An Interactive Online Curriculum in Phase Equilibria

  1. Credits: This course is being developed by Prof. Paul Asimow with funding from the U.S. National Science Foundation. It is the result of an ongoing collaboration with a series of undergraduate students that have developed most of the software, including Asitha Mallawaarachchi, Hoang Banh, Yuh Han John Liow, and Nizar Almoussa. The inspiration for developing this course, and much guidance in the sequence of topics presented, comes from Prof. James B. Thompson.
  2. How to use this curriculum: The curriculum is structured as an interactive tutorial, in which the concepts are developed in sequence with explanatory text in a series of html files (beginning with this one) with links to a platform-independent visualization applet for the binary part or to a Windows-only application that handles the three-dimensional graphics for the ternary part. Students can work through the course in tutorial mode; Professionals may want to jump directly to the new binary applet or, coming soon, download and run the ternary Windows application. Other useful entry points include the Table of Contents below, the glossary of thermodynamic terms, or the old binary demonstration applet.
  3. Under construction: This course is an ongoing project and is subject to update at any time. Although it begins at the beginning, it does not yet reach the end.

Table of Contents

  1. Underlying thermodynamics
    1. What's this all about?
    2. First and second laws, equilibrium, Gibbs free energy
    3. Chemical potential
    4. The graphical representation of equilibrium conditions
    5. The phase rule
  2. Binary systems
    1. One phase
      1. One phase, including critical point
    2. Two phases
      1. Simple phase loop
      2. Phase loop with minimum coincidence
      3. Phase loop with maximum coincidence
      4. Phase loop with minimum coincidence, miscibility gap, and critical end point
    3. Three phases
      1. Three phases with eutectic-type univariant line
      2. Three phases with peritectic-type univariant line and singular point
    4. Four phases
      1. Four phase invariant point, first orientation
      2. Four phase invariant point, second orientation
    5. Degeneracy and final remarks
      1. MgO-SiO2 system, approximately
  3. Ternary systems

I. Underlying thermodynamics

A. What's this all about?

This curriculum teaches the elementary principles of phase equilibria, a branch of chemical thermodynamics. This knowledge is useful to anyone dealing with chemical systems that may involve more than one phase, especially when there is also more than one component. This includes geologists, since rocks are generally multiphase systems (more than one mineral coexists), especially in igneous, metamorphic, and hydrothermal systems where fluid phases and high temperatures contribute to the achievement of thermodynamic equilibrium. It also includes all branches of materials engineering, including metallurgy, ceramics, composites, and chemical engineering; if a phase change (freezing, boiling, precipitation, recrystallization, etc.) occurs then the principles of phase equilibria are needed to understand it.

The approach of this curriculum emphasizes the use of geometric constructions that allow a visual interpretation of the rules of phase stability. The basic idea is to understand how we predict what phases will be stable in a system when we prescribe the temperature, pressure, and composition, and how those stability relations will change as the prescribed parameters are varied. If all the definitions, equations, and/or graphical representations ever start to make your head spin, come back and read this paragraph again to remind yourself what we are doing here.

In order to derive the equations underlying the graphical constructions and understand their meaning, we must go all the way back to the foundations of thermodynamics and build up some definitions and concepts.

B. First and second laws, equilibrium, Gibbs free energy

The First and Second laws of thermodynamics allow us to define criteria for whether or not a chemical system is at equilibrium. For a closed system, the first law can be written

,
where E is the internal energy of our system, q is heat entering the system, and w is work done on the system. See the notes on exact differentials to understand what is meant by dE. Considering only mechanical work associated with changes in the volume of the system, the work term may be written w = -PdV.

