Polymers, Liquid Crystals, and Low-Dimensional Solids

This book deals with three related areas having both fundamental and technological interest. In the first part, the objective is to provide a bird's­ eye view on structure in polymeric solids. This is then complemented by a chapter, directly technological in its emphasis, dealing with the influence of processing on polymeric materials. In spite of the technological interest, this leads to some of the current fundamental theory. Part II, concerned with liquid crystals, starts with a discussion of the physics of the various types of material, and concludes with a treatment of optical applications. Again, aspects of the theory are stressed though this part is basically phenomenological in character. In Part III, an account is given first of the use of chemical-bonding arguments in understanding the electronic structure of low-dimensional solids, followed by a comprehensive treatment of the influence of dimen­ sionality on phase transitions. A brief summary of dielectric screening in low-dimensional solids follows. Space-charge layers are then treated, including semiconductor inversion layers. Effects of limited dimensionality on superconductivity are also emphasized. Part IV concludes the volume with two specialized topics: electronic structure of biopolymers, and topological defects and disordered systems. The Editors wish to acknowledge that this book had its origins in the material presented at a course organized by the International Centre for Theoretical Physics, Trieste.

I. Polymers.- 1. Introduction to Polymeric Structure and Polymers.- 1.1. Classification.- 1.1.1. Linear Chains and Networks.- 1.1.2. Periodic and Aperiodic Polymers.- 1.1.3. Homopolymers and Copolymers.- 1.1.4. Single-Component Polymers.- 1.2. Main Types of Polymerization Reaction.- 1.2.1. Condensation Polymerization.- 1.2.2. Addition Polymerization.- 1.3. Imperfection Types in a Linear Homopolymer Chain.- 1.3.1. End Groups.- 1.3.2. Molecular Length (Weight) Distribution.- 1.3.3. Isomerism.- 1.4. Formulas of Some Important Polymers.- 1.4.1. Condensation Polymers.- 1.4.2. Addition Polymers.- 1.5. Melting Range of Polymers; Specialty Materials.- 1.5.1. Conventional Polymers.- 1.5.2. Specialty Materials.- 1.6. The Physical State.- 1.6.1. Resumé of the Amorphous State.- 1.6.2. The Usefulness of Polymers in Terms of Their Physical State.- 2. Crystallinity and Kinetics of Crystallization.- 2.1. Basic Classifications.- 2.1.1. Generalities.- 2.1.2. Sources of Lattice Imperfections in Polymers.- 2.1.3. Modes of Crystallization.- 2.2. Crystal Structure.- 2.2.1. The Unit Cell.- 2.2.2. Chain Conformations.- 2.2.3. Chain Packing.- 2.2.4. Crystal Structure Determination.- 2.3. Degree of Crystallinity.- 2.3.1. The Principle of the Determination.- 2.3.2. Methods of Determination.- 2.3.3. An Appreciation of the Different Methods.- 2.4. Kinetics of Crystallization.- 2.4.1. Rates of Crystallization.- 2.4.2. The Xc vs t Curves.- 2.4.3. Morphological vs True Crystallinity.- 2.4.4. Primary and Secondary Crystallization.- 2.4.5. Textures.- 2.4.6. Analytical Treatment of Crystallization Kinetics.- 2.5. Spherulites.- 2.5.1. Optical Properties of Spherulites.- 2.5.2. Morphology of Spherulites.- 2.5.3. The Fine Structure of Spherulites.- 3. The Basic Crystal Unit.- 3.1. Single-Crystal Lamella.- 3.1.1. Discovery and Description.