Harris criterion

In studies of the phase-transition phenomena, the systems considered before were assumed to be perfectly homogeneous. In real physical systems, however, some defects or impurities are always present. Therefore, it is natural to consider what effect impurities might have on the phase-transition phenomena. As we have seen in the previous chapter, the thermodynamics of the second-order phase transition is dominated by large-scale fluctuations. The dominant scale, or the correlation length, Rc ∼ |T/TC – l|–v grows as T approaches the critical temperature Tc, where it becomes infinite. The large-scale fluctuations lead to singularities in the thermodynamical functions as |τ| ≡ |T/Tc – 1| → 0. These singularities are the main subject of the theory.

If the concentration of impurities is small, their effect on the critical behavior remains negligible so long as Rc is not too large, i.e. for T not too close to Tc. In this regime the critical behavior will be essentially the same as in the perfect system. However, as |τ| → 0 (T → Tc) and Rc becomes larger than the average distance between impurities, their influence can become crucial.

As Tc is approached the following change of length scale takes place. First, the correlation length of the fluctuations becomes much larger than the lattice spacing, and the system ‘forgets’ about the lattice. The only relevant scale that remains in the system in this regime is the correlation length Rc(τ).

In this chapter we introduce several advanced topics. The first deals with different approaches that can be taken to develop overall mass balances within a vessel. This can be used to simplify the analysis of a complex process, as we will see. We then turn to the problem of multi-phase resistances. Up to now we have been able to make assumptions about which of two phases in contact dominates the mass transfer kinetics across the interface between them. However, there are situations in which both phases may contribute to the overall mass transfer, at least over a limited range of conditions. We will develop an approach for dealing with this situation. Finally, we will consider a specific case involving topochemical reactions in porous solids.

Overall mass balance

We have used mass balances extensively throughout this book. Most often we have used them to develop an expression for the motion of an interface separating two reacting phases. In fluids, however, this process can become quite complex. Consider, for example, what happens when a gas stream is passed through or over a bed containing some reactive species. The reaction occurs slowly as the gas stream passes.

In this book we have studied various effects produced by quenched disorder on thermodynamical properties of statistical systems. Considering different types of model, the emphasis has been made on the demonstration of the basic theoretical approaches and ideas. Although the considered systems and the corresponding problems involved (such as spin glasses, critical phenomena, directed polymers etc.) may at first look quite different, the aim of the book was to demonstrate that basically all these problems are deeply interconnected. I was trying to convince the reader that to work successfully on any particular problem in this field one needs to be familar with all the methods and ideas of statistical field theory. The physics of both the spin-glass state and critical phenomena in weakly disordered systems involve the ideas of the scaling theory of phase transitions, and the basic concepts of the replica theory of spin glasses. The aim of this book was to take the reader, starting from fundamentals and demonstrating well-established solutions of various problems, to the frontier of modern research. Here we are facing quite a few fundamental problems, both long-standing and new ones, still waiting for their solutions.

The most appealing problem in the scope of spin glasses had remained a question for almost two decades: whether or not the mean-field RSB physical picture (described in Chapters 2–6) is valid, at least at the qualitative level, for realistic spin glasses with finite-range interactions.

This chapter introduces a wide range of examples related to the heat treatment of binary alloys. The process involved is very simple–an alloy of fixed overall composition is subjected to a given temperature–time cycle. However, the analysis can be quite complex. Our guide to the various possibilities is the appropriate binary phase diagram, which summarizes the equilibrium conditions for the system. We will almost always assume that local equilibrium is established at interfaces, the boundary conditions thus being given by the phase diagram. The problems of interest include the dissolution and the growth of precipitates and the growth of lamellar structures such as pearlite. We will briefly consider how the analysis method can be modified to treat systems involving a third component.

Introduction

We now wish to consider what happens when a multi-component material, initially at equilibrium, is subjected to a change of temperature. This is clearly related to the process of heat treatment of materials. Depending on the temperature change involved particles may dissolve, they may be precipitated or they may change in size and volume fraction.

In this chapter we address a range of issues related to mass transport when counter diffusion is possible. This can occur in solid alloys containing a relatively large concentration of a substitutional solute, such that solute diffusion requires a significant compensating diffusion of the solvent. It also occurs in fluids in which both counter diffusion and convective flow can occur. In this chapter we will consider only quiescent liquids (i.e. those in which convection due to external forces is absent). We will see that counter diffusion and convection are in fact similar in their impact on mass transport. This will enable us to develop a general framework in which it is possible to treat a wide range of problems. One process we will encounter in this chapter involves the use of diffusion couples, in which two materials are placed in contact such that inter diffusion occurs. Diffusion couples represent the second most common process for micro structural manipulation in the solid state (the first being heat treatment, as discussed in Chapter 6). We will also consider a range of problems in which a species diffuses into a binary mixture and reacts with one of the elements in this mixture at a well-defined front. Examples include evaporation and internal oxidation.

In this chapter we continue our study of mass transfer in fluids by relaxing the assumption of quiescent flow. This raises considerably the level of mathematical complexity involved in obtaining solutions, to the point where analytical solutions are not available in most cases. Instead we introduce the concept of a mass transfer coefficient, analogous to the diffusion coefficient for simple diffusion. We will see that a wide range of correlations can be made which predict the mass transfer coefficient as a function of the geometry of the interface, the nature of fluid flow and other material parameters. These correlations are generally obtained by a combination of experimentation and numerical simulation. However, in some simple cases rudimentary analytical models provide useful approximations that we can use. In the latter part of the chapter we will return to the problem of reactions occurring within fluids. We will see that analytical solutions are not possible unless the reaction occurs at a well-defined front.

Transient diffusion in fluids

In the last chapter we worked mostly with flux equations analogous to Fick's First Law. We have noted previously, however, that this form of the diffusion law is most useful when working on steady-state problems. For transient problems, it is more convenient to use an equation which contains an explicit time dependence of concentration.