Thermodynamic treatment of kinetics of internal processes

In Chapter 1 we considered spontaneous processes inside a system when discussing the second law but later in that chapter we only considered equilibria. We shall now discuss the thermodynamic treatment of the kinetics of such processes. This field of thermodynamics is often called irreversible thermodynamics but the full term should rather be thermodynamics of irreversible processes. The word irreversible is often replaced by the word spontaneous. A process occurring inside a system may be caused by a change imposed upon the system by some external action, but it will here be regarded as a spontaneous result of the new conditions inside the system. All processes inside a system that actually occur will thus be regarded as spontaneous. It would really be unnecessary to use either of the terms irreversible and spontaneous processes if it were not for the need to distinguish them from the limiting case of a cyclic process, e.g. the Carnot cycle, when it is carried out in such a way that the internal processes it gives rise to produce a negligible amount of entropy. Since a cyclic process is controlled by actions from the outside and they could be performed in the reverse direction, it is possible to run the cycle in the reverse direction. All the internal processes it gives rise to will also reverse and if their entropy production is again negligible the two cases will be identical in the limit, except for the sign.

The requirement of the second law that the internal entropy production must be positive for all spontaneous changes of a system results in the equilibrium condition that the entropy production must be zero for all conceivable internal processes. Most thermodynamic textbooks are based on this condition but do not discuss the magnitude of the entropy production for processes. In the first edition the entropy production was retained in the equations as far as possible, usually in the form of Dd ξ where D is the driving force for an isothermal process and ξ is its extent. It was thus possible to discuss the magnitude of the driving force for a change and to illustrate it graphically in molar Gibbs energy diagrams. In other words, the driving force for irreversible processes was an important feature of the first edition. Two chapters have now been added in order to include the theoretical treatment of how the driving force determines the rate of a process and how simultaneous processes can affect each other. This field is usually defined as irreversible thermodynamics. The mathematical description of diffusion is an important application for materials science and is given special attention in those two new chapters. Extremum principles are also discussed.

A third new chapter is devoted to the thermodynamics of surfaces and interfaces. The different roles of surface energy and surface stress in solids are explained in detail, including a treatment of critical nuclei.

Thermodynamics is an extremely powerful tool applicable to a wide range of science and technology. However, its full potential has been utilized by relatively few experts and the practical application of thermodynamics has often been based simply on dilute solutions and the law of mass action. In materials science the main use of thermodynamics has taken place indirectly through phase diagrams. These are based on thermodynamic principles but, traditionally, their determination and construction have not made use of thermodynamic calculations, nor have they been used fully in solving practical problems. It is my impression that the role of thermodynamics in the teaching of science and technology has been declining in many faculties during the last few decades, and for good reasons. The students experience thermodynamics as an abstract and difficult subject and very few of them expect to put it to practical use in their future career.

Today we see a drastic change of this situation which should result in a dramatic increase of the use of thermodynamics in many fields. It may result in thermodynamics regaining its traditional role in teaching. The new situation is caused by the development both of computer-operated programs for sophisticated equilibrium calculations and extensive databases containing assessed thermodynamic parameter values for individual phases from which all thermodynamic properties can be calculated. Experts are needed to develop the mathematical models and to derive the numerical values of all the model parameters from experimental information.

Introduction

In this chapter we shall model the thermodynamic effect of some physical phenomena. In each case we shall start by defining an internal variable representing the extent of the physical phenomenon to be discussed. We shall proceed by deriving an expression for one of the characteristic state functions in terms of the internal variable together with a set of external variables. The choice of characteristic state function depends upon what set of external variables is most convenient. Then we shall calculate the equilibrium value of the internal variable by putting the driving force for its change equal to zero. Finally, we shall try to eliminate the internal variable by inserting the expression for its equilibrium value in the characteristic state function.

Our derivation of an expression for the characteristic state function will usually be based upon two separate evaluations, one concerned with the entropy due to the disorder created by the physical phenomenon and the other concerned with what may be called the non-configurational contribution. The entropy will be evaluated from Boltzmann's relation which is here preferred because it is felt that it gives a better physical insight than the more general and elegant method of statistical thermodynamics based upon the use of partition functions. The purpose of statistical thermodynamics is to model the thermodynamic properties of various types of systems from statistical considerations on the atomic level. The relation proposed by Boltzmann can be derived from such considerations.

Surface energy and surface stress

By cutting a piece of material in two one can create two fresh surfaces and it is evident that they represent an increase of the energy of the system because bonds between atoms or molecules have been broken. Admittedly, the energy may then decrease somewhat by relaxation in the surface layer. The net effect can be defined as the surface energy or rather surface free energy or surface Gibbs energy under the usual isobarothermal conditions. We shall simply use the term surface energy and apply the same term to real surfaces as well as interfaces. Specific surface energy, i.e. the energy per surface area, will be denoted by σ.

However, the energy of the system may decrease further by minimizing the surface area. Primarily, there would be a tendency of the two new pieces to minimize the surface area by a shape change of the material and for an isotropic material the final shape would be spherical. That could happen quickly if the material is liquid but it could be an extremely slow process for a piece of solid material. The decrease of energy during the shape change is easily calculated for an isotropic material because its specific surface energy, σ, the energy per area, is constant.

Secondarily, the surface could contract further without a shape change by compressing the material in the sphere. It will thus be put under an increased pressure, formally caused by a stress in the surface.