- 3.1.2. The Chain-Folding Model.- 3.1.3. Fold Length l.- 3.2. Theories of Chain Folding.- 3.2.1. Framework of the Kinetic Theories.- 3.2.2. Further Developments and Problems.- 3.2.3. Growth Rates.- 3.2.4. Melting Behavior as a Function of l.- 3.2.5. Comparison of Crystallization from Solution and Melt.- 3.2.6. Some New Perspectives in Crystallization Theories.- 3.3. Morphology of Chain-Folded Crystals.- 3.3.1. Monolayer Crystals.- 3.3.2. Multilayer Crystals.- 3.4. Fold Structure — Nature of Amorphous Material.- 3.4.1. The Issues.- 3.4.2. Experimentation in Aid of the Fold-Surface Problem.- 3.4.3. Outcome of the Enquiries.- 3.5. Neutron Scattering Experiments: The Chain Trajectory.- 3.5.1. Technique and Potential.- 3.5.2. Angular Ranges.- 3.5.3. Some Results.- 3.6. Alternative Morphologies.- 3.6.1. Extended-Chain-Type Crystals.- 3.6.2. Micellar Crystals (Crystal Gels).- 4. Other Classes of Crystallization.- 4.1.Crystallization Concurrent with Polymerization (Nascent Polymers).- 4.2. Orientation-Induced Crystallization.- 4.2.1. General.- 4.2.2. Morphological Background.- 4.2.3. Mode of Chain Extension.- 4.2.4. Structure of Shish-Kebabs.- 4.2.5. Properties of Shish-Kebabs.- 4.2.6. Some Practical Consequences.- 5. Hierarchical Nature of Macromolecular Structure.- 5.1. Introduction.- 5.1.1. Crystalline Constituents.- 5.1.2. Amorphous Constituents.- 5.2. Crystal Defects.- 5.2.1. Defects Within the Crystal Lattice.- 5.2.2. Defects Beyond the Level of the Lattice.- 5.3. Thermal Behavior.- 5.3.1. Amorphous Material.- 5.3.2. Crystal Lattice.- 5.3.3. Melting Range.- 5.4. Deformation.- 5.4.1. Polymers as Self-Structured Composites.- 6. Influence of Processing on Polymeric Materials.- 6.1. Polymeric Processing.- 6.2. General Comments on Influence of Flow.- 6.3. Dumbbell Model.- 6.3.1. Shear Flow.- 6.3.2. Elongational Flow.- 6.3.3. Other Flows.- 6.4. Multiplicity of Friction Points: Rouse-Zimm Model.- 6.5. Concentrated Systems.- 6.6. Effects of Flow on Crystallization.- 6.7. Recent Developments.- References and Bibliography for Part I.- II. Liquid Crystals.- 7. Structural Classification of Thermotropic Liquid Crystals.- 7.1. Introduction.- 7.2. Rod-Like Molecules.- 7.2.1. Effect of Pressure on Polymorphism.- 7.2.2. The Reentrant Phenomenon.- 7.3. Disk-Like Molecules.- 8. Nematic Liquid Crystals.- 8.1. Elastic Properties.- 8.1.1. Basic Equations.- 8.1.2. Determination of the Elastic Constants: The Freedericksz Effect.- 8.1.3. Orientational Fluctuations and Light Scattering.- 8.1.4. Disclinations.- 8.2. Viscous Properties.- 8.2.1. Experimental Determination of the Viscosity Coefficients.- 8.2.2. Viscous Torques.- 8.2.3. Orientational Relaxation.- 8.3. Nematic-Isotropic Transition.- 8.3.1. Molecular Theories of the Nematic Phase.- 8.3.2. Short-Range Order Effects in the Isotropic Phase: The Landau-de Gennes Model.- 8.3.3. Near-Neighbor Correlations.- 9. Cholesteric Liquid Crystals.- 9.1. Optical Properties.- 9.2. Flow Properties.- 10. Smectic Liquid Crystals.- 10.1. Smectic A.- 10.1.1. Continuum Theory of Smectic A.- 10.1.2. The Smectic A — Nematic Transition.- 10.1.3. Bilayer Smectic A and the Reentrant Phenomenon.- 10.2. Smectic.- 10.3. Flexoelectricity in Liquid Crystals.- 10.4. Ferroelectric Liquid Crystals.- 11. Optical Applications of Liquid Crystals.- 11.1. Why Use Liquid Crystals for Optical Applications?.- 11.1.1. Mechanical Properties of Liquid Crystal Phases.