The Second Law is an inequality if we consider both reversible and spontaneous processes:

.
Solving for q and substituting the second law into the first, we obtain what is sometimes called the fundamental equation of thermodymamics,
.
Though fundamental, this equation is not very useful: it tells us that at constant entropy (dS = 0) and volume (dV = 0), the internal energy will decrease in any spontaneous process and will reach a minimum at equilibrium. But it is very hard to imagine a natural process or construct an experiment in which entropy and volume are controlled. We therefore perform a change of variables by introducing the Gibbs free energy, G = E + PV - TS. Now the total differential of G is
,
from which we get a much more useful criterion for equilibrium:
.
This equation shows why Gibbs free energy is so important: at constant temperature (dT = 0) and pressure (dP = 0), conditions which are easy to impose in the laboratory or to imagine in nature, it becomes dG < 0, where the less than sign applies to irreversible or spontaneous processes (it is inherited from the second law) and the equals sign applies to reversible processes or equilibrium states. In other words, if we control P and T, then the direction of approach to equilibrium is always a decrease in Gibbs free energy, until equilibrium is achieved when Gibbs free energy reaches a minimum. If there are no lower values of G accessible to the system (a global minimum), the equilibrium is stable; if a perturbation could bump the system out of a local minimum in G and allow it to evolve down to a lower minimum, then we were at a metastable equilibrium.

Though we will here explore only the common situation of constant T and P, wherein equilibrium is defined by minima in G, it is worth keeping always in mind that there are other situations where the equilibrium criterion is different. At constant temperature and volume, equilibrium is found at the minimum in Helmholtz free energy F = E - TS. At constant pressure and entropy, equilibrium is found at the minimum in enthalpy H = E + PV.

For a phase, the Gibbs free energy is a function of P, T, and composition. Like all energies in thermodynamics, G is expressed not in absolute terms but relative to a standard state, usually the energy difference relative to the elements at 1 bar and 298.15 K. If the enthalpy of formation and third law entropy are measured, then we get G from G = H - TS. If we know G at one temperature and pressure, then since it is an exact differential, we can see from the expression for dG above that for reversible paths

and ,
which allows us to calculate G at any P and T from calorimetric and volumetric data.

The intensive Gibbs free energy of a phase, which we denote , is also, in general, a function of composition. All real phases have some range of variability in composition, though some are always nearly pure. Thermodynamically, that is to say that their free energy may increase very fast as one tries to add other components to such a phase. The extent to which a phase will vary in composition in equilibrium with other phases is apparent from a graph of its intensive Gibbs free energy vs. composition; for a binary system with one intensive compositional variable, X, this is a graph of (X).

For a binary phase to be stable, it is necessary that the (X) curve be concave-up. You can prove this to yourself by considering what happens if the curve is concave down: if a homogenous phase at composition X were to unmix at constant temperature, pressure, and bulk composition into two phases of composition X+e and X-e, then the Gibbs free energy would change from (X) to ((X+e) + (X-e))/2. But the second derivative is defined as the limit as e goes to zero of [((X+e) + (X-e)) - 2(X)]/e2, so if the second derivative is less than zero than the unixing lowers the Gibbs free energy. Hence the phase is unstable to decomposition by an infinitesimal perturbation where one region becomes slightly richer in one component and another slightly poorer. The situation of concave-down free energy surfaces does in fact arise; the boundary between a region where the free-energy surface is concave-up (stable) and concave-down (unstable) is called the spinodal and leads to the phenomenon of exsolution. At any pressure, there is generally a maximum temperature where the spinodal terminates, which is called a critical point. The family of critical points at various pressures forms a critical line.

In ternary and higher-order systems, there is more than one independent intensive compositional variable and the the simple criterion d2/dX2 > 0 is not sufficient to define stability. The criterion is still that the surface be concave-up, but to define this idea mathematically we must consider the matrix of second derivatives of , also called the Hessian matrix. If this matrix is positive definite (has all positive eigenvalues and a positive determinant), the surface is concave-up and the phase is stable at this point. If any of the eigenvalues are negative, the surface is either hyperbolic (concave-down along some directions) or concave-down and hence unstable. On the boundary between these regions, the spinodal, the determinant of the Hessian matrix is zero.

C. Chemical potential

In multicomponent systems, where G depends on composition, we need to modify our equilibrium criterion so we can examine both the composition of the stable phases and how the stable phase assemblage varies with bulk composition. This requires introducing a quantity called the chemical potential, . Chemical potential has the important property that the chemical potential of each component is the same in all phases coexisting at equilibrium. It is easy to understand why: the definition of is the change in G that results from adding an increment of component mass to a phase at constant temperature, pressure, and masses of the other components. If for some component is higher in one phase than in another, it follows that we can lower the overall G of the system by moving a mass increment of that component from the high phase to the low phase. Hence G was not at a minimum. But being at a minimum of G is a criterion for equilibrium.

