# Equilibrium Phase Diagrams

Most of the materials that we use are not pure but consist of a mixture of one or more constituents. Each of the three material states of gases, liquids and solids may consist of a mixture of different components, e.g. in alloys of two metals.

These components are called phases, with each phase being homogeneous. We need a scheme that allows us to summarise the influences of temperature and pressure on the relative stability of each state (and, where necessary its component phases) and on the transitions that can occur between these. The time-honoured approach to this is with equilibrium diagrams. Note the word equilibrium.

Thermodynamics tells us that this is the condition in which the material has minimum internal energy. By definition, equilibrium diagrams tell us about this minimum energy state that a system is trying to reach, but when using these we should bear in mind that it will always take a finite time for a transition from one state to another to occur or for a chemical reaction to take place.

Sometimes, this time is vanishingly small, as when dynamite explodes. At other times, it can be a few seconds, days or even centuries. Glass made in the Middle Ages is still glass and shows no sign of crystallising.

So, not every substance or mixture that we use has reached thermodynamic equilibrium. We only have space here to introduce some of the elements of the great wealth of fundamental theory underlying the forms of equilibrium diagrams.

# Single-component Equilibrium Phase Diagrams

The temperature–pressure diagram for water (Fig. 1) is an important example of a single-component diagram, and we can use this to establish some ground rules and language for use later. The diagram is in ‘temperature–pressure space’ and a number of lines are marked which represent boundary conditions between differing phases, i.e. states of H2O.

The line AD represents combinations of temperature and pressure at which liquid water and solid ice are in equilibrium, i.e. can coexist. A small heat input will alter the proportions of ice and water by melting some of the ice.

However, it is absorbed as a change in internal energy of the mixture, the latent heat of melting. The temperature is not altered, but if we put in large amounts of heat, so that all the ice is melted and there is some heat left over, the temperature rises and we end up with slightly warmed water.

Similarly, line AB represents the equilibrium between liquid water and gaseous steam, and line AC the equilibrium between solid ice and rather cold water vapour. It is helpful to consider what happens if we move around within the diagram.

First, let us start at point X, representing −5°C at atmospheric pressure. We know we should have ice and, indeed, the point X lies within the phase field labelled ice. Adding heat at constant pressure takes the temperature along the broken line.

This crosses the phase boundary, AD, at 0°C (point Y) and the ice begins to melt as further heat is added. Not until all the ice has melted does the temperature continue to rise. We now have liquid water until we reach 100°C (point B).

Now, again, heat has to be added to boil the water but there is no temperature increase until all the liquid water has gone. We now have steam and its temperature can be increased by further heat input.

Next think of keeping temperature constant and increasing pressure, again starting at point X. If the pressure is raised enough, to about 100 atmospheres (≈10 MPa, point D) we reach the ice–water equilibrium and the ice can begin to melt.

This accounts for the low friction between, for example, an ice skate and the ice itself: local pressures cause local melting. It is a factor that engineers need to consider when contemplating the use of locally refrigerated and frozen ground as coffer dams or as foundations for oil rigs in Alaska.

The Gibbs phase rule is a formal way of summarising the relationship between the number of phases (P) that can coexist at any given point in the diagram and the changes brought about by small changes in temperature or pressure. This states that:

P + F = C + 2 ….(1)

Here, C is the number of components in the system; in this case we have only H2O so C = 1. F is the number of degrees of freedom allowed to change.

To illustrate, at point X in Fig. 1 there is just one phase, ice, so P = 1 and F = 2. This means that both temperature and pressure can be changed independently without bringing about a significant change to the material.

At Y both solid and liquid can coexist, so P = 2 and F = 1. To maintain the equilibrium, temperature and pressure must be changed in a co-ordinated way so that the point Y moves along the boundary AD.

At A, all three phases can coexist so P = 3, therefore F = 0, i.e. any change at all will disturb the equilibrium.

## Two-component Phase Diagrams

We now go on to look at two-component diagrams, such as we get with alloys between two metals or between iron and carbon. We now have a further variable, composition and, strictly, we should consider the joint influences of this variable in addition to temperature and pressure.