- 11.1.2. Coupling to External Fields.- 11.1.3. Coupling to Light.- 11.1.4. Conclusion.- 11.2. Optical Properties of Textures.- 11.2.1. Orientation of Uniform Textures.- 11.2.2. Optical Properties of Twisted Textures.- 11.2.3. Use of Dichroic Dyes in Solution.- 11.3. Texture Distortions Under the Action of an Electric Field.- 11.3.1. Electrically Controlled Birefringence.- 11.3.2. Twisted Nematic Valve.- 11.3.3. Dynamic-Scattering Mode.- 11.3.4. Texture Changes in Cholesterics.- 11.4. Multiplexing of Liquid-Crystal Displays.- 11.4.1. Response Time of Liquid-Crystal Texture Instabilities.- 11.4.2. Multiplexing Liquid Crystals.- 11.4.3. Parallel Addressing.- 11.5. Applications of Smectics.- 11.5.1. Mechanical Properties of Smectic Textures.- 11.5.2. Electric Field Effects on Smectic A.- 11.5.3. The Thermally Excited Optical Smectic Valve.- 11.5.4. The Thermal Electric X-Y Addressed Smectic.- 11.5.5. Ferroelectric C*. The Bistable Optical Switch.- References for Part II.- III. Low-Dimentional Solids.- 12. Chemical Bonding.- 12.1. Outline.- 12.2. Linear Polyenes.- 12.2.1. Free-Electron Model: Absorption Spectra as Function of Chain Length.- 12.2.2. Linear Combination of Atomic Orbitals (LCAO) Hückel Approximation.- 12.2.3. Alternating Bond Lengths.- 12.2.4. Effect of Electron-Electron Interactions.- 12.3. Platinum Chain Compounds.- 12.3.1. Tight-Binding (LCAO) Results.- 12.3.2. Role of Ligands.- 12.4. Two-Dimensional Layer Compounds I.- 12.4.1. Structure of Transition Metal-Dichalcognide Layer Compounds.- 12.4.2. Trigonal-Prismatic Coordination.- 12.4.3. Crystal-Field Splitting of d-Levels.- 12.4.4. Molecular-Orbital Calculations.- 12.4.5. Effect of ?-Bonding.- 12.4.6.Role of d-Covalency in Stabilizing Trigonal- Prismatic Coordination.- 12.5. Two-Dimensional Layer Compounds II.- 12.5.1. Electronic Structure of Borazole.- 12.5.2. Broadening of ?-Levels into Bands.- 12.6. Molecular Scattered Wave Calculations I: Treatment of TTF-TCNQ.- 12.6.1. Molecular Structure of Cation and Anion.- 12.6.2. Energy Levels of TCNQ- and TTF+.- 12.6.3. Approximation to Energy Levels of a Crystal.- 12.6.4. Photoemission Spectrum.- 12.7. Molecular Scattered Wave Calculations II: Treatment of Polymer (SN)X.- 12.7.1. Square and Open S2N2.- 12.7.2. S4N4 Chain.- 12.7.3. X-Ray Photoelectron Spectrum.- 12.7.4. Interaction of Two S4N4 Chains.- 12.7.5. Band-Structure of (SN)X.- 12.8. Electron Density Description of the Ground State of Molecules.- 12.8.1. Simplest Theory of Inhomogeneous Electron Gas.- 12.8.2. Value of the Chemical Potential in a Neutral Atom or Molecule.- 12.8.3. Relation Between Total Energy and Sum of Orbital Energies.- 12.8.4. Inclusion of Density Gradients and Electron Correlation.- Appendix A12.1. Alternation of Bond Lengths in Long Conjugated Chain Molecules.- Appendix A12.2. Partitioning of the Space of a Molecule (and One-Body Potential).- 13. Phase Transitions and Dimensionality.- 13.1. Cooperative Behavior and Phase Transitions.- 13.1.1. Basic Aspects of Cooperative Behavior and Phase Transitions.- 13.1.2. Criticality.- 13.1.3. Correlation Functions.- 13.2. Systems and Models Exhibiting Transitions.- 13.2.1. Magnetic Systems.- 13.2.2. Alloys.- 13.2.3. Liquid-Gas Systems.- 13.2.4. Melting.- 13.2.5. ?-Transition in He4.- 13.2.6. Superconductivity.- 13.2.7. Peierls’ Transition.- 13.2.8. Mott and Anderson Transitions.- 13.2.9. Structural Transitions; Jahn-Teller Transition.