Dielectric Relaxation and Dynamics of Polar Molecules

The following sections are included:

• PREFACE

• ACKNOWLEDGEMENTS

• CONTENTS

• LIST OF SYMBOLS

The following sections are included:

• Displacement and Polarization Vectors for a Constant External Field

• Relative Complex Permittivity and Susceptibility

• Complex Refractive Index and Absorption Coefficient

• Debye Model of Rotational Diffusion
  • Debye Formula for the Complex Permittivity
  • Sketch of the Debye Relaxation
  • Restrictions on Applicability of the Debye Theory

• About the Dynamic Method
  • Instantaneous Collisions
  • Complex Power
  • Linear Response
  • The Effective Susceptibility
  • Transition from Anisotropy to Isotropy
  • The Low Frequency Approximation
  • Spectrum of a Harmonic Oscillator

• Classification of Semi-microscopic Molecular Models

The following sections are included:

• Gaseous and gas-like states
  • Rotational Spectrum in a Gas State
  • Quasi-Rotational Spectrum in a Liquid
  • The Effect of Molecular Rotation on the Infrared Vibration Spectrum

• Diluted solutions
  • Absorption Spectra
  • Vibration-Rotational Spectra

• Individual Nonassociated Liquids
  • The Evolution of Dielectric Spectra due to the Temperature Alteration
  • Temperature Dependence of the Relaxation Time and of the Lifetime
  • The Self-Diffusion Coefficient and Translational Motion

• H-Bonded Network
  • Anomalies of Liquid Water
  • Dielectric Spectra of Liquid Water
  • Dielectric Spectra of Ice Ih
  • Molecular Model of Reorientation in Liquid Water

• Water Bounded by a Macromolecule
  • Critical Humidity of a Bound Water
  • Dielectric Properties of Humidified Collagen as a Model of a Skin
  • Hemoglobin Molecule in an Aqueous Solution

• Aqueous solutions
  • FIR Spectra of Water, Modified by Ions
  • Aqueous Solution of Dimethyl Sulfoxide (DMSO): the Concentration Dependence

The following sections are included:

• Maxwell's Equations and Wave Equation for a Plane Electromagnetic Wave

• The t₀ Theorem

• The Average Perturbation (AP) Theorem

• Rotation of a Polar Molecule in an Axisymmetric Potential
  • Linear Molecule: Equations of Motion
  • Linear Molecule: the Law of Free Rotation
  • Linear Molecule: Rotation in a Plane
  • Equations of Motion of a Symmetric-Top Molecule
  • Some Symmetry Properties of the Orientational Autocorrelation Function
  • The Law of Free Rotation of a Symmetric-Top Molecule
  • Motion of a Symmetric Top: a More Detailed Picture

• Libration of a Polar Molecule in a Parabolic Potential Well
  • Harmonic Librations in a Plane
  • Harmonic Librations in Space

The following sections are included:

• The Dispersion Equation
  • The Dispersion Equation in Terms of Discrete Charges
  • The Dispersion Equation in Terms of Integral ∫Qdt

• The Steady State and Induced Distributions

• The Effective Susceptibility

• Susceptibility / Permittivity Relation for the Case of Isotropy

The following sections are included:

• Representation in Terms of Perturbed Motion of a Dipole in Radiation Field

• Representation in Terms of Undamped Motion for Thermal Equilibrium
  • Transition to the Spectrum of the Autocorrelation Function

• Relation to the Direction of Alternating External Field
  • Planar and Spatial Ensembles
  • The Quasi-Spatial Ensemble

• The Spectral Functions in the Case of an Isotropic Medium
  • Reorientation in Space
  • Reorientation in a Plane
  • The Quasi-Spatial Ensemble

• Fourier Series for the Spectral Function: Planar motion
  • General Case of Periodic Libration/Rotation
  • Free Rotation between Strong Collisions

• Fourier Series for the Spectral Function: Motion in Space
  • Longitudinal Orientation of External Field
  • Transverse Orientation of External Field
  • Frequencies of Resonant Interaction
  • The Static SF and the Susceptibility at Rare Collision
  • About an Ensemble Averaging
  • Transition to Ensembles, Having Reduced Number of Degrees of Freedom

• Integrated Absorption
  • Rotation in Space
  • Rotation in a Plane
  • The Quasi-Spatial Ensemble

• The Kramers-Kronig Rule

• Landau Damping in Polar Medium
  • Relation of the Loss Susceptibility to the Steady State Distributions
  • The Loss Susceptibility of Free Planar Rotator

The following sections are included:

• The Effective Susceptibility at an Arbitrary Non-equilibrium Induced Distribution F

