Diffusion

Diffusion is associated with the random, thermal motion of atoms that produces a change in the macroscopic concentration profile. This process occurs in gases, liquids, amorphous and crystalline solids of metals, ceramics, polymers, semiconductors, etc. The investigation of diffusion provides valuable information about the atomic structure of materials and the defects within them. Perhaps, most importantly, diffusion controls the rates of a wide range of kinetic processes associated with the synthesis of materials, processes by which we modify materials, and processes by which materials fail. The most common driving force for diffusion in a single-phase systems is associated with the entropy of mixing of its constituents (recall that we showed that the entropy of mixing of gases and the components of an ideal solution are always positive—see Sections 1.2.6 and 3.3). Since diffusional processes occur through the thermal motion of atoms (see below), it will not be surprising to learn that the rate of diffusion increases with increasing temperature. However, note that while the mechanisms of thermal motion in gases (random collision of atoms with each other) and liquids (e.g. Brownian motion) necessarily lead to mixing, the mechanisms of mixing within a solid are not as obvious. In solids, thermal motion corresponds to the vibrations of atoms near their equilibrium positions. Since the amplitude of such vibrations is much smaller than the nearest-neighbor separation, it would seem that such thermal motions cannot lead to mixing. Thus, the question ‘‘how do atoms migrate in solids’’ is not so simple. The equations describing diffusion were suggested by the physiologist Fick in 1855 as a generalization of the equations for heat transfer suggested by Fourier in 1824. Fick’s equations for diffusion can be obtained by analogy with Fourier’s equations for heat transfer by replacing heat with the number of atoms, temperature with concentration, and thermal conductivity with diffusivity. Fick’s first law provides a relationship between atomic currents and concentration gradients. As discussed above, this relationship can be understood by analogy with thermal conductivity or electrical conductivity.

Kinetics of homogeneous chemical reactions

Kinetics considers the rates of different processes. Chemical kinetics refers to the rates and mechanisms of chemical reactions and mass transfer (diffusion). Recall that since thermodynamic equilibrium implies that the rates of all processes are zero, time is not a thermodynamic variable. Rather, time is the new parameter introduced by the consideration of kinetic processes. The rate of a kinetic process and how it depends on time is determined, in part, by the degree of the deviation from equilibrium. If the deviation from equilibrium is small, the rate decreases (without changing sign) as the system approaches equilibrium. If the deviation from equilibrium is large, the situation is more complicated. For example, non-monotonic (including oscillatory) processes are possible. The sign of the rate can change during such processes; that is, the reaction can proceed in one direction and then the other. Additionally, if the deviation from equilibrium is large, small changes to the system can produce very large changes in the rate of the kinetic process (i.e. chaos). Non-equilibrium, yet nearly stationary states of the system can arise (i.e. states that exist for a very long time). Finally, if the deviation from equilibrium is very large, the system can explode (i.e. the process continues to accelerate with time). In this chapter, we develop a formal description of the kinetics of rather simple chemical reactions. Consecutive and parallel reactions will also be considered here. A more general approach (irreversible thermodynamics) will be considered in Chapter 9. In Chapter 10, we examine diffusive processes. Then, in Chapter 11, we consider the kinetics of heterogeneous processes. In order to start the study of chemical reaction kinetics, we must first define what we mean by the rate of reaction. Consider the following homogeneous reaction: . . . Cl2 + 2NO → 2NOCl. (8.1) . . .