- 13.2.10. Ferroelectric Transitions.- 13.2.11. Percolation.- 13.2.12. Summary.- 13.2.13. Bose Condensation.- 13.3. Mean-Field Theory.- 13.3.1. Mean-Field Theory for Magnetic Systems.- 13.3.2. Ornstein-Zernicke Theory.- 13.3.3. Crntical Exponents of Mean-Field and Ornstein-Zernicke Theories.- 13.3.4. Mean-Field Theory for Superconductivity.- 13.3.5. Mean-Field Theory for Jahn-Teller Transitions.- 13.3.6. Critique of Mean Field Approaches.- 13.4. Excitations.- 13.4.1. Ground States.- 13.4.2. Low-Lying Excitations.- 13.4.3. Random-Phase Approximation.- 13.4.4. Broken Symmetry.- 13.5. Instabilities and Fluctuations.- 13.5.1. Instabilities in Heisenberg Magnets.- 13.5.2. Instabilities in Ising Magnets.- 13.5.3. Lower Critical Dimensionality for Bose-Einstein Condensation.- 13.5.4. Instability of Low-Dimensional Crystal Lattices.- 13.5.5. Melting.- 13.6. Peierls’ Transition, Charge-Density Waves, Solitons, and Phasons.- 13.7. Quasi-Low-Dimensional Systems.- 13.7.1. Introduction.- 13.7.2. Weakly-Coupled-Layer Heisenberg Ferromagnet.- 13.8. Transitions Without Usual Long-Range Order; Kosterlitz-Thouless Transition.- 13.8.1. Introduction.- 13.8.2. Two-Dimensional XY Model and Superfluid.- 13.9. Critical Behavior.- 13.9.1. Exponents and Universality.- 13.9.2. Upper Critical Dimensionality, Power Counting.- 13.9.3. Homogeneity and Scaling.- 13.9.4. Renormalization Group Method.- 13.10. Critical Behavior of Low-Dimensional Systems.- 13.10.1. One-Dimensional Magnetic Systems.- 13.10.2. Two-Dimensional Magnetic Systems.- 13.10.3. Percolation and Other Disorder Problems.- 13.11. Concluding Remarks.- 14. Many-Electron Effects.- 14.1. Response to Electric and Magnetic Fields.- 14.1.1. Some General Formulas for the Dielectric Function.- 14.1.2. The Dielectric Function of a Uniform System.- 14.1.3. The Magnetic Susceptibility.- 14.2. Self-Consistent Independent-Particle Models.- 14.3. The Electron Liquid.- 14.3.1. Approximate Dielectric Functions.- 14.3.2. Collective Modes and Screening Effects.- 14.3.3. Comparison Between Different Approximations to the Dielectric Function.- 14.3.4. The One-Electron Spectrum.- 14.3.5. Effect of Correlations on Spin Susceptibility.- 14.3.6. Different Types of Ground States: Ferromagnetic, Wigner Lattice, Spin Density Waves, Charge Density Waves.- 14.4. The Two-Dimensional Electron Liquid.- 14.5. Partially Localized Electrons.- 14.5.1. Electron-Electron Interactions in Van der Waals Crystals.- 14.5.2. The Hubbard Hamiltonian.- 14.5.3. Discussion of the Hubbard Model.- 15. Space Charge Layers.- 15.1. Screening in One- and Three-Dimensional Systems.- 15.1.1. Screened Point Charge in Three Dimensions.- 15.1.2. Screening with One Spatial Variable.- 15.2. Physical Examples.- 15.2.1. p-n Junctions.- 15.2.2. The Jellium-Vacuum Interface.- 15.2.3. Semiconductor Surfaces.- 15.2.4. Schottky Barriers.- 15.2.5. Metal-Insulator-Semiconductor Systems.- 15.2.6. Elections on Liquid Helium.- 15.3. Experimental Probes of Space-Charge Layers.- 15.3.1. Capacitance.- 15.3.2. Conductance.- 15.3.3. Photoeffects.- 15.3.4. Interface Characterization.- 15.4. Semiconductor Inversion Layers.- 15.4.1. Quantum Effects.- 15.4.2. Transport Properties.- 15.4.3. Optical Properties.- 15.4.4. Impurity Bands and Localization.- 15.5. Two-Dimensional Physics: Some Examples.- 15.5.1. Coulomb Scattering.- 15.5.2. Bound States.