We would therefore need three-dimensional diagrams, but to simplify things we usually take pressure to be constant. After all, most engineering materials are prepared and used at atmospheric pressure, unless you work for NASA! This leaves us with a composition–temperature diagram, the lifeblood of materials scientists.

The alloys formed between copper, Cu, and nickel, Ni (Fig. 2) produce an example of the simplest form of two-component diagram. This is drawn with composition as the horizontal axis, one end representing pure (100%) Cu, the other pure (100%) Ni. The vertical axis is temperature.

Let us think about an alloy that is 50%Cu:50%Ni by mass. At high temperatures, e.g. at A, the alloy is totally molten. On cooling, we move down the composition line until we arrive at B. At this temperature, a tiny number of small crystals begin to form.

Further reduction in temperature brings about an increase in the amount of solid in equilibrium with a diminishing amount of liquid. On arriving at C, all the liquid has gone and the material is totally solid. Further cooling brings no further changes.

Note that there is an important difference between this alloy and the pure metals of which it is composed.

Both Cu and Ni have well defined unique melting (or freezing) temperatures but the alloy solidifies over the temperature range BC; metallurgists often speak of the ‘pasty range’. We now need to examine several matters in more detail.

First, the solid crystals that form are what is known as a ‘solid solution’. Cu and Ni are chemically similar elements and both, when pure, form face-centred cubic crystals.

In this case, a 50:50 alloy is also composed of face-centred cubic crystals but each lattice site has a 50:50 chance of being occupied by a Cu atom or a Ni atom.

If we apply Gibbs’s phase rule at point A, C = 2 (two components, Cu & Ni) and P = 1 (one phase, liquid) and so F = 3 (i.e. 3 degrees of freedom). We can therefore independently alter composition, temperature and pressure and the structure remains liquid.

But remember, we have taken pressure to be constant and so we are left with 2 practical degrees of freedom, composition and temperature. The same argument holds at point D, but, of course, the structure here is the crystalline solid solution of Cu and Ni.

At a point between B and C we have liquid and solid phases coexisting, so P = 2 and F = 2. As before, we must discount one degree of freedom because pressure is taken as constant. This leaves us with F = 1, which means that the status quo can be maintained only by a coupled change in both composition and temperature.

Therefore, it is not only that the structure is two phase, but also that the proportions of liquid and solid phases remain unaltered. We can find the proportions of liquid and solid corresponding to any point in the two phase field using the so-called Lever rule.

The first step is to draw the constant temperature line through the point X, Fig. 2. This intersects the phase boundaries at Y and Z. The solid line containing Y represents the lower limit of 100% liquid, and is known as the liquidus.

The solid line containing Z is the upper limit of 100% solid and known as the solidus. Neither the liquid nor solid phases corresponding to point X have a composition identical with that of the alloy as a whole. The liquid contains more Cu and less Ni, the solid less Cu and more Ni.

The composition of each phase is given by the points Y and Z, respectively. The proportions of the phases balance so that the weighted average is the same as the overall composition of the alloy. It is easy to show that:

(Weight of liquid of composition Y) × YX = (Weight of solid of composition Z) × XZ

This is similar to what would be expected of a mechanical lever balanced about X, hence the name Lever rule. One consequence of all this can be seen by reexamining the cooling of the 50:50 alloy from the liquid phase.

Consider Fig. 3. At point X1 on the liquidus, solidification is about to begin. At a temperature infinitesimally below X1 there will be some crystals solidifying out of the liquid; their composition is given by Z1.

At a temperature about halfway between solidus and liquidus (X2), we have a mixture of solid and liquid of compositions Z2 and Y2. In general, the proportion of liquid to solid halfway through the freezing range need not be ≈50:50, but in this case it is.

Finally, at a temperature infinitesimally above X3, which is on the solidus, we have nearly 100% solid of composition Z3 together with a vanishingly small amount of liquid of composition Y3.

When the temperature falls to just below X3, the alloy is totally solid and Z3 has become identical with X3.

