical Shells	378	(1)
2.6.3 Forced Crumpling of Elastic Sheets	379	(2)
3 Fluid Membranes and Vesicles	381	(18)
3.1 Spontaneous Curvature Model and Area-Difference-Elasticity Model for Bilayer Vesicles	381	(1)
3.2 Randomly-Triangulated-Surface Models for Fluid Membranes	382	(1)
3.2.1 Dynamic Triangulation	382	(1)
3.2.2 Bending Energy	382	(2)
3.3 Phase Diagram of Fluid Vesicles at Low Bending Rigidities	384	(2)
3.4 Quasi-Spherical Vesicles	386	(2)
3.5 Renormalization of the Bending Rigidity	388	(1)
3.5.1 Renormalization Group Theory	388	(1)
3.5.2 Scaling of the Vesicle Volume	389	(1)
3.5.3 Undulation Modes of Quasi-Spherical Vesicles	389	(2)
3.6 Fluctuations of Non-Spherical Vesicles	391	(2)
3.7 Dynamics of Vesicles in External Fields	393	(1)
3.7.1 Elongational and Shear Flow	394	(1)
3.7.2 Vesicles in Micro-Channels	395	(1)
3.8 Fluid Membranes with Edges	395	(2)
3.9 Two-Component Fluid Membranes	397	(1)
3.9.1 Strong-Segregation Limit and Domain-Induced Budding	397	(1)
3.9.2 Triangulated-Surface Models	397	(1)
3.9.3 Phase Separation and Budding Dynamics of Two-Component Membranes	398	(1)
4 Crystalline and Hexatic Membranes	399	(10)
4.1 Melting in Two Dimensions	399	(1)
4.1.1 Theory of Kosterlitz, Thouless, Halperin, Nelson and Young (KTHNY)	399	(2)
4.1.2 Simulation Results for Network Models in Two Dimensions	401	(1)
4.2 Freezing of Flexible Membranes	402	(1)
4.2.1 Continuum Model and Renormalization Group Results	402	(3)
4.2.2 Simulation Results for Triangulated Surfaces	405	(2)
4.3 Budding of Crystalline Domains in Fluid Membranes	407	(2)
5 Membranes of Fluctuating Topology	409	(11)
5.1 Microemulsion and Sponge Phases	409	(3)
5.2 Theoretical Predictions	412	(1)
5.2.1 Gaussian Random Fields	412	(2)
5.2.2 Small-Scale Membrane Fluctuations, Scale-Dependent Rigidity, and Phase Behavior	414	(2)
5.3 Triangulated-Surface Models for Membranes with Fluctuating Topology	416	(1)
5.4 Simulation Results	417	(2)
5.5 Comparison with Experiments	419	(1)
6 Summary and Outlook	420
References	421