Statistical Mechanics for Chemistry and Materials Science

Biman Bagchi

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9780429833601
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physical & Theoretical Chemistry

This book covers the broad subject of equilibrium statistical mechanics along with many advanced and modern topics such as nucleation, spinodal decomposition, inherent structures of liquids and liquid crystals. Unlike other books on the market, this comprehensive text not only deals with the primary fundamental ideas of statistical mechanics but also covers contemporary topics in this broad and rapidly developing area of chemistry and materials science.

1Preliminaries

Overview

Statistical Mechanics is a major scientific discipline that aims at explaining a large number of experimental phenomena in many different areas, such as phase transitions, protein folding, enzyme kinetics, pattern formation, viscosity anomaly in binary mixtures, to name a few. The theory is based on two postulates and one hypothesis. Armed with these, and a given inter-molecular potential between two atoms or molecules, Statistical Mechanics (SM) attempts to explain many complex phenomena. This elaborate and well-developed formalism connects the macroscopic world and thermodynamics to the microscopic world, and thus provides the bridge between the two. SM was primarily developed by Maxwell, Boltzmann, Gibbs and Einstein. However, the theory can be formidable at places, as discussed in the subsequent chapters. The question then naturally arises: why should one study and try to master a subject that requires considerable time and effort on the part of the student or researcher? Here we articulate the great scope and the enormous reach of Statistical Mechanics, also with emphasis on recent interest. We then briefly discuss laws of thermodynamics and stability conditions. The latter proves to be of great value in the study of phase transitions.

1.1Why Study Statistical Mechanics?

The aim of Statistical Mechanics (SM) is to provide a microscopic description of collective phenomena occurring in the universe, such as phase transitions or protein folding. These phenomena involve many atoms and molecules and are consequences of interactions among them. That is, a system of non-interacting atoms, such as classical ideal gas, does not show any phase transition. There are many interesting questions and observations in the natural world that need understanding. For example, why do we have to supercool most liquids below their freezing point to grow crystals? Why does liquid sodium freeze into a body centered cubic (bcc) crystalline phase instead of a face centered cubic (fcc) phase, while liquid iron freezes into an fcc phase and not a bcc phase? Or, why and how does a protein fold to its unique native state?

For many other processes, such as a chemical reaction in solution, the reaction coordinate, i.e., the coordinate along which primary changes occur, can be coupled to many other solvent degrees of freedom. So, if you are interested in studying the viscosity dependence of rotational and translational motion of a solute molecule, for example of an iodine molecule in solution, you would need to consider many body interactions. Vibrational relaxation of a given bond in solution is also coupled to many degrees of freedom. Statistical Mechanics is used to understand all such phenomena.

Statistical Mechanics is usually divided into two parts: equilibrium and non-equilibrium Statistical Mechanics, and both are active areas of research. They can be interpreted as the microscopic generalization of the two divisions of Mechanics: Statics and Dynamics. One outstanding triumph of equilibrium Statistical Mechanics is that it provides us with microscopic calculable expressions (and interpretations) of thermodynamic functions like entropy (S) and free energy. As you know, despite its perfection and enormous utility, thermodynamics is kind of lame because while it provides fundamental relations among important functions, it does not allow microscopic calculations of quantities. Statistical Mechanics not only provides exact expressions for such quantities, but also provides valuable additional insight into the functions such as specific heat, compressibility, dielectric constant, magnetic susceptibility, which are collectively called the response functions of the system. Moreover, phase transition is an important area of equilibrium Statistical Mechanics.

Non-equilibrium Statistical Mechanics deals with dynamics. It deals with relaxation phenomena of various kinds, dynamics of phase transitions, kinetics of chemical ...

Table of contents

• Cover
• Half Title Page
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Author
• Chapter 1 Preliminaries
• Chapter 2 Probability and Statistics
• Chapter 3 Fundamental Concepts and Postulates of Statistical Mechanics
• Chapter 4 Liouville Theorem and Liouville Equation
• Chapter 5 Ensembles and Partition Functions: From Postulates to Formulation
• Chapter 6 Fluctuations and Response Functions
• Chapter 7 Ideal Monatomic Gas: Microscopic Expression of Translational Entropy
• Chapter 8 Ideal Gas of Diatomic Molecules: Microscopic Expressions for Rotational and Vibrational Entropy and Specific Heat
• Chapter 9 Quantum Statistics and Bose-Einstein Condensation
• Chapter 10 Lattice Models Including Ising
• Chapter 11 Distribution Function Theory of Liquids
• Chapter 12 Hard Core Interactions
• Chapter 13 Perturbation Theories of Liquids
• Chapter 14 Cell Theory and the Concept of Free Volume and Communal Entropy in Liquids
• Chapter 15 Scaled Particle Theory and Calculation of Chemical Potential
• Chapter 16 Phase Transition: Elementary Concepts
• Chapter 17 Landau Theory of Phase Transitions: Order Parameter Expansion and Free Energy Diagrams
• Chapter 18 Critical Phenomena: Universality, Scaling, and Renormalization Group
• Chapter 19 Cluster Expansion and Mayer’s Theory of Condensation
• Chapter 20 Yang-Lee Theory of Phase Transition
• Chapter 21 Classical Density Functional Theory
• Chapter 22 Surface Phenomena and Surface Tension
• Chapter 23 Theory of Melting and Freezing and Glass Transition
• Chapter 24 Nucleation, Ostwald Step Rule, and Nanomaterial Synthesis
• Chapter 25 Spinodal Decomposition and Pattern Formation: Evolution of Structure through Dynamics
• Chapter 26 Binary Mixtures: Towards Understanding Non-Ideality and Osmotic Pressure
• Chapter 27 Inherent Structures and Energy Landscape View of Liquids and Glasses
• Chapter 28 Polymers in Solutions and Polymer Collapse
• Chapter 29 Computer Simulation Methods in Statistical Mechanics
• Chapter 30 Computational Methods for Free Energy Calculation and Study of Rare Events
• Epilogue
• Index