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The purpose of this chapter is to continue the unification that was begun in Chapter 8. There, first and second derivatives of normal stress with respect to orientation were used; we now examine the idea that the chemical potential of a component at a point can be a multivalued direction-dependent scalar like the normal-stress magnitude, and that it too can have a gradient with respect to orientation. The essence of a nonhydrostatic stress is that different planes through a point are subject to different normal compressive stresses: σn varies with the orientation of the plane considered. Let us focus on a plane i across which the normal compressive stress is σi: then we put forward the assertion that an equilibrium state that can be associated with plane i is a hydrostatic state whose pressure has the same magnitude as σi. For illustration, see Figure 9.1. (For the present, we take a cautious stance: each hydrostatic state in the figure is certainly an equilibrium state, and each is certainly associated with a plane, but is it the associated equilibrium state that properly belongs with that plane according to the precepts of, for example, de Groot (1951, p. 11)? For now, we make no attempt to prove that it is so: we simply use the assertion and explore its consequences. Fortunately its consequences include large amounts of classical mechanics so that it counts as a “successful assertion” on those grounds, but at least for now it lacks any underpinnings.) An immediate consequence of the assertion illustrated in Figure 9.1 is the relation in Figure 9.2.

Chemical potential is defined, and its physical meaning explored. Examples in solids and gases are given. An illustrative calculation for an ideal gas presents the isothermal atmosphere. The Saha equation is obtained and discussed.

This chapter presents a coupled theory for transport of a single atomic (or molecular) chemical species through a solid that deforms elastically. Consideration is limited to isothermal conditions and circumstances in which the deformations are small and elastic, and the changes in species concentration from a reference concentration are small --- a framework known as the theory of linear chemoelasticity. Underlying the presented approach is the notion that the solid can deform elastically but it retains its connectivity and does not itself diffuse. To account for the energy flow due to species transport, the notion of chemical potential of the species is introduced. First the basic equations of the fully-coupled linear theory of anisotropic linear chemoelasticity are derived, and then these equations are specialized for the case of isotropic materials.

This chapter is an introduction to classical thermodynamics that does not assume any knowledge of the subject. The significance of thermodynamic equilibrium in materials is discussed keeping in mind that it is rarely achieved in practice. The concepts of thermodynamic systems, components, work, energy, phase, absolute temperature, heat, potential energy, internal energy, state variables, intensive and extensive variables are introduced and defined. The first and second laws of thermodynamics are introduced. The concept of entropy is discussed in terms of irreversibility, the direction of time and microstates of the system. Configurational entropy is illustrated with the example of a binary alloy. The Helmholtz and Gibbs free energies are introduced and their physical significance is discussed in terms of the conditions for a material to be in equilibrium with its environment. This leads to a discussion of chemical potentials, the Gibbs-Duhem relation for each phase present and the phase rule.

Although the law of entropy increase governs the direction in which things change, we don’t observe entropy directly. Instead we observe three quantities—temperature, pressure, and chemical potential—that tell us how the entropy of a system changes as it interacts in three different ways with its surroundings. This chapter shows how these three quantities are mathematically related to a system’s entropy, energy, volume, and number of particles. These relations complete the foundation of macroscopic thermodynamics. Moreover, for the three model systems whose entropies are calculated explicitly in the previous chapter, these relations lead to detailed testable predictions of thermal behavior.

“The chemical potential for a semiconductor” deals with the way in which the chemical potential (Fermi level) of a semiconductor is affected: by the densities of states in the bands; by temperature; and by doping. The electron–hole product is usually independent of doping but sensitive to temperature. The chemical potential is worked out numerically for an example case, and is shown to be most sensitive to doping.