Thermodynamics of stressed systems

As every school child knows, the difference between a solid and a liquid is that a liquid takes the shape of the container in which it is placed while the shape of a solid is independent of the shape of the container (providing the container is big enough). In other words, we must apply a force in order to change the shape of a solid. However, the thermodynamic functions described heretofore have no terms that depend on shape. In this chapter, we extend the thermodynamics discussed above to include such effects and therefore make it applicable to solids. However, since this is a thermodynamics, rather than a mechanics text, we focus more on the relationship between stress and thermodynamics rather than on a general description of the mechanical properties of solids. We start out discussion of mechanical deformation by describing the change of shape of a solid. We define the displacement vector at any point in the solid u(x, y, z) as the change in location of the material point (x, y, z) upon deformation: that is, ux(x, y, z) = x' - x, where the prime indicates the coordinates of the material that was at the unprimed position prior to the deformation. In linear elasticity, we explicitly assume that the displacement vector varies slowly from point to point within the solid where i and j denote the directions along the three axes, x, y, and z. Consider the small parallel-piped section of a solid with perpendicular edges shown in Fig. 7.1. We label the first corner as O, located at position (xO, yO, zO) and subsequent corners as A, B, . . . located at positions (xA, yA, zA), (xB, yB, zB), . . . The edge lengths are Δx, Δy, and Δz such that, for example, xA = xO + Δx. As a result of the deformation, the material originally at point O is displaced to point O' with coordinates (x'O, y'O, z'O).

Thermodynamic theory of solutions

A solution is a homogeneous mixture consisting of two or more components in which the composition can be continuously varied (within some range) with no change of phase. Solutions can be gases, liquids, or solids. We have already considered the properties of gaseous solutions (when we considered a mixture of ideal gases). In this chapter, we focus on condensed phases (i.e. liquids and solids). The composition of a solution can be described in several ways. Here are the most common: 1. The molar fraction of the ith component, xi, is the ratio of the number of moles of component i, ni, to the total number of moles of all species within the solution, n: 2. The weight fraction of the ith component, [i], is the ratio of the mass of component i, wi, to the total mass of all species within the solution, w: the weight fraction is often written as a weight percent [wt%] = 100[i]. 3. The molarity of the ith component, ci, is the number of moles of component i, ni per liter of solution, V: 4. The molality of ith component (used only for dilute solutions), mi, is the number of moles of component i, in 1 kg of solvent. There are several other definitions used to describe the composition of a solution, but we shall only refer to those described above in this text. The reason that there are so many definitions of the composition is related to how the term ‘‘concentration’’ is applied. For example, from the physical–chemical point of view, the molar fraction is the most convenient definition of the concentration since it is on an atomic basis. However, from the point of view of someone who has to prepare solutions from separate solutes and solvents, the mass fraction is the most convenient definition since it is directly related to the mass of the components, rather than the number of moles of the component. The former is easily measured, while the latter requires the additional knowledge of the molecular weight (and a trivial calculation).

Basic laws of thermodynamics

In this chapter, we first introduce the basic laws of thermodynamics and R17the most important thermodynamic functions. Even though many of the concepts introduced here will be familiar to many readers with a background in elementary physics, this chapter should not be ignored as it presents these concepts in the language of physical chemistry. Since these concepts form the basis of physical chemistry, this subject will make no sense without a firm footing in these fundamentals. Thermodynamics focuses on the thermal behavior of macroscopic systems (i.e. systems containing a very large number of particles). Thermal processes include both heat exchange between a system and its surroundings and work. The general scheme of a thermodynamic description of such processes can be described as in the picture: Thermodynamic descriptions are usually based upon experimental observations. Experiments can characterize the thermodynamic state of the system in terms of a small number of measurable parameters (e.g. temperature T and pressure p). The generalization of these measurements yields thermodynamics laws. Thermodynamic laws identify state functions that describe the system behavior solely in terms of the system parameters and not on how the system came to be in a particular state. Changes in the state functions during some process depend on only the intial and final states of the system but not on the path between them. Therefore, these changes can be determined from calculations based on a very small set of data. Thermodynamics can be used to answer such questions as (1) is a particular process possible? (2) can the system spontaneously evolve in a particular direction?, and (3) what is the final or equilibrium state? all under a given set of conditions. Equilibrium can be understood as the state in which the system parameters no longer evolve, there are no fluxes of matter or energy through the system, and for which all small disturbances decay. According to the zeroeth law of thermodynamics any isolated system will eventually evolve to an equilibrium state and will never spontaneously leave this state (without a substantial external disturbance).