Deviation from local equilibrium

As discussed in Section 7.8 it is common to assume that the rate of a phase transformation in an alloy is controlled by the rate of diffusion. The local compositions at the phase interfaces are then used as boundary conditions for the diffusion problem and they are evaluated by assuming local equilibrium at the interfaces. That is a very useful approximation but there are important exceptions. It is necessary to realize that the exceptions are of two different types and they have opposite effects. The first type of exception is caused by a limited mobility of the interface. In order to keep pace with the diffusion, the interface requires a driving force which is subtracted from the total driving force and decreases the driving force for the diffusion process. Due to this effect, a partitionless transformation, which would otherwise be completely diffusion-controlled but rapid due to a very short diffusion distance, requires an increased supersaturation of the parent phase, as shown in Section 7.8. Formally, this case was treated by assuming a pressure difference between the two phases as if the interface were curved more than it actually is, and the local equilibrium assumption was modified to this case.

The other type of exception will instead decrease the driving force needed by decreasing the need for diffusion and will thus result in a higher rate of transformation and make it possible for an alloy with a lower supersaturation to transform.

Molar axes

If one starts from a potential phase diagram, one may decide to replace one of the potentials by its conjugate variable. However, the potential phase diagram has no information on the size of the system and one should thus accept introducing a molar quantity rather than its extensive variable. By replacing all the potentials with their conjugate molar variables, one gets a molar diagram. One would like to retain the diagram's character of a true phase diagram, which means that there should be a unique answer as to which phase or phases are stable at each location. In this chapter we shall examine the properties of molar diagrams and we shall find under what conditions they are true phase diagrams. Only then may they be called molar phase diagrams. However, we shall start with a simple demonstration of how a diagram changes when molar axes are introduced.

Figure 9.1(a)–(d) demonstrates what happens to a part of the T, P potential phase diagram for Fe when Sm and Vm axes are introduced. Initially the P axis is plotted in the negative direction because V is conjugate to −P. It can be seen that the one-phase fields separate and leave room for a two-phase field. It can be filled with tie-lines connecting the points representing the individual phases in the two-phase equilibrium. It is self-evident how to draw them when one axis is still a potential but they yield additional information when all axes are molar (Fig. 9.1(d)).

Schreinemakers' projection of potential phase diagrams

Another method of reducing the number of axes is based on projection. By projecting all the features onto one side of the phase diagram, one will retain all the features, but the features of the highest dimensionality will no longer be visible because the dimensionality of a geometrical element will decrease by one unit by projection and they may thus overlap each other and also overlap features of the next-higher dimensionality. As an example, Fig. 10.1(b) shows a P, T diagram obtained by projection of Fig. 8.11 (shown again as Fig. 10.1(a)) in the μB direction. Such a P, T diagram is called Schreinemakers' projection [16]. In a system with c components it is obtained by projecting in the directions of c − 1 μi axes. It will show invariant equilibria with c + 2 phases as points, univariant equilibria with c + 1 phases as lines and in the angles between them there will be surfaces representing divariant equilibria with c phases. Using a short-hand notation developed by Schreinemakers, the coexistence lines for c − 1 phases are here identified also by giving in parentheses the phases from the invariant equilibrium which do not take part. For example, the (α) curve represents the α-absent equilibrium, i.e. β + γ + δ. By comparison with Fig. 10.1(a) it can be seen that the angle between (α) and (β) is covered by the γ + δ surface but also by the α + δ surface which extends to the (γ) line and by the β + γ surface which extends to the (δ) line.

Sublattice solution phases

In the substitutional solutions discussed in Section 20.4 all lattice sites were equivalent and a solution was formed from a pure substance by substituting new kinds of atoms for the initial one. However, relatively few crystalline phases belong to this class. The great majority have different kinds of lattice sites and can be described by using two or more sublattices. Examples of such phases will be discussed in this chapter. It will be demonstrated that a great variety of such phases can be modelled in a very direct way using an approach often called the compound energy model or formalism. It is a crude model in the sense that it assumes random mixing within each sublattice. The expression for the entropy of such phases is simple and was presented in Section 19.8, ‘Restricted random mixtures’, but the excess Gibbs energy can easily become very complicated. However, it should be realized that actual calculations of equilibria, and even of whole phase diagrams, can now be carried out with sophisticated computer programs which only require that the expression for the molar Gibbs energy of each phase is defined.

Section 19.8 gave the expression for the entropy assuming random mixing of all the components present in each sublattice. The result was expressed in terms of the site fraction variable, yi, and in Section 19.10 it was then applied to interstitial solutions, which are a special case of solution phases with sublattices.

In the previous chapter we introduced the one-particle irreducible effective action by collecting the one-particle irreducible vertex functions into a generator whose argument is the field, the one-state amplitude in the presence of the source. The effective action thus generates the one-particle irreducible amputated Green's functions. We shall now enhance the usability of the non-equilibrium effective action by establishing its relationship to the sum of all one-particle irreducible vacuum diagrams. To facilitate this it is convenient to add the final mathematical tool to the arsenal of functional methods, viz. functional integration or path integrals over field configurations. We are then following Feynman and instead of describing the field theory in terms of differential equations, we get its corresponding representation in terms of functional or path integrals. This analytical condensed technique shall prove powerful when unraveling the content of a field theory. The loop expansion of the non-equilibrium effective action is developed, and taken one step further as we introduce the two-particle irreducible effective action valid for non-equilibrium states. As an application of the effective action approach, we consider a dilute Bose gas and a trapped Bose–Einstein condensate.

Functional integration

Functional differentiation has its integral counterpart in functional integration. We shall construct an integration over functions and not just numbers as in elementary integration of a function. We approach this infinite-dimensional kind of integration with care (or, from a mathematical point of view, carelessly), i.e. we base it on our usual integration with respect to a single variable and take it to a limit.