Introduction
Study rates at which various processes occur
Complexity of problem reduced by introducing approximations such as assumption of local equilibrium.
Knowledge of kinetics leads to prediction of rates
Mechanisms of change are important in determining kinetics
1.1 Thermodynamics and Kinetics
Two broad topics in the study of materials science:
- Thermodynamics: study of equilibrium states in which state variables of a system do not change with time.
- Kinetics: study of the rates at which systems that are out of equilibrium change under the influence of various forces.
Thermodynamics provides information about the final state of a system
Kinetics concerns itself with the paths and rates adopted by systems approaching equilibrium
Much of the machinery of thermodynamics can be applied locally under an assumption of local equilibrium.
1.1.1 Classical Thermodynamics and Construction of Kinetic Theories
Thermodynamics grew out of studies of systems that exchange energy
Poincare coined the term thermodynamics to refer to insights that developed out of the first and second laws.
From Gibbs's careful and rigorous derivations of equilibrium conditions of matter, the modern subjects of chemical and material thermodynamics were born
Thermodynamics is precise, but is strictly applicable to phenomena that are unachievable in finite systems in finite amounts of time.
Two fundamental results from classical thermodynamics that form the basis for kinetic theories in materials:
1. If an extensive quantity can be exchanged between two bodies, a condition necessary for equilibrium is that the conjugate potential, which is an intensive quantity, must have the same value throughout both bodies.
There are an infinite number of ways that a potential can differ from its equilibrium value. The task of describing and analyzing nonequilibrium systems is more complex than describing equilibrium systems
2. If a closed system is in equilibrium with reservoirs maintaining constant potentials (e.g. P and T), that system has a free-energy function (e.g., G(P, T) that is minimized at equilibrium. Therefore, a necessary condition for equilibrium is that any variation in G must be nonnegative(dG pt >= 0)
This leads to classical geometric constructions of thermodynamics.
General statements regarding the approach to equilibrium that are based on thermodynamic functions necessarily involve extrapolation away from equilibrium conditions. Useful models and theories can be developed.
Another approach is to build kinetic theories empirically.
1.1.2 Averaging
It is possible to construct interatomic potentials and forces between atoms that approximate real systems in a limited number of atomic configurations.
Just as statistical mechanics overcomes difficulties arising from large numbers of interacting particles by constructing rigorous methods of averaging, kinetic theory also uses averaging.
Many theories developed in this book are expressed by equations or results involving continuous functions. Materials systems are fundamentally discrete and do not have an inherent continuous structure from which continuous functions can be constructed.
1.2 Irreversible Thermodynamics and Kinetics
The extensive state of a material or system not at equilibrium will change consisten with the second law of thermodynamics.
The chemical potential cannot be defined when the system is not at equilibrium, but it is useful to extend the concept of the chemical potential to systems close to but not at equilibrium
It is reasonable to take a continuum limit of an idealized measurement and refer to temperature at a point.
1.3 Mathematical Background
A few basic physical and mathematical concepts are essential to study kinetics
1.3.1 Fields
A field associates a physical quantity with a position. It may be time dependend, and the simplest case is a scalar field.
A vector field requires specification of a magnitude and a direction in reference to a fixed frame.
Every sufficiently smooth scalar field has an associated natural vector field.
1.3.2 Variations
The gradient of c points in the direction of maximum rate of increase of the scalar field: the magnitude of the gradient vector is equal to the rate of increase
1.3.3 Continuum Limits and Coarse Graining
Materials are comprised of discrete atoms, which complicates the definition of local concentration when the volume sampled becomes comparable to the mean distance between atoms being counted.
Consider the behavior of the concentration of i as the volume shrinks toward the point where the concentration is evaluated. The limiting value used to define the concentration is not a well-defined limit of the function of the concentration
A local convolution function can be defined with a convolution function, which specifies, at a position the weight to assign to assign to a particle located at that position.
In this book, it is assumed that the continuum limits exist and coarse-grained functions can be obtained that do not depend significantly on the choice of zeta.
1.3.4 Fluxes
The flux describes the rate at which i flows through a unit area fixed with respect to a specified coordinate system.
1.3.5 Accumulation
The amount that accumulates in a volume is equal to the amount flowed in minus the amount that flowed out plus the amount produced inside during the time interval.
The rate of accumulation of the extensive quantities density is minus the divergence of the flux minus the rate of production.
The divergence theorem has a geometrical interpretation. If the volume is comprised of many neighboring cells, the total accumulation in the volume is the sum of accumulations in all the cells
1.3.6 Conserved and Nonconserved Quantities
A conserved quantity cannot be created or destroyed and therefore has no sources or sinks.
1.3.7 Matrices, Tensors, and the Eigensystem
If x is an applied force and y is the material response to the force, M is a rank-two-material-property tensor.
Many materials properties are anisotropic
When anisotropic materials properties are characterized, the values used to represent the properties must be specified with respect to particular coordinate axes.
The components of tensor quantities transform in specified ways with changes in coordinate axes; such transformation laws distinguish tensors from matrices.
Linear transformations-such as rotations, reflections, and affine distortions-can be performed on vector forces and responses by matrix multiplication to describe force-response relations in different coordinate systems.
It is often convenient to select the coordinate system for which the only nonzero elements of the property tensor lie on its diagonal. This is the eigensystem.
The pattern of a rank-one tensor transformed by a single matrix multiplication and a rank-two tensor transformed by two matrix multiplications--holds for tensors of any rank.
Square matrices and tensors can be characterized by their eigenvalues and eigenvectors.
The eigenvectors of a matrix can be interpreted geometrically as the set of vectors that do not change direction when multiplied by the matrix. They are instead scaled by a constant.
A rank-two property tensor is diagonal in the coordinate system defined by its eigenvectors. Rank-two tensors transform like 3 x 3 square matrices. Nearly all rank-two property tensors can be represented by 3 x 3 symmetric matrices and necessarily have real eigenvalues.