Introduction to statistical thermodynamics of condensed matter

In this Chapter we apply statistical thermodynamics to condensed matter. We start with a description of the structure of liquids and the relation between this structure and its thermodynamic properties. Taking the low density limit, we derive a general equation of state appropriate for both liquids and gases. Next, we turn to a statistical thermodynamic description of solids. Finally, we consider the statistical theory of solutions. Recall that interactions between molecules in an ideal gas can be ignored for the purpose of determining thermodynamic properties. Therefore, we can assume that the spatial position of a molecule is independent of the positions of all of the other molecules in the gas. In real gases under high pressure and, even more so, in condensed matter, the intermolecular interactions play an important role and the positions of molecules are not independent. In other words, intermolecular interactions lead to the formation of correlations in the location of the molecules or, equivalently, to the development of structure. The energy of the system and the other thermodynamic properties depend on this structure. Therefore, we now turn to a discussion of structure. There are two distinct approaches to this problem. The first approach is designed for crystalline materials and is based upon a description of crystal symmetry. The description of this method is outside the scope of this text. The second is based upon the introduction of probability functions for atom locations and is applicable to disordered systems such as dense gases, liquids, and amorphous solids. Consider, as we are apt to do, the ideal gas. In this case, the probability of finding s molecules at points r1, r2, . . . , rs is simply In contrast with the ideal gas, the positions of molecules in high density gases or condensed matter are not independent of each other. Therefore, we write where Fs(r1, . . . , rs) is called the s-particle correlation function. Note three obvious properties of such functions. First, the system does not change when we exchange two molecules. This implies that the correlation functions should be symmetric with respect to their arguments.

Kinetics of heterogeneous processes

Most practical reactions that occur in synthesizing or processing materials are heterogeneous. These include oxidation, reduction reactions, dissolution of solids in liquids, and most solid-state phase transformations. Consider the oxidation of a metal by exposure of a solid metal to an atmosphere with a finite partial pressure of oxygen. In order for oxidation to occur, molecular oxygen must dissociate into atomic oxygen on the metal surface. In some cases, atomic oxygen diffuses into the metal and reacts to form an internal oxide, while in others, the reaction occurs at the surface. In the latter case, thickening of the oxide layer requires either metal or oxygen diffusion through the growing oxide layer. This example demonstrates that heterogeneous processes commonly involve several steps. The first step is usually the transport of a reactant through one of the phases to the interface. The second is the adsorption (segregation) or chemical reaction on the interface. Finally, the last third step is the diffusion of the products into the growing phase or the desorption of the product. Since the entire heterogeneous process is a type of complex reaction, there is usually one step that controls the rate of the process, that is, is the rate-determining step. Recall that the rate-determining step is the slowest (fastest) step for a consecutive (parallel) reaction (see Sections 8.2.1 and 8.2.2). Consider the case of a consecutive heterogeneous reaction in which one of the reactants is transported through the fluid phase to the solid–fluid interface, where a first-order reaction takes place. The reaction rate ωr in such a case is ωr=kcx, where cx is the concentration of the reactant on the interface. Since the reactant is consumed at the interface, cx is smaller than the reactant concentration far from the interface, c0. It is usually easier to measure the reactant concentration in the bulk fluid. Therefore, it is convenient, to rewrite the reaction rate in terms of the bulk concentration in the fluid and an effective rate constant . . . ωr = kcx = keffc0. (11.1) . . . It is easiest to see the relation between keff and k by considering the steady-state case.