• The Boltzmann Induced Distribution F_B

• The Debye Induced Distribution F_D

• The Self-Consistent Induced Distributions F_G, F_K, F_T
  • The Gross Collision Model
  • The Correlation-Orientation Distribution F_K
  • The Isothermal Collision Model (F = F_T)

• The Spectral Function K(z). The Summary Table for the Susceptibility

• The Low-Frequency Approximation
  • Parameters of the Debye Relaxation Expressed in Terms of the Molecular-Model Parameters
  • Interpretation of Debye Relaxation in Terms of Statistical Averages
  • Other Parameters of the Low-Frequency Behavior

• The Resonance Phenomena (The Poley Absorption Region)
  • The Gordon Sum Rule for the Loss χ"(ω)
  • The Estimate of the Dipole Moment from the Experimental value of the Integrated Absorption
  • The Calculation Scheme for the ACFs
  • The Quasi- Resonant Absorption

• The Extension of the Theory to Multicomponent Media
  • Systems Possessing a Discrete Set of Relaxation Times
  • Media with a Single Relaxation Time (Media, Homogeneous on Microscopic Scale)

• Models of Collisions for an Isotropic Medium (Continuation)
  • Models with a Single Well Potential
  • The Averaged Susceptibility

The following sections are included:

• The t₀ theorem

• The averaged perturbation (AP) theorem

This part consists of three chapters. We shall describe:

The classical extended diffusion (ED) model (Chap.7), in which the potential U is homogeneous (not depending on coordinates); we may put U = 0. In the ED model molecules rotate freely by inertia during the time interval t_v between “strong collisions”, the mean value 〈t_v 〉 = τ, termed “lifetime”, being the free parameter of the model. In the ED model we take into account the form of a molecule by considering dielectric response of symmetric top molecules…

The following sections are included:

• Ensemble of Linear Molecules
  • Geometry and the Law of Motion
  • The Spectral Functions
  • Some Approximate Expressions for the Complex Functions w(z) and E₁(z)
  • Dielectric Spectra

• Ensemble of Symmetric Top Molecules
  • The Spectral Function
  • The Long Lifetime Limit and The Debye Relaxation for y <<1
  • Dielectric Spectra

The following sections are included:

• The Planar CR Model
  • Law of Motion and Steady State Distribution
  • Spectral Functions

• The Simplified Spatial CR Model

• Dielectric Spectra of an Isotropic Medium for the DWP configuration

• The Peak Absorption and Debye Relaxation Time as Functions of the Lifetime
  • The Estimated Spectral Parameters as Functions of the Normalized Lifetime τ/η

• The Cone Confined Rotator (CCR) Model
  • Geometry
  • The Steady State Distribution Function and the Ensemble Averages
  • The Spectral Functions
  • Transition to Free Rotation
  • Rotation in Hemisphere
  • The Approximation of Small-Amplitude Librations
  • Transition to the Planar Confined Rotator Model
  • Dielectric Behavior
  • The Comparison of Exact and Approximate Theories

• The Cone Confined Rotator Model with a Single Potential Well: Dielectric Behavior

The following sections are included:

• The Mean Molecular Parameters at Equilibrium

• The Spectral Functions K(z) and L(z)

• Evolution of Dielectric Spectra with the Rise of the Rectangular Potential

• The Hybrid Model 2 (HM2)
  • The Spectral Function of Librators: a Rigorous Description
  • Librational Subensemble: Averaging of some Parameters
  • Another Representation for the Spectral Function of Librators
  • Approximate Representation of the SF in the Form of Ordinary Integrals
  • Rigorous Theory for the Large β Case
  • Evolution of Dielectric Spectra due to the Rise of the Cone Angle

The concept of the intermolecular potential, employed here, is quite different as compared with that, considered in the previous Part. Indeed, the «walls» of the well lose their «mechanical» clearness, since instantaneous reflections of dipoles from the walls of the rectangular box are replaced by continuous interaction of reorienting dipoles with non-homogeneous potential field. As a result, the libration amplitude β becomes dependent on the energy h of a dipole: the greater h, the greater β. We shall show, that dielectric behavior substantially depends on the steepness of the potential and on its depth U₀…

The following sections are included:

• The Spectral Function of the Planar Ensemble

• The Spectral Function of the Spatial Ensemble
  • The Longitudinal SF
  • The Transverse SF
  • The Spectral Function L(z) of an Isotropic Medium
  • Detailed Calculation of the Fourier Amplitudes b_n

• The Complex Susceptibility of Linear Molecules Subjected to the Influence of the Parabolic Potential

• The Poley Absorption

• The Loss Frequency Dependence for the Single Well Potential

• The Loss Frequency Dependence for the Double Well Potential

The following sections are included:

