A Modern Course in Statistical Physics

4

Thermodynamics of Phase Transitions

L.E.Reichl

''A Modern Course in Statistical Physics''

Univ of Texas Press (80)

A. Introductory Remarks

Thermodynamic systems can exist in a number of phases, each of which can exhibit dramatically different macroscopic behavior. Generally, systems become more ordered as temperature is lowered. Forces of cohesion tend to overcome thermal motion, and atoms rearrange themselves in a more ordered state. Phase changes occur abruptly at some critical temperature although evidence that one will occur can be found on a microscopic scale as the critical temperature is approached. The study of the transition region between phases is one of the most intcresting fields of statistical physics and one that we shall return to often throughout this book. In this chapter we will be concerned solely with the thermodynamics of phase transitions, that is, the description of phase transitions in terms of macroscopic variables. In later chapters we shall study them from a microscopic point of view.

The first step in trying to understand the phase changes that occur in a system is to map out the phase diagram for the system. At a transition point, two (or more) phases can coexist in equilibrium with each other. The condition for equilibrium between phases is obtained from the equilibrium conditions derived in Chap. 2. Since phases can exchange matter, equilibrium occurs when the chemical potentials of the phases become equal for given values of Y and T. From the equilibrium condition, we can determine the maximum number of phases that can coexist and, in principle, find equations for the region of coexistence (e.g., the Clausius- Clapeyron equation).

At phase transitions the chemical potentials of the phases, and therefore the Gibbs free energy, must change continuously. However, phase transitions can be divided into two classes according to the behavior of derivatives of the Gibbs free energy. Phase transitions which are accompanied by a discontinuous change of state (discontinuous first derivatives of the Gibbs free energy) are called first- order phase transitions. Phase transitions which are accompanied by a continuous change of state (higher order derivatives of the Gibbs free energy will be discontinuous) are called continuous phase transitions. We give examples of both in this chapter.

Classical fluids provide some of the most familiar examples of first- order phase transitions. The vapor- liquid, vapor- solid, and liquid- solid transitions are all first order. We shall discuss the phase transitions in classical fluids in some detail. For the vapor- solid and vapor- liquid transitions, we can use the Clausius- Clapeyron equation to find explicit equations for the coexistence curvcs. Since the vapor- liquid transition terminates in a critical point, we will focus on it and compare the observed behavior of the vapor- liquid coexistence region to that predicted by the van der Waals equation.

A binary mixture of molecules provides a different example of a phase transition. For that system below a certain critical temperature, we can have a physical separation of the mixture into two fluids, each rich in one of the types of molecules. Following Hildebrand, it is possible to write a rather simple expression for the Gibbs frre energy of the mixture (on the level of the van der Waals equation) which contains the essential features of the phase transition. We then can use stability arguments to study the phase transition in some detail.

Most phase transitions have associated with them a critical point (the liquid- solid transition is one of the few which does hot). That is, there is a well- defined temperature above which one phase exists, and as the temperature is lowered a ncw phase appears. When a new phase appears as we lower the temperature, it often has different symmetry properties and some new variable, called the order parameter, appears which characterizes the new phase. For first- order phase transitions, there need not be a connection between the symmetries of the high- temperature phase and of the low- temperature phase. For a continuous transition, since the state changes continuously, there generally will be a well defined connection between symmetry properties of the two phases. Ginzburg and Landau were able to construct a completely general theory of continuous symmetry- breaking phase transitions which involves a power series expansion of the free energy in terms of the order parameter. We shall discuss the Ginzburg- Landau theory in this chapter and show how it can be applied to magnetic systems at the Curie point and to superfluid systems.

Superconductor are especially interesting from the standpoint of thermodynamics because they give us quite a different application of the Clausius- Clapeyron equation and they provide a very clear example of the uses of the Ginzburg- Landau expansion. Many features of the Ginzburg- Landau expansion for superconductors can be taken over directly to superfiuid and, with some major complications, to superfluid . We therefore discuss superconductors in some detail in this chapter.

Since a great deal of this chapter is devoted to classical fluids, we are including a brief discussion of the phase diagrams for liquid and liquid , which are quantum fluids. These systems give a good illustration of the third law of thermodynamics.

The critical point plays a unique role in the theory of phase transitions. As a system approaches its critical point, from high temperatures, it begins to adjust on a microscopic level. Large fluctuations occur which signal the emergence of a new order parameter which finally does appear at the critical point itself. At the critical point, some thermodynamic variables can become infinite. Critical points occur in a huge variety of systems, but regardless of the particular substance or mechanical variable involved, there appears to be a great similarity in the behavior of all systems as they approach their critical points. For this reason, there is a whole field of physics devoted to the study of critical phenomena.

One of the best ways to characterize the behavior of systems as they approach the critical point is by means of critical exponents. We shall define critical exponents in this chapter and give explicit examples of some of them for the liquid- vapor transition in simple fluids and for the Curie point. We shall also compute critical exponents for the liquid- vapor transition using the van der Waals equation. This will give us another means of comparing the van der Waals equation with the experimentally observed behavior of real fluids.

In this chapter we will discuss several equations which describe the behavior of systems around the critical point. Among them are the van der Waals equation, the equation for regular binary mixtures, and the Ginzburg- Landau expansion. All these theories are known as mean field theories because they can be derived by assuming that each particle moves in the mean field of all other particles. Mean field theories do not give the correct value for the critical exponents, because they do not correctly take into account short- ranged correlations which are important near the critical point. However, they give qualitatively correct behavior near critical points and therefore are useful in building our intuition.