Thermodynamics of irreversible processes

The thermodynamics of irreversible processes, formulated by Onsager and Prigogine, considers small deviations from equilibrium in open systems. Despite the fact that the name contains ‘‘thermodynamics,’’ this is a type of kinetic theory that describes the rates of irreversible processes. Since there are no currents of any type in thermodynamic equilibrium, the concept of a current is never used in classical thermodynamics. On the other hand, the thermodynamics of irreversible processes introduces currents as the rates at which processes proceed: the heat or energy current (measured in J/s), matter current (measured in mole/s or kg/s), charge or electrical current (measured in C/s or Amps). Since these currents have a direction and magnitude, they are vectors. The thermodynamics of irreversible processes also considers scalar currents (e.g. rates of chemical reactions) and tensor currents (e.g. momentum currents). In this text, we will focus on current densities or fluxes (that is the current per unit area) rather than currents themselves. The dimensions of the currents described above can be converted to the dimensions of fluxes by dividing through by area or m2. Associated with each flux is a driving force. These forces are known as thermodynamic forces. How can we determine these driving forces? What is the relation between fluxes and driving forces? The answers to these questions can be found in the thermodynamics of irreversible processes briefly described in this chapter. Onsager’s first postulate states that the flux of property i ( ji) is a linear function of all thermodynamic forces, Xk, acting in the system where Lik are called Onsager (or kinetic) coefficients. This postulate was formulated as a generalization of a wide body of experimental observations. In fact, long before Onsager’s work it was known that the heat fluxes are proportional to temperature gradients (Fourier’s law, 1824), charge fluxes are proportional to electric potential gradients (Ohm’s law, 1826), and matter fluxes are proportional to concentration gradients (Fick’s law, 1855). However, Onsager’s contribution was the inclusion of the word ‘‘all’’ in his first postulate.

Interfacial phenomena

An interface is a surface across which the phase changes. Interfaces must be present in all heterogeneous systems, such as those discussed above. Interfacial properties necessarily differ from those of the bulk phases since the atomic bonding/structure of an interface represents a compromise between those of the phases on either side of the interface. For example, an atom at a free surface, which is an interface between a condensed phase and a gas (or a vacuum), generally has fewer neighbors with which to bond than it would have if it were in the bulk, condensed phase. In an equilibrium multi-component system, the chemical potential of each species must be the same in all phases, as well as at the interface. Not surprisingly, the chemical composition of the interface will, in general, differ from that of the bulk. For example, molecules in a gas (or solute in a condensed phase) can adsorb (segregate) onto the surface (interface) of a condensed phase. Interfacial processes play important roles in all areas of materials science and in many (most) areas of modern technology. As the trend toward miniaturization in microelectronics continues and interest in nanoscale structures grows, interfacial phenomena will become even more important. Clearly, the ratio of the number of atoms at surfaces and interfaces to those in the bulk grows as system size decreases (70% of the atoms in a nanometer diameter particle are on a surface!). Therefore, the thermodynamic properties of a system become increasingly dominated by interfacial properties as the dimensions of the system shrink. We can distinguish several types of interfaces: solid–liquid, liquid–gas, solid–gas, solid phase α–solid phase β, and grain boundaries. The meaning of the first four types of interface is self-explanatory. Grain boundaries represent a special class of interfaces; interfaces across which the phase does not change. What does change abruptly across this interface is the spatial orientation of the crystallographic axes. Most crystalline materials are polycrystalline, which means that they are composed of a large number of grains, each with a unique crystallographic orientation with respect to some laboratory frame of reference.

Thermodynamics of chemical reactions

This chapter is devoted to chemical equilibrium. We will use thermodynamics to answer two main questions: (1) ‘‘In which direction will a chemical reaction proceed?’’ and (2) ‘‘What is the composition of the system at equilibrium?’’ These are the oldest and most important questions in all of chemical thermodynamics for obvious reasons. The answers to these questions represent the foundation upon which all modern chemical technologies rest. Consider the following chemical reaction: . . . aA = bB ⇆ cC + dD. (5.1) . . . A, B, C, and D represent the chemical species participating in the reaction and a, b, c, and d are the stoichiometric coefficients of these species. We refer to the species on the left side of this chemical equation as reactants and those on the right as products. The reaction in Eq. (5.1) can either go forward, from left to right (reactants to products), or backward, from right to left (products to reactants). Therefore, we see that the definition of which we call reactants and which products is arbitrary. Assume that Eq. (5.1) occurs at constant temperature and pressure. Under these conditions, the direction of the reaction is determined by the sign of the change of the Gibbs free energy.