The phenomenon of atomic and ionic migration in crystals is called solidstate diffusion, and its study has shed light on many problems of technological and scientific importance. Diffusion is intimately connected to the strength of metals at high temperature, to metallurgical processes used to control alloy properties, and to many of the effects of radiation on nuclear reactor materials. Diffusion studies are important in understanding the ionic conductivity of the materials used in fuel cells, the fabrication of semiconductor integrated circuits, the corrosion of metals, and the sintering of ceramics. When two miscible materials are in contact across an interface, the quantity of diffusing material which passes through the interface is proportional to the concentration gradient. The atomic flux J is given by where J is measured per unit time and per unit area, c is the concentration of the diffusing material per unit volume, and Z is the gradient direction. The proportionality factor D, the diffusion coefficient, is measured in units of m2/s. This equation is sometimes referred to as Fick’s First Law. It describes atomic transport in a form that is analogous to electrical resistivity (Ohm’s Law) or thermal conductivity. There are several objections to Fick’s Law, as discussed in Section 19.5. Strictly speaking, it is valid only for self-diffusion coefficients measured in small concentration gradients. Since J and Z are both vectors, the diffusion coefficient D is a second rank tensor. As with other symmetric second rank tensors, between one and six measurements are required to specify Dij, depending on symmetry. The relationship between structure and anisotropy is more apparent in PbI2. Lead iodide is isostructural with CdI2 in trigonal point group.m. The self-diffusion of Pb is much easier parallel to the layers where the Pb atoms are in close proximity to one another. Diffusion is more difficult along Z3 = [001] because Pb atoms have a very long jump distance in this direction. The mineral olivine, (Mg, Fe)2SiO4, is an important constituent of the deeper parts of the earth’s crust.

A thermodynamic phase specifies a substance or mixture of substances that occupies a limited volume with a characteristic temperature, T, a definite pressure, p, at its boundary, and a composition that can be described at any moment by the masses of its constituents. A well-mixed sample of air or of sea water in contact with our instruments, therefore, is defined as a phase. Different phases, brought together in a thermodynamic system, tend to change until an equilibrium has been established. This involves an exchange of energy and matter between the originally different phases. If the process is isolated from external influences, then it results in an increase of entropy. In fact, entropy is always generated by mixing (i.e., by the transport of a conservative property down its own gradient). This can be said to include the transport of momentum by viscosity. Entropy can be diminished locally when gradients are sharpened by a flux in the opposite direction. In nature this can be brought about by a coupling of the transports of different properties. The potential for change that exists in a physical system is the subject of the thermodynamics of irreversible processes. The ocean and the atmosphere are open systems which can exchange matter through their boundaries. Though the system that they form together is closed—at least on the time scales with which we are concerned—it is not isolated, being subject to heat exchange with the surrounding universe. The inhomogeneity and variable temperature of both oceans and atmospheres cause further complications. Although equilibrium thermodynamics alone cannot provide information about the rate or the mechanism of energy transformations, it does provide, in the First Law, a constraint that must always be obeyed. It also allows us to predict the general direction of development in limited regions that only interact slowly with their surroundings. The properties or “coordinates” which specify a thermodynamic phase are not independent. Anyone of them can be expressed as a function of all the others by an equation of state. In a liquid, the state is strongly affected by molecular interaction.

The preceding chapter focused on the dislocation core structure at zero temperature obtained by energy minimization. In this chapter we will discuss a case study of dislocation motion at finite temperature by molecular dynamics (MD) simulations. MD simulations offer unique insights into the mechanistic and quantitative aspects of dislocation mobility because accurate measurements of dislocation velocity are generally difficult, and direct observations of dislocation motion in full atomistic detail are still impossible. The discussion of this case study is complete in terms of relevant details, including boundary and initial conditions, temperature and stress control, and, finally, visualization and data analysis. In Section 3.1 we discussed a method for introducing a dislocation into a simulation cell. It relies on the linear elasticity solutions for dislocation displacement fields. To expand our repertoire, let us try another method here. The idea is to create a planar misfit interface between two crystals, such that subsequent energy minimization would automatically lead to dislocation formation.