B. Coexistence of Phases: Gibbs Phase Rule

Most systems can exist in a number of different phases, each of which can exhibit different macroscopic behavior. The particular phase that is realized in nature for a given set of independent variables is the one with the lowest free energy. It can happen that, for certain values of the independent varibles, two or more phases of a system can coexist. There is a simple rule, called the Gibbs phase rule, which tells us the number of phases that can coexist. Generally, coexisting phases are in thermal and mechanical equilibrium and can exchange matter. Under these conditions, the temperature and chemical potentials of the phases must be equal (cf. Sec. 2.H) and there will be another condition between mechanical variables expressing mechanical equilibrium. For example, for a simple PVT system, the pressures of the two phases may be equal (this need not be the case if they are separated by a surface which is not free to move.)

For simplicity, let us first consider a YXT system which is pure (composed of one kind of particle). For a pure system, two phases, I and II, can coexist at a fixedvalue of Y and T if their respective chemical potentials are equal:

(4.1)

(The chemical potentials are functions only of intensive variables.) Eq. (4.1) gives a relation between the values of Y and T for which the phases can coexist,

(4.2)

and in the Y-T plane defines a coexistence curve for the two phases. If the pure system has three phases, I, II, and III, they can only coexist at a single point in the Y-T plane (the triple point). Three coexisting phases must satisfy the equations

(4.3)

Since we have two equations and two unknowns, the triple point is uniquely determined. For a pure system, four phases cannot coexist, because we would then have three equations and two unknowns and there would be no solution.

For a mixture of l different types of particles, phases can coexist. To show this, we note that if there are l types of particles in each phase then there will be independent variables for each phase, namely, ( ) where is the mole fraction of particles of type i. If we have several phases coexisting, the chemical potentials for a given type of particle must be equal in the various phases. Thus, if there are r coexisting phases at a given value of Y and T, the condition for equilibrium is

(4.4)

(4.5)



(4.6)

Eqs.(4.4)-(4.6) give equations to determine unknowns. For a solution, the number of equations cannot be greater than the number of unknowns. Thus, we must have  or . The number of coexisting phases must be less than or equal to , where l is the number of different types of particle. For a pure state and as we found before. For a binary mixture and , and, at most, four different phases can coexist.

As an example of the Gibbs phase rule, we show the coexistence curves for various solid phases of water (cf. Fig. 4.1). We see that, although water exists in many solid phases, no more than three phases can coexist at a given temperature and pressure.

C. Classification of Phase Transitions

As we change the independent intensive variables ( ) of a system, we reach values of the variables for which a phase change can occur. At such points the chemical potentials (which are functions only of intensive variables) must be equal and the phases can coexist.

In Sec.2.F, we found that the Gibbs free energy is closely related to the chemical potentials. The fundamental equation for the Gibbs free energy is

(4.7)

where is the chemical potential per mole, and for processes which occur at constant Y and T,

(4.8)

Thus, at a phase transition, the Gibbs free energy of each phase must have the same value, and the derivatives must be equal. However, no restriction is placed on the derivatives and . The behavior of these derivatives is used to classify phase transitions. If the derivatives and are discontinuous at the transition point (that is, if the extensive variable X and the entropy S have different values in the two phases), the transition is called "first- order." If the derivatives and are continuous at the transition but higher ordcr derivatives are discontinuous, then the phase transition is continuous. (The terminology "nth-order phase transition" was introduced by Ehrenfest to indicate a phase transition for which the nth derivative of G was the first discontinuous derivative. However, for some systems higher order derivatives are infinite, and the theory proposed by Ehrenfest breaks down for those cases. )

Let us now plot the Gibbs free energy for first-order and continuous transitions in a PYT system. For such a system the Gibbs free energy must be a concave function of P and T (cf. Sec.2.H). The Gibbs free energy and its first derivatives are plotted in Fig. 4.2 for a first-order phase transition. A discontinuity in means that there is the discontinuity in the volume of the phases,

(4.9)

and a discontinuity in means there is a discontinuity in the entropy of the two phases,

(4.10)

Since the Gibbs free energy is the same for both phases at the transition, the fundamental equation shows that the enthalpy of the two phases is different,

(4.11)

for a first order phase transition; is called latent heat of the transition.

For a continuous phase transition, the Gibbs free energy is continuous but its slope changes rapidly. This in turn leads to a peaking in the heat capacity at the transition point. An example is given in Fig. 4.3. For a continuous transition, there is no abrupt change in the entropy or the extensive variable (as a function of Y and T) at the transition.

In the subsequent sections we shall give examples of first-order and continuous phase transitions.

D. Pure PVT Systems

1. Phase Diagrams

A pure PVT systcm is a system composed of only one type of moleculc. The molecules generally have a repulsive core and a short-range attractive region outside the core. Such systems have a number of phases: a gas phase, a liquid phase, and various solid phases. A familiar example of a pure PVT system is water.

A typical set of coexistence curves for pure substances is given in Fig. 4.4. (Note that Fig. 4.4 does not describe the isotopes of helium, or , which have superfluid phases, but it is typical of most other pure substances.) Point A on the diagram is the triple point, the point at which the gas, liquid, and solid phases can coexist. Point C is the critical point, the point at which the vapor pressure curve terminates. The fact that the vapor pressure curve has a critical point means that we can go continuously from a gas to a liquid without ever going through a phase transition, if we choose the right path. The fusion curve does not have a critical point (one has never been observed). We must go through a phase transition in going from the liquid to the solid state. This difference between the gas-liquid and liquid-solid transitions indicates that there is a much greater fundamental difference between liquids and solids than between liquids and gases — as one would expect. Solids exhibit spatial ordering, while liquids and gases do not. (We use "vapor" and "gas" interchangeably.)