• The Steady State Law of Motion

• Distribution Functions

• The Field Dependence of the Steady State Parameters
  • Statistical Weights of Librators and Hindered Rotators
  • The Mean Libration Amplitude
  • The Mean Fundamental Frequencies of a Periodic Motion
  • The Reverse Value of the Collision Time
  • The Mean Values of Direction Cosines

• Rigorous Expressions for the Spectral Functions
  • Fourier Amplitudes a_n, b_n of Periodic Motion
  • The Spectral Function of Librating Dipoles
  • The Spectral Function of Hindered Rotors

• The Quasi-harmonic Approximation (QHA)
  • The Quasi-Harmonic Approximation for Librators
  • The Quasi-Harmonic Approximation of Hindered Rotators

• Dielectric Behavior
  • Dielectric Spectra of an Isotropic Medium
  • Dielectric Behavior of an Anisotropic Medium
  • The Limiting Loss /Absorption Lines

• The Stratified Approximation

The following sections are included:

• The Law of Motion

• The Steady State

• The Spectral Function: Rigorous Theory
  • Fourier Amplitudes
  • The SF of Librating Particles
  • The SF of Hindered Rotors

• The Spectral Function: the Quasi-Harmonic Approximation

• Limiting Lines
  • Librators
  • Hindered Rotors

• Influence of the Field parameter on Dielectric Spectra

• The Stratified Approximation

The following sections are included:

• The Law of Motion

• The Steady-State Distribution W = C exp(−h)

• Fourier Amplitudes of Librating Molecules

• The Spectral Function of Librators

• The Spectral Function of Free Rotating Molecules

• The Effect of the Potential Well Steepness on Loss / Absorption Spectra

• Statistical Parameters

• The Curved-Brim Hat Model

The following sections are included:

• Liquid Fluoromethane CH₃F

• The Rotational Cell Model (RCM)

• Liquid Trifluoromethane CHF₃
  • Application of the CR Model with Fitted Parameters
  • Application of the CR Model / Rotational Cell Model

• Prediction of the Critical Temperature
  • Determination of T_cr from the Condition, Imposed on the Libration Amplitude
  • Determination of T_cr from the Minimum τ_D(τ) Condition
  • On the Nature of Liquid → Solid Phase Transition

• Molecular Interpretation of the “Rotational” Lifetime τ in the Light of the Debye Rotational Diffusion Theory

• Relation of the Potential Welldepth to Dielectric Spectra
  • Dielectric Spectra and the Potential Welldepth
  • Dependence of the Welldepth on Temperature

• The Polarizability A"(v) Spectra

• Distortion of the Cole-Cole Diagram due to Micro Heterogeneity

• Isotropic Potential and the Self-Diffusion Coefficient

The following sections are included:

• Liquid Water
  • The Experimental Data
  • Dielectric Spectra
  • Translations as a Potential-Controlled Process
  • Prediction of Wideband Spectra
  • The Autocorrelation Function
  • The Theory of an Isotopic Effect (IE)

• Electrolytes in Aqueous Solution
  • NaCl - H₂O Solution as a Medium, Characterized by the Positive Hydration
  • Media with Negative Hydration

• Water in a Bound State
  • A Protein Molecule in Aqueous Solution
  • Humidified Protein Molecule
  • Weakly Humidified Collagen

• Nonelectrolyte in Aqueous Solution

The following sections are included:

• The Quasi-classical Approach
  • Rotational Frequencies
  • The Set of Lorentz Lines
  • The Set of Gross Lines

• Resonance phenomena in Biatomic Gases
  • Molecular Parameters of a Gas State
  • Dielectric Behavior of Carbon Monoxide CO
  • The Semi-Classical Approach in Terms of Dipolar ACF

• Classical Theory of Collisions
  • Collision Frequency of Binary Collisions
  • Calculation of Collision Integrals

• Gas-like Liquids

The following sections are included:

• ABOUT SOME NEW PROBLEMS

Here we mean models elaborated by the author with his collaborators.

THE CONFINED ROTATOR (CR) MODEL. We have already mentioned in Addendum I that within the frames of the dynamic method the planar CR model by (Gaiduk, Kalmykov, and Tseitlin 1979) was the first one, applied to polar fluids. The Debye collision model (i.e. with F − F_D) was envisaged, that is characterized by the relaxation time τ_D. It should be mentioned that the formula for the susceptibility was derived earlier in (Higasi, Minami, Takahashi, and Ohmo 1973), (Minami and Ohmo 1973) by considering ACF of dipoles, librating in the rectangular potential well. Unfortunately, the theory, elaborated in the aforecited works, is not applicable in the FIR region…

The following sections are included:

• REFERENCES

• SUBJECT INDEX

• SERIES IN CONTEMPORARY CHEMICAL PHYSICS