It can be shown that the intensive Gibbs free energy either of the system or of a phase is related to the sum of the mass fractions of the components times their partial specific Gibbs free energies or chemical potentials; for the binary case, there are two components (say, 1 and 2) and:

.
Since the two mass fractions sum to unity in the binary case, however, let us define X = x2 = 1 - x1 and simplify the above to
,
which is the equation of a line in -X space with intercept 1 and slope 2-1. In a plot of vs. X at constant T and P, this line is tangent to the (X) curve. Do not misinterpret the above equation as an expression that tells how varies with X: the chemical potentials of the components are themselves functions of X. Now, if two or more binary phases are in equilibrium, the chemical potential of both components must be the same in the coexisting phases (see previous paragraph), so their (X) curves must share a common tangent line, whose intercept and slope are given by the above equation. Furthermore, the compositions that coexist are given by the tangency points where this "chemical potential line" touches the free energy-composition curves. This is the basis of the visual construction that we use to find equilibria. This concept is easily extended to systems of more components, where it remains true that the chemical potentials of all components must be equal among all phases at equilibrium. In the ternary system, the tangent plane to a free-energy surface is defined by the chemical potentials and all coexisting phases must share a common tangent plane in (X1, X2) space.

D. The graphical representation of equilibrium conditions

Figure 1

Consider the -X diagram in Figure 1. This diagram always shows the properties of two phases of a binary system as a function of composition (X) at contant temperature and pressure. In Figure 1 we show the individual (X) curves for two phases. If we imagine fixing P, T and total X of the system, how do we find the equilibrium state of a system in which these are the only two possible phases, and how do we see the changes in that state as a function of X? By state of the system we mean what phases are present, how much of each, and the composition of each phase.
Figure 2

Recall what we said in the previous paragraph about chemical potential: it is a necessary and sufficient condition for equilibrium between the two phases that the chemical potential of the each component be equal between the phases, and this is shown graphically by both curves sharing a common tangent line, as shown in Figure 2. The intercept of the tangent line at X = 0 (remember X is the mass fraction of component 2) is 1 in phase A and in phase B, and the intercept of the tangent line at X = 1 is 2 in phase A and in phase B. These two points are sufficient to define a line, and the line is tangent to both (X) curves. But there is more information here: not just any composition of phase A can be in equilibrium with phase B. Only phase A with the composition marked XA(B), which is the point where the tangent line touches the A(X) curve, has the correct chemical potential. Likewise only if phase B has the composition marked XB(A), where the tangent line touches the B(X) curve, can it be in equilibrium with A under these conditions.

There is still more information on these diagrams. We have shown that there exists an equilibrium where A and B of particular phase composition will coexist. But how do we decide what the stable phase assemblage will be at given bulk composition X? Well, we need to find the configuration with minimum Gibbs free energy. There are two possibilities: either (1) the system will contain only one homogeneous phase whose phase composition equals the bulk composition, or (2) the system will contain a mechanical mixture of more than one phase which are all in equilibrium with each other and whose compositions add up to the bulk composition. In case (1), the intensive Gibbs free energy of the system is equal to the intensive Gibbs free energy of the single phase and we can read this directly off the diagram from the (X) curve for the phase. In case (2), begin by imagining there are two coexisting phases A and B, making up mass fractions fA and fB of the system, respectively, and having composition XA(B) and XB(A). Then the bulk composition of the system is X = fAXA(B) + fBXB(A), and the intensive Gibbs free energy of the system is (X) = fAA(XA(B)) + fBB(XB(A)). Now look again at Figure 2: these two equations define a line segment that connects the two points (XA(B), A(XA(B))) [when fA=1 and fB=0] and (XB(A), B(XB(A))) [when fA=0 and fB=1]. But this line segment is exactly the same as the segment between the points of tangency of the common tangent line defined by the chemical potentials of the coexisting phases. Let's call this part of the tangent line an "interior tangent segment". When two phases coexist, we can read the bulk (X) off the diagram by taking the where the tangent line between their curves crosses the bulk composition of interest. The part of the common tangent line exterior to the two tangency points is still useful for extrapolating to the chemical potentials at X=0 and X=1, but it does not represent a physically achievable free energy for the system, since one of the phase proportions would have to be negative to obtain a point along the line exterior to the tangency points. Figure 3

Now we have what we need to read the stable phase assemblage off a binary -X diagram. Recall that the criterion for equilibrium at prescribed P, T and X is that be a minimum; graphically this means finding the lowest physically achievable state on the diagram for a given X (the diagram already represents prescribed P and T). A physically achievable state is either the entire system as a single phase or a mechanical mixture of phases that can coexist with one another. In the case of a single phase, the phase composition equals the bulk composition and the value of is given by the curve for that phase. So, on our plot, at any X, the lowest available is represented either by the (X) curve of a phase (these are colored red on the diagram) or by the interior segment of a tangent line connecting two such curves (this is colored green on the diagram). If a curve is lower than any interior tangent segment, then we are in a one-phase region in which the phase with the lowest (X) curve exists alone and the phase composition equals the bulk composition. On the other hand, if at some X an interior tangent segment is lower than any curve, then a mechanical mixture of the phase compositions at the two endpoints will have lower free energy than any single phase with composition X. Hence we will be in a two-phase region with the compositions of the two phases fixed at the endpoints of the interior tangent segment. The relative proportions of the two phases follow from the lever rule,

.
In Figure 3, we have colored red the sections of the A(X) and B(X) curves where A and B are stable alone in one-phase regions, and green the interior tangent segment where A and B coexist. We have also labelled the sequence of stable phase assemblages (A, A+B, B) across the bottom and divided the X-axis into the three regions.

This sequence, and the points where the phase assemblage changes across the X-axis, contain the essential information derived from this diagram: once we have found the minimum Gibbs free energy assemblages we usually do not care any further about . Therefore, since it is easiest to make graphs in two dimensions, it is not an efficient use of graph paper (or computer screen) to show a whole plot which only describes the system at a single pressure and temperature. Instead, we make T-X (isobaric) or P-X (isothermal) diagrams that show a stack of the X-axis stability sequences derived from -X diagrams at a sequence of temperatures and equal pressure (T-X) or a sequence of pressures and equal temperature (P-X). The one-phase and two-phase intervals of these segments link up with those at adjacent T or P to form regions, bounded by the compositions of phases that coexist in two-phase regions. Note that with two phases whose (X) curves are everywhere concave-up, there are only a few possible sequences: either one (X) curve is lower than the other everywhere and one phase will be stable alone at all X; or the curves cross once, in which case the sequence will be like (A,A+B,B); or the curves cross twice, in which case the sequence will be (A,A+B,B,B+A,A).

Much of what we have discussed so far can be seen by exploring the old binary applet; just click on some random points in the P-T projection at upper left. The applet will bring up a -X diagram corresponding to the P-T point you clicked. There are four phases involved (or three in the simplified version), but if you do not click on a line or intersection of lines, you will see that no more than two phases ever share a common tangent, and the sequences of stability across the X-axis all follow the rules we have just developed. If you wish to continue with the old binary applet, proceed to Page 2 of the old tutorial. However, we recommend that instead you work through the series of examples below that make use of the new applet. When you finish with binary systems, you can proceed to ternary systems.

E. The phase rule

At this point we need to step aside a minute and consider the apparently unrelated issues of how many phases can coexist at equilibrium and how many variables need to be fixed to determine the state of the system. We have shown already that in a binary system we can have one-phase regions where the phase can have any composition at fixed P and T or two-phase regions where both phase compositions are fixed. This suggests that actually the number of phases and the number of free variables is related. This relationship is explicitly captured by the Gibbs Phase Rule, which relates the variance (f) of an assemblage to the number of phases , number of components c, and other restrictions imposed. The phase rule is best understood by thinking of the conditions of equilibrium as a set of simultaneous equations that nature must solve. We learn in elementary algebra that if we have the same number of unknowns (variables) and equations (constraints) then we should expect to find a unique solution, i.e. there are no remaining degrees of freedom in the unknown variables. On the other hand, if we have more unknowns than equations, there is usually a family of solutions, and the dimension of the solution space is the number of extra unknowns. Finally, if there are more equations than unknowns, then unless we are very lucky there will be no solution at all. In the case of the phase rule, the unknowns are P, T, and (c-1)* independent phase compositions. The constraints are the (-1)*c statements of equality of chemical potential among phases. Hence the dimension of the allowed solution space is f = 2 + (c-1)* - (-1)*c = c + 2 - . Once it is given that we have an equilibrium assemblage in a system of c components containing particular phases, the variance is the number of remaining variables we have to fix to determine the state of the system.