Note two important features. First, Z3 is the same as the average composition we started with, X1. Second, solidification takes place over a range of temperatures, and as it occurs the compositions of liquid and solid phases change continuously. For this to actually happen, substantial amounts of diffusion must occur in both liquid and solid.

Diffusion in solids is very much slower than that in liquids and is the source of some practical difficulty. Either solidification must occur slowly enough for diffusion in the solid to keep up or strict equilibrium conditions are not met.

The kinetics of phase transformations is therefore of interest, but for the moment, we will continue to discuss very slowly formed, equilibrium or near equilibrium structures.

## Eutectic Systems

Let us now examine another diagram, that for aluminium–silicon (Al–Si) alloys (Fig. 4). Pure Al forms face-centred crystals but Si has the same crystal structure as diamond. These are incompatible and extensive solid solutions like those for Cu:Ni cannot be formed. Si crystals can dissolve only tiny amounts of Al.

For our purposes, we can ignore this solubility, although we might recognise that the semiconductor industry makes great use of it, small as it is. Al crystals can dissolve a little Si, but again not very much, and we will ignore it. Thus, two solid phases are possible, Al and Si. When liquid, the elements dissolve readily in the melt in any proportions.

Consider the composition Y. On cooling to the liquidus line at A, pure (or nearly pure) crystals of Si begin to form. At B we have solid Si coexisting with liquid of composition LB in proportions given by the Lever rule. At C we have solid Si in equilibrium with liquid of composition close to E.

Now consider alloy X. The sequence is much the same except the first solid to form is now Al. When the temperature has fallen to almost TE we have solid Al in equilibrium with liquid of composition close to E. Note that both alloy X and alloy Y, when cooled to TE, contain substantial amounts of liquid of composition E. An infinitesimal drop in temperature below TE causes this liquid to solidify into a mixture of solid Al and solid Si.

At E we have 3 phases which can coexist; liquid, solid Al and solid Si. The system has two components and thus the phase rule gives us no degrees of freedom once we have discounted pressure. E is an invariant point; any change in temperature or composition will disturb the equilibrium. The point E is known as the eutectic point and we speak of the eutectic composition and the eutectic temperature, TE.

This is the lowest temperature at which liquid can exist and the eutectic alloy is that which remains liquid down to TE. It solidifies at a unique temperature, quite unlike Cu–Ni or Al–Si alloys of other compositions.

Alloys close to the eutectic composition (≈13%Si) are widely used because they can be easily cast into complex shapes, and the Si dispersed in the Al strengthens it. Eutectic alloys in other systems find similar uses (cast-iron is of near eutectic composition) as well as uses as brazing alloys etc.

## Intermediate Compounds

Often, the basic components of a system can form compounds. In metals we have CuAl2, Fe3C and many more. Some other relevant examples are:

• SiO2 and corundum, Al2O3, which form mullite, 3(Al2O3)2(SiO2), an important constituent of fired clays, pottery and bricks. Figure 5 shows the SiO2–Al2O3 diagram. It can be thought of as two diagrams, one for ‘SiO2-mullite’ and the other for ‘mullite–Al2O3’, joined together. Each part diagram is a simple eutectic system like Al–Si;
• lime, CaO, and silica, SiO2, which form the compounds 2(CaO)SiO2, 3(CaO)SiO2 and others, which have great technological significance as active ingredients in Portland cement. In a similar way to mullite, the lime (CaO)–silica (SiO2) diagram (Fig. 6) can be thought of as a series of joined together eutectic systems.

In many cases we do not have to think about the whole diagram. Figure 7 shows the Al-CuAl2 diagram, again a simple eutectic system. A notable feature is the so-called solvus line, AB, which represents the solubility of CuAl2 in solid crystals of Al. This curves sharply, so that very much less CuAl2 will dissolve in Al at low temperatures than will at high temperatures.

This is a fortunate fact that underlies our ability to alter the microstructures of some alloys by suitable heat-treatments. We have not yet considered the iron–carbon diagram, which is perhaps the most important diagram for nearly all engineers. This is of particular relevance in civil and structural engineering since steel in all its forms is used extensively. We will discuss this in another article.