Phase Diagrams

Objectives:

When you finish this module (especially the appropriate textbook sections and problems), you will be able to:
· Describe how to control the mechanical properties of materials by the addition of point defects
· State the Hume-Rothery rules
· Use tie line calculations to find percentages of the phases present
· Read percent compositions from phase diagrams

References: §9.1 - 9.9; 10.1 - 10.9 of The Science and Engineering of Materials, 3^rd Edition, Askeland, PWS Publishing

Co.  You should also look at the web reference section in your Course Map.

Phases: A phase has a) the same structure or atomic arrangement throughout; b) uniform composition and properties; c) a clear boundary between it and another phase. Pure materials have typically a single phase at a specific temperature and pressure, although it is certainly possible to choose a temperature at which two or three phases exist. For example, ice and water coexist at a pressure of 1 atm and 0^o K; the solid, liquid, and gaseous phases are in equilibrium at the triple point. As another example, magnesium vapor, liquid, and solid are in equilibrium at approximately 720^oC and a pressure which is less than 1 atm (see the unary phase diagram - Fig. 9-2 page 237).

The Gibbs Phase Rule describes the state of the material: F = C - P + 2, where C is the number of components, F is the number of degrees of freedom, and P is the number of phases present. Compare points A, B, and C in Figure 9-2 (page 237):

At A, the magnesium is liquid - there is one phase present. The number of components is also one (there's only magnesium). F = 1 - 1 + 2 = 2 degrees of freedom. That tells us that we can change the pressure and the temperature, and still remain in the liquid region. Or, we could say that we must fix both temperature and pressure to place the point A on the diagram.

At B, liquid and solid coexist; there are two phases. There's still only one component -magnesium. F = 1 - 2 +2 = 1 degree of freedom. That means this: if we change the temperature, the pressure is determined, and vice versa; if we try to change botharbitrarily, we'll move off the curve and into either the solid or the liquid region.

At point C (the triple point), we have no degrees of freedom; i.e., we can't change anything or the magnesium won't be at C.

Copper and nickel are two atoms which can dissolve in each other at all concentrations (see the phase diagram - Fig. 9-8a, page 245). Appendix A gives the lattice parameter for copper as 3.62 A and for nickel as 3.52; both are fcc. The phase diagram shows that at, for example, 40 w/o nickel (w/o is commonly used for weight percent), the solution passes from a single phase (liquid) at 1400^oC to a solid/liquid two phase mixture (1280^o to 1240^o) to a single solid phase (below 1240^o). Greek letters are commonly used to represent solid phases; since there is only one solid phase here, the first Greek letter (alpha) is used. The liquidus is the point at which solidification begins; it ends at the curve called the solidus. Thus, an alloy solidifies over a range of temperatures, unlike a pure material. These temperature-time curves may illustrate this better. Notice that the pure material's solidification is represented by a horizontal line - a constant temperature?

Solid solutions need not be restricted to metals. For example, consider the phase diagram in Fig. 9-9 (page 246), which is the solution of alumina in chromium oxide (or v.v.) and the design criteria for such a solution.

Compositions and Tie Lines: Please look at Fig. 9-11 (page 249). Again we consider the cooling of the liquid from 1300^o, where we have the single phase (liquid, 40 w/o Ni). At 1270^o, which is just below the liquidus, we have a small amount of solid and lots of liquid. While the overall composition is still 40 w/o Ni, the liquid (from the liquidus) has a composition of 37 % Ni, whereas the solid has a composition of 50 % Ni. (Notice that you can't average these two numbers to get 40 %? There's a lot more liquid than solid!) A horizontal line is drawn - it's called a tie line - from which we read the compositions. But the lengths of the sections of the tie line tell us how much of each phase there is. Since it's difficult to measure anything on this tie line, your text has expanded the section in Fig. 9-12 (page 250) and changed the temperature to 1250^oC. At that temperature, the liquid has a composition of 32 % Ni and the solid is 45 % Ni. The left hand section of the tie line has a "length" of 40 - 32 = 8, whereas the entire "length" of the tie line is 45 - 32 = 13. 8/13 = 0.62, so there's 62 % solid (and 38 % liquid). Notice that the measurements are "opposite"; the left hand section relates to the right hand curve - the solidus.