Consider, e.g., the one component system H2O. If one phase — say, liquid water — is present, then f = 1 + 2 - 1 = 2, and indeed we can freely vary both temperature and pressure within a two-dimensional but bounded stability region without change of phase. However, if we insist that water and ice are coexisting (so f = 1 + 2 - 2 = 1) and we fix one more variable — say make the pressure 1 atm — then the temperature becomes fixed (it must be 0 °C). Ice+liquid in the H2O system is an example of a univariant assemblage, which is restricted to a one-dimensional array (i.e., a line or curve) in P-T-X space. Furthermore, although it is slightly more remote from everyday experience, there is a single point in (P,T) space at P = 6 mbar, T = 0.01 °C where ice, liquid water, and water vapor (steam) can all coexist. This is called the triple point and is an example of an invariant condition f = 1 + 2 - 3 = 0 where merely declaring the number of phases and components has entirely determined the state.

There can be other constraints in the phase rule that lower the variance without adding more phases. For example, if we require that two phases are equal in composition or that three phases are collinear in a three-or-more component system, this takes away one degree of freedom. Hence in the binary system if a solid phase coexists with a liquid of the same composition as the solid (a congruent melting point) this assemblage is univariant: f = 2 + 2 - 2 - one extra restriction = 1. Likewise, specifying that one of the phases is at its critical point reduces the variance by two extra restrictions. And there can be situations where some of the phases are restricted to specific compositions or subspaces of the composition space; this leads to degenerate equilibria where the variance is actually higher than if the phases were free to vary in composition.

II. Binary systems

We now begin specifically applying the concepts we have learned so far to binary systems, with the help of the binary visualization applet. We will work upwards in complexity from systems with one phase through systems of four or more phases. In each case we will focus on the special situations that can arise as the number of phases increase. Thus, when we get to two phases the possibility of the two phases coexisting with equal composition arises, so we will look at coincidences. When we get to three phases, we can have univariant assemblages, so we will focus on such cases. Finally, when we get four phases we can have an invariant point. We begin, however, with one phase by itself, but already some interesting phenomena can arise.

A. One phase

1. The situation of a binary system with a single phase that contains a critical point is explored using the new binary visualization applet on the Example 1 page.

B. Two phases

2. The simplest situation that can arise with two phases is a binary phase loop, with degenerate coincidences at the bounding pure systems, but only a divariant field in the binary. Go to the Example 2 page to learn more.

3. When the phases are not ideal solutions, the possibility can arise that they have a coincidence point or azeotrope in the binary system. We first look at the case where this coincidence point forms a minimum in temperature: Example 3.

4. Next comes the case where this coincidence point forms a maximum in temperature: Example 4.

5. Perhaps the most complicated situation that can arise involving only two phases occurs when the two-phase loop has a coincidence point as in example 3 and furthermore one of the phases has a critical point and miscibility gap as in example 1. The intersection of these phenomena generates three new features: a univariant curve involving two instances of the phase with a miscibility gap and one instance of the other phase; a critical end-point where the univariant line terminates against the critical line and the critical line becomes metastable; and a singular point where the coincidence encounters the univariant, at which point the coincidence becomes metstable and the unvariant changes from eutectic-type to peritectic-type. All this is visible in Example 5.

C. Three phases

6. Example 6. Explanation to come.

7. Example 7. Explanation to come.

D. Four phases

8. Example 8. Explanation to come.

9. Example 9. Explanation to come.

E. Degeneracy and final remarks

10. Example 10. Explanation to come.

III. Ternary systems

Under construction.


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