- 16. Superconductivity via Electron-Phonon and Electron-Exciton Interactions.- 16.1. Electron-Phonon Hamiltonian.- 16.2. Canonical Transformation and Cooper Pair.- 16.3. Superconductivity: The Ground State.- 16.4. The Bogoliubov Equation.- 16.5. The Transition Temperature.- 16.6. Landau-Ginzburg Theory.- 16.7. Excitonic Mechanisms.- 16.8. Effects of Limited Dimensionality.- References for Part III.- IV. Special Topics.- 17. Biopolymer Electronic Phenomena.- 17.1. Ab-initio SCF-LCAO Crystal-Orbital Formalism.- 17.1.1. Simple Translational Symmetry.- 17.1.2. The SCF-LCAO-CO Method in the Case of a Combined Symmetry Operation.- 17.2. Semiempirical SCF-LCAO Crystal-Orbital Methods.- 17.2.1. The Semiempirical SCF-Electron (Pariser-Parr-Pople) Crystal-Orbital Method.- 17.2.2. The Semiempirical SCF All-Valence Electron (CNDO/2) Crystal-Orbital Method.- 17.3. Correction of the Virtual Levels and for Long-Range Correlation.- 17.3.1. Application of the Excitation Hamiltonian (ÔÂÔ) Method to Polymers.- 17.3.2. Correlation Corrections to the Hartree Fock Bands on the Basis of the Electron-Polaron Model.- 17.4. Applications to Periodic DNA and Protein Models.- 17.4.1. Periodic DNA Models.- 17.4.2. Periodic Protein Models.- 17.5. Treatment of Correlation in Polymers.- 17.5.1. Intermediate Exciton Theory for Excited States of Polymers.- 17.5.2. Discussion of the Correlation in the Ground State of Polymers.- 17.6. Methods for Treatment of Aperiodic Polymers.- 17.6.1. The Virtual Crystal, Coherent Potential Approximation (CPA), and SCF Resolvent Method.- 17.6.2. Negative-Factor Counting Techniques.- 17.7. Transport Properties of Biopolymers.- 17.8. From Electronic Structures to Biological Functions.- 17.8.1. Hypotheses for Tumor Development on the Molecular Level.- 17.8.2. Possible Local Effects of Carcinogens.- 17.8.3. Possible Nonlocal Effects of Carcinogens Bound to DNA.- 17.8.4. DNA-Protein Interaction and Its Possible Change Due to the Binding of a Carcinogen.- 18. Topological Defects and Disordered Systems.- 18.1. Classical Theory of Dislocations.- 18.1.1. Two Everyday Examples.- 18.1.2. Translation Dislocation Lines.- 18.1.3. Burgers Circuit and Burgers Vector.- 18.1.4. Disclinations.- 18.2. Point Defects.- 18.2.1. Defects in “Magnetic” Systems.- 18.2.2. Nematic Liquid Crystals.- 18.3. Homotopy Theory of Defects.- 18.3.1. The Problems.- 18.3.2. Mathematical Tools.- 18.3.3. Connection with Defects.- 18.3.4. General Homotopy Groups.- 18.4. Applications of the Topological Theory.- 18.4.1. Vector Order Parameter.- 18.4.2. Nematics.- 18.4.3. New Problems.- 18.4.4. The Many-Defect Problem.- 18.4.5. Limitations of the Method.- 18.5. Disordered Systems: Dilution and Competition.- 18.5.1. Dilution-Type Disorder.- 18.5.2. Competition-Induced Disorder.- 18.6. Percolation and Related Problems.- 18.6.1. The “Classical Theory”.- 18.6.2. Limitations of the Classical Theory.- 18.6.3. Approaches that Work.- 18.6.4. Phenomenological Renormalization.- 18.6.5. Related Problems.- 18.7. Frustration.- 18.7.1. Local Gauge Invariance.- 18.7.2. Is There a Spin-Glass Transition?.- 18.7.3. Connection with Lattice Gauge-Field Theories.- 18.8. Mean-Field Theory of Spin Glasses.- 18.8.1. The Sherrington-Kirkpatrick Model.- 18.8.2. Replica-Symmetry Breaking.- 18.8.3. The Random-Energy Model.- 18.8.4. The Projection Hypothesis for the S-K Model.- 18.9. Conclusion.- References for Part IV.