A note about sample preparation...A piece of the material is cut from a rod, inserted into an apparatus with powdered plastic and placed under pressure and heated until the plastic sets. Then the cross section of the material is at the surface of the plastic (the plastic is there to make it easier to handle the metal sample). Now the metallic surface is polished to remove the top several thousandths of an inch, since the operation of cutting the rod disturbs the surface. After polishing, it's treated with HCl (acid). The acid preferentially reacts at the grain boundaries, since it's there that the atoms haven't completed their preferred structure (fcc, bcc, whatever) and are therefore more reactive. Then one looks at the sample under the microscope.

You need to realize that phase diagrams are equilibrium diagrams. That means that it is implied that one allows lots of time for diffusion to take place. For example, as our 40 % Ni solution cooled from the liquidus to the solidus temperatures, the solid solution changed in composition. The first solid to come out has a different composition than the last! If the cooling is rapid, diffusion can't take place completely, and we get the segregated structure shown in Fig. 9-16 (page 255).  The dashed line (not the horizontal tie lines which are also dashed) indicates compositions for a particular (rapid) cooling rate.

Two-Phase Systems: Lead and tin are not completely soluble in each other at all compositions. (Compare the lattice parameters from Appendix A; 4.95 vs 6.49 A. That's a larger difference than for Cu-Ni!) In Figure 10-8 (page 272), two Greek letters are used for the solid phases representing a lead-rich solution (alpha, on the left) and a tin-rich solution (beta, on the right).

For a 2 % tin solution in lead, Fig. 10-9 shows the transition from liquid to alpha-phase solid floating in liquid to alpha-grains. But for a 10 w/o tin solution (Fig. 10-10, page 273), when the temperature drops below the solvus curve (perhaps at 30^oC), the lead structure cannot keep as much tin in solution as at a higher temperature. So the tin precipitates out (much like a supersaturated solution of sugar in water will precipitate out sugar crystals as the temperature is lowered). Remember, this is slow, and it won't happen if the cooling takes place quickly! This precipitate reduces the capability of the cells to slip under stress, and we have what's called "dispersion strengthening."

Next, let's consider a 61.9 w/o Sn solution in lead (Fig. 10-11, page 274). Notice that this has the lowest solidification temperature? This is called the "eutectic", or the "eutectic temperature." That's from the Greek eutektos, meaning "easily melted." The eutectic is the one composition for which an alloy behaves like a pure metal - it solidifies at one temperature. Also, we don't get alpha grains floating in liquid as before. Instead (look at the tie line which goes through the eutectic from the alpha region to the beta region) both alpha and beta solidify simultaneously in a"lamellar" structure (alternating layers of alpha and beta). Another name for this is the "eutectic microconstituent." You should study Example 10-4 very carefully.

As a third example, consider the hypoeutectic alloy (less than eutectic composition) of 30 % tin (Fig 10-14 page 276). Nothing unusual at, say, 220^oC - alpha grains floating in liquid. But as the temperature is lowered to just above 184^o (the eutectic temperature), the liquid that remains has the eutectic composition and comes out in the characteristic lamellar structure (the alternating alpha and beta phases are called lamellae). A similar structure would occur for a hypereutectic alloy (for example, 80 % tin), except that the result would be beta grains surrounded by lamellae.

The two metal system (Al - Sb) is shown in Figure 10-2a (page 266). Because a stoichiometric compound represents a fixed ratio of one element to the other, a vertical line (and not a region) must be used to show this on a phase diagram. In the Al-Sb phase diagram, gamma has been used to represent the compound AlSb (one-to-one ratio).

Nonstoichiometric intermetallic compounds probably shouldn't be called compounds at all. But they are! They are sometimes called intermediate solid solutions, and can have a range of compositions. For example, in Fig. 10-2b (page 266), the gamma region (note the almost-vertical boundaries?) is a nonstoichiometric compound of molybdenum and rhodium, whose composition can vary from 45 % to 83 % Rh at 1600^oC.  